Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph
We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052-3132], and are expected to provid...
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| citation_txt | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph / C. Cuenca // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. |
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| description | We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052-3132], and are expected to provide tools for the study of statistical mechanical models, representation theory, and random matrices. As the first application of our formulas, we characterize the boundary of the (q,t)-deformation of the Gelfand-Tsetlin graph when t=qθ and θ is a positive integer.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 001, 66 pages
Asymptotic Formulas for Macdonald Polynomials and
the Boundary of the (q, t)-Gelfand–Tsetlin Graph
Cesar CUENCA
Department of Mathematics, Massachusetts Institute of Technology, USA
E-mail: cuenca@mit.edu
URL: http://math.mit.edu/~cuenca/
Received April 21, 2017, in final form December 09, 2017; Published online January 02, 2018
https://doi.org/10.3842/SIGMA.2018.001
Abstract. We introduce Macdonald characters and use algebraic properties of Macdonald
polynomials to study them. As a result, we produce several formulas for Macdonald cha-
racters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43
(2015), 3052–3132], and are expected to provide tools for the study of statistical mechanical
models, representation theory and random matrices. As first application of our formulas,
we characterize the boundary of the (q, t)-deformation of the Gelfand–Tsetlin graph when
t = qθ and θ is a positive integer.
Key words: Branching graph; Macdonald polynomials; Gelfand–Tsetlin graph
2010 Mathematics Subject Classification: 33D52; 33D90; 60B15; 60C05
1 Introduction
Macdonald polynomials are remarkable two-parameter q, t generalizations of Schur polynomials.
They were first introduced by Ian G. Macdonald in [31]; the canonical reference is his classical
book [32]. The Macdonald polynomials are very interesting objects for representation theory and
integrable systems, due to their connections with quantum groups, e.g., [20, 33], double affine
Hecke algebras, e.g., [13, 28], etc. More recently, and releveant for us, Macdonald polynomials
have been heavily used to study probabilistic models arising in mathematical physics and random
matrix theory. The important work [4] of Borodin and Corwin showed how to use the algebraic
properties of these polynomials to obtain analytic formulas that allow asymptotic analysis
of the so-called Macdonald processes. Remarkably, by specialization or degeneration of the
parameters q, t defining Macdonald processes, the paper [4] yields tools that can be used to
analyze interacting particle systems [5], beta Jacobi corners processes [6], probabilistic models
from asymptotic representation theory [3], among others; see the survey [9] and references therein.
It should be noted that the special case t = q of Macdonald processes are known as Schur
processes and they have the special property of being determinantal point processes, thus allowing
much more control over their asymptotics. Schur processes were introduced by Okounkov and
Reshetikhin, as generalizations of the classical Plancherel measures, several years before the work
of Borodin and Corwin [35, 38]. The Schur processes, though a very special case of Macdonald
processes, produced various applications to statistical models of plane partitions and random
matrices, see for example [27, 39]. However, most of the physical models that were studied with
the Macdonald processes do not have a determinantal structure, and therefore they could not
have been analyzed solely by means of the Schur processes machinery.
The conclusion from the story of Schur and Macdonald processes is that studying the more
complicated object can allow one to tackle more complicated questions, despite losing some
integrability (such as the determinantal structure in the case of Schur processes). We follow this
philosophy in our work, by introducing and studying Macdonald characters, two-parameter q, t
mailto:cuenca@mit.edu
http://math.mit.edu/~cuenca/
https://doi.org/10.3842/SIGMA.2018.001
2 C. Cuenca
generalizations of normalized characters of unitary groups. The normalized characters of the
unitary groups are expressible in terms of Schur polynomials, reason why we will call them Schur
characters, whereas the generalization we present involves Macdonald polynomials.
Our main results are asymptotic formulas for Macdonald characters, which are generalizations
of those for Schur characters, proved in [24] by different methods. As it is expected, the
asymptotic formulas for Schur characters are simpler and they involve certain determinantal
structure, whereas the formulas for Macdonald polynomials are more complicated and the
determinantal structure is no longer present. However the advantage of our work on Macdonald
polynomials, much like the advantage of Macdonald processes over Schur processes, is that we are
able to access a number of asymptotic questions that are more general than those given in [24].
The tools we obtain in this paper are therefore very exciting, given that the formulas for Schur
characters have already produced several applications to stochastic discrete particle systems,
lozenge and domino tiling models, and asymptotic representation theory [10, 11, 12, 22, 24, 44].
The paper [15] is this article’s companion, in which the author studies Jack characters,
a natural degeneration of Macdonald characters and obtains their asymptotics in the Vershik–
Kerov limit regime. The approach to the study of asymptotics of Jack characters is different
from the approach we use here to study the asymptotics of Macdonald characters; in particular,
it relies heavily on the Pieri integral formula, see [15] for further details. The tools from this
paper and [15] afford us a very strong control over the asymptotics of Macdonald characters,
Jack characters and Bessel functions [36, 43], if the number of variables remains fixed and
the rank tends to infinity. As another application of the developed toolbox, we have studied
a Jack–Gibbs model of lozenge tilings in the spirit of [7, 25]. The author was able to prove
the weak convergence of statistics of the Jack–Gibbs lozenge tilings model near the edge of the
boundary to the well-known Gaussian beta ensemble, see, e.g., Forrester [1, Chapter 20] and
references therein. This result, and its rational limit concerning corner processes of Gaussian
matrix ensembles, will appear in a forthcoming publication.
We proceed with a more detailed description of the results of the present paper.
1.1 Description of the formulas
The main object of study in this paper are the Macdonald characters, which we define as follows.
For integers 1 ≤ m ≤ N , a Macdonald character of rank N and m variables is a polynomial, with
coefficients in C(q, t), of the form
Pλ(x1, . . . , xm;N, q, t)
def
=
Pλ
(
x1, . . . , xm, 1, t, . . . , t
N−m−1; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
) ,
where Pλ(x1, . . . , xN ; q, t) is the Macdonald polynomial of N variables parametrized by the
signature λ = (λ1 ≥ λ2 ≥ · · · ≥ λN ) ∈ ZN . Macdonald characters, under the specialization t = q,
turn into q-Schur characters, which have appeared previously in [21, 24]. The reason behind the
use of the word “character” is that q-Schur characters turn into normalized characters of the
irreducible rational representations of unitary groups, after the degeneration q → 1.
We make one further comment about terminology. Macdonald characters, as defined here,
are two-parameter q, t degenerations of normalized and irreducible characters of unitary groups.
One could also consider a two-parameter degeneration of the characters of the symmetric groups,
in the spirit of Lassalle’s work [29], where a one-parameter degeneration of symmetric group
characters was considered. Thus a better name for our object would be Macdonald unitary
character. For convenience, we simply will use the name Macdonald character. We remark
that Macdonald symmetric group characters have not been considered yet, to the author’s
best knowledge. However, there have been many articles studying the structural theory and
Asymptotic Formulas for Macdonald Polynomials 3
asymptotics on Jack symmetric group characters, notably several recent works by Maciej Do lȩga,
Valentin Féray and Piotr Śniady, see, e.g., [16, 17, 18, 46].
The main theorems of this paper fall into two categories:
(A) Integral representations for Macdonald characters of one variable and arbitrary rank N .
The initial idea that led to the integral representations in this paper is due to Andrei Okounkov,
see [24, remark following Theorem 3.6]. An example of the integral formulas we prove is the
following theorem. Observe that the integrand is a simple expression in terms of q-Gamma
functions and can be analyzed by well known methods of asymptotic analysis, such as the method
of steepest descent or the saddle-point method [14]. In this paper, we study the regime in which
the signatures grow to infinity, whereas the remaining parameters are fixed.
Theorem 1.1 (consequence of Theorem 3.2). Assume q ∈ (0, 1) and θ > 0. Let N ∈ N, λ ∈ GTN
and x ∈ C \ {0}, |x| ≤ qθ(1−N). The integral below converges absolutely and the identity holds
Pλ
(
x;N, q, qθ
)
=
ln q
1− q
(
xqθ; q
)
∞(
xq1+θ(1−N); q
)
∞
Γq(θN)
2π
√
−1
×
∫
C+
(
xqθ(1−N)
)z N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
dz, (1.1)
where C+ is a certain contour described in Theorem 3.2, and which looks as in Fig. 2. In the
formula above, we used the q-Pochhammer symbol (z; q)∞ and the q-Gamma function Γq(z); see
Appendix A.
(B) Formulas expressing Macdonald characters of m variables (and rank N) in terms of
Macdonald characters of one variable (and rank N).
These formulas will involve certain q-difference operators. Formulas of this kind will be called
multiplicative formulas.1 One of the simplest multiplicative formulas we prove is the one below
that expresses a Macdonald character of two variables in terms of those of one variable; the
general formula is given below in Theorem 4.1.
Theorem 1.2 (reformulation of Corollary 4.2). Let θ ∈ N, N ∈ N, λ ∈ GTN . Then
Pλ
(
x1, x2;N, q, qθ
)
=
q−(N− 3
2
)θ2+ 1
2
θ(1− q)θ
θ∏
i=1
(1− qθN−i)
1
θ(N−1)−1∏
i=1
(x1 − qi−θ)(x2 − qi−θ)
×
(
1
x1 − x2
◦ (Dq,x2 −Dq,x1)
)θ
2∏
i=1
Pλ(xi;N, q, qθ) θN−1∏
j=1
(
xi − qj−θ
) ,
where Dq,xi , i = 1, 2, are the linear operators in C(q)[x1, x2] acting on monomials by Dq,xi(x
m1
1 xm2
2 )
= 1−qmi
1−q (xm1
1 xm2
2 ), i = 1, 2.
Observe that the multiplicative formula above requires θ ∈ N. It is somewhat surprising that
all the identities we prove in this paper, even the integral representations, behave better for
θ ∈ N.
1The reason for the name is that analogous formulas for Schur characters were used to prove statements of
the form lim
N→∞
Fλ(N)(x1, . . . , xm) =
m∏
i=1
lim
N→∞
Fλ(N)(xi), where the functions F are certain normalizations of Schur
characters, see, e.g., [24, Corollaries 3.10 and 3.12]
4 C. Cuenca
1.2 The boundary of the (q, t)-Gelfand–Tsetlin graph
As an application of our formulas, we characterize the space of central probability measures
in the path-space of the (q, t)-Gelfand–Tsetlin graph, when t = qθ and θ is a positive integer.
To state our result, we first introduce a few notions. Assume that q, t ∈ (0, 1) are generic real
parameters for the moment.
The Gelfand–Tsetlin graph, or simply GT graph, is an undirected graph whose vertices are
the signatures of all lengths GT =
⊔
N≥0
GTN ; we also include the empty signature ∅ as the only
element of GT0 for convenience. The set of edges is determined by the interlacing constraints,
namely the edges in the GT graph can only join signatures whose lengths differ by 1 and µ ∈ GTN
is joined to λ ∈ GTN+1 if and only if
λN+1 ≤ µN ≤ λN ≤ · · · ≤ λ2 ≤ µ1 ≤ λ1.
If the above inequalities are satisfied, we write µ ≺ λ. If µ ∈ GTN and λ ∈ GTN+1 are joined by
an edge, then we consider the expression ΛN+1
N (λ, µ) given by
ΛN+1
N (λ, µ) = ψλ/µ(q, t)
Pµ
(
tN , . . . , t2, t; q, t
)
Pλ
(
tN , . . . , t, 1; q, t
) ,
where ψλ/µ(q, t) is given in the branching rule for Macdonald polynomials, see Theorem 2.5 below.
If µ ∈ GTN is not joined to λ ∈ GTN+1, set ΛN+1
N (λ, µ) = 0. One can easily show, see, e.g.,
Theorem 2.3 below,
ΛN+1
N (λ, µ) ≥ 0, ∀λ ∈ GTN+1, µ ∈ GTN ,∑
µ∈GTN
ΛN+1
N (λ, µ) = 1, ∀λ ∈ GTN+1.
Thus, for any N ∈ N, λ ∈ GTN+1, ΛN+1
N (λ, ·) is a probability measure on GTN . For this reason,
we will call the expressions ΛN+1
N (λ, µ) cotransition probabilities.
Next we define the path-space T of the GT graph as the set of infinite paths in the GT graph
that begin at ∅ ∈ GT0: T = {τ = (∅ = τ (0) ≺ τ (1) ≺ τ (2) ≺ · · · ) : τ (n) ∈ GTn ∀n ∈ Z≥0}.
Each finite path of the form φ =
(
∅ = φ(0) ≺ φ(1) ≺ · · · ≺ φ(n)
)
defines a cylinder set
Sφ =
{
τ ∈ T : τ (1) = φ(1), . . . , τ (n) = φ(n)
}
⊂ T . We equip T with the σ-algebra generated
by the cylinder sets Sφ, over all finite paths φ. Equivalently, the σ-algebra of T is its Borel
σ-algebra if we equip T with the topology it inherits as a subspace of the product
∏
n≥0
GTn. Each
probability measure M on T admits a pushforward to a probability measure on GTm via the
obvious projection map
Projm : T ⊂
∏
n≥0
GTn −→ GTN ,
τ =
(
τ (0) ≺ τ (1) ≺ τ (2) ≺ · · ·
)
7→ τ (N).
We say that a probability measure M on T is a (q, t)-central measure if
M
(
S
(
φ(0) ≺ φ(1) ≺ · · · ≺ φ(N−1) ≺ φ(N)
))
= ΛNN−1
(
φ(N), φ(N−1)
)
· · ·Λ1
0
(
φ(1), φ(0)
)
MN
(
φ(N)
)
,
for all N ≥ 0, all finite paths φ(0) ≺ · · · ≺ φ(N), and for some probability measures MN on GTN .
It then automatically follows that MN = (ProjN )∗M are the pushforwards of M ; moreover, they
satisfy the coherence relations
MN (µ) =
∑
λ∈GTN+1
MN+1(λ)ΛN+1
N (λ, µ), ∀N ≥ 0, ∀µ ∈ GTN .
Asymptotic Formulas for Macdonald Polynomials 5
We denote by Mprob(T ) the set of (q, t)-central (probability) measures on T ; it is clearly
a convex subset of the Banach space of all finite and signed measures on T . Let us denote by
Ωq,t = Ex(Mprob(T )) the set of its extreme points. From a general theorem, we can deduce
that Ωq,t ⊂Mprob(T ) is a Borel subset. We call Ωq,t, with its inherited topology, the boundary
of the (q, t)-Gelfand–Tsetlin graph. The theorem stated below, which is our main application,
completely characterizes the topological space Ωq,t. Before stating it, let us make a couple of
relevant definitions.
Consider the set of weakly increasing integers N = {ν = (ν1 ≤ ν2 ≤ · · · ) : ν1, ν2, · · · ∈ Z} and
equip it with the topology inherited from the product Z∞ = Z × Z × · · · of countably many
discrete spaces. For each k ∈ Z, we can define the automorphism Ak of N by ν 7→ Akν =
(ν1 + k ≤ ν2 + k ≤ · · · ). Clearly Ak has inverse A−k. There is a similar automorphism of GT,
given by λ 7→ Akλ = (λ1 + k ≥ λ2 + k ≥ · · · ), ∅ 7→ Ak∅ = ∅, which restricts to automorphisms
GTm → GTm, for each m ∈ Z≥0.
For any k ∈ Z, one can easily show that µ ∈ GTm interlaces with λ ∈ GTm+1 iff Akµ ∈ GTm
interlaces with Akλ ∈ GTm+1, that is, µ ≺ λ iff Akµ ≺ Akλ. This allows us to define
automorphisms Ak of T by
Ak : T −→ T ,
τ =
(
∅ ≺ τ (1) ≺ τ (2) ≺ · · ·
)
7→ Akτ =
(
∅ ≺ Akτ (1) ≺ Akτ (2) ≺ · · ·
)
.
One can similarly obtain maps Ak on the set of finite paths of length n by φ = (φ(0) ≺
φ(1) ≺ · · · ≺ φ(n)) 7→ Akφ = (Akφ
(0) ≺ Akφ(1) ≺ · · · ≺ Akφ(n)). Consequently we can also define
automorphisms on cylinder sets by AkSφ = SAkφ, for all finite paths φ = (φ(0) ≺ φ(1) ≺ · · · ≺ φ(n)),
in the natural way.
We named several maps above by the same letter Ak, but there should be no risk of confusion.
For our main theorem, we make the assumption θ ∈ N, t = qθ. We believe the theorem can be
generalized for any θ > 0, but we do not have a proof at the moment.
Theorem 1.3. Assume q ∈ (0, 1), θ ∈ N and set t = qθ.
1. There exists a homeomorphism N : N → Ωq,t sending each ν ∈ N to the (q, t)-central
probability measure Mν ∈ Ωq,t determined by the relations∑
λ∈GTm
Mν
m(λ)
Pλ
(
x1, x2t, . . . , xmt
m−1; q, t
)
Pλ
(
1, t, . . . , tm−1; q, t
) = Φν
(
x1t
1−m, . . . , xm−1t
−1, xm; q, t
)
,
∀m ∈ N, ∀ (x1, . . . , xm) ∈ Tm. (1.2)
In (1.2), we denoted by {Mν
m}m≥1 the corresponding sequence of pushforwards of Mν under
the projection maps Projm : T → GTm. The left side in (1.2) is absolutely convergent
on Tm, T = {z ∈ C : |z| = 1}, and the functions Φν in the right side are defined in (5.2)
and (5.8). The probability measure Mν is determined uniquely by the relations (1.2).
2. For each k ∈ Z, the probability measures Mν and MAkν are related by
MAkν(SAkφ) = Mν(Sφ), for all finite paths φ =
(
φ(0) ≺ φ(1) ≺ · · · ≺ φ(n)
)
. (1.3)
Moreover the (q, t)-coherent sequences {Mν
m}m≥0 and {MAkν
m }m≥0 are related by
MAkν
m (Akλ) = Mν
m(λ), ∀m ≥ 0, λ ∈ GTm. (1.4)
Another main result of this article is Theorem 7.9, where we characterize the Martin boundary
of the (q, t)-Gelfand–Tseltin graph for t = qθ and θ ∈ N. In fact, we first prove that the Martin
boundary is homeomorphic to N and then show that the minimal boundary Ωq,t coincides with
the Martin boundary. See Sections 7.1 and 7.2 for the definition and characterization of the
Martin boundary.
6 C. Cuenca
1.3 Comments on Theorem 1.3 and connections to existing literature
Our first comment is that Theorem 1.3 is a generalization of the main theorem in the article
of Vadim Gorin [21], which is the special case θ = 1 of our theorem, and characterizes the
boundary of the q-Gelfand–Tsetlin graph. Some ideas in the proofs are the same, especially the
overall scheme of using the ergodic method of Vershik–Kerov, see [47], but we need many new
arguments as well. For example, [21] makes heavy use of the shifted Macdonald polynomials, in
particular the binomial formula for shifted Macdonald polynomials at t = q, [34], while we do
not use them at all. Moreover, in order to prove that the boundary of the q-Gelfand–Tsetlin
graph is homeomorphic to N , [21] made use of the following closed formula for the shifted-Schur
generating function of Mν,θ=1
N , in the case that ν1 ≥ 0:
∑
λ∈GT+
N
Mν,θ=1
N (λ)
s∗λ
(
qN−1x1, . . . , q
N−1xN ; q−1
)
s∗λ
(
0, . . . , 0; q−1
) = Hν(x1) · · ·Hν(xN ), (1.5)
where
Hν(x) =
∞∏
i=0
(1− qit)
∞∏
j=1
(1− qνj+j−1t)
.
In the formula above, s∗λ(x1, . . . , xN ; q) is the shifted Macdonald polynomial at t = q. In addition
to the usefulness of the closed formula (1.5) above, the multiplicative structure is surprising. It
would be interesting to find a closed formula for the shifted Macdonald generating function of
the measures Mν
N , for general θ ∈ N, and find out if the multiplicative structure still holds in
this generality.
It is shown in [21] that their main statement is equivalent to the characterization of certain
Gibbs measures on lozenge tilings. A conjectural characterization of positive q-Toeplitz matrices
is also given in that paper. Finally, it is mentioned that the asymptotics of q-Schur functions is
related to quantum traces and the representation theory of Uε(gl∞). It would be interesting to
extend some of these statements to the Macdonald case, especially to connect the asymptotics of
Macdonald characters to the representation theory of inductive limits of quantum groups.
Several other “boundary problems” have appeared in the literature in various contexts. For
instance, in the limiting case t = q → 1, the problem of characterizing the boundary Ωq,t
becomes equivalent to characterizing the space of extreme characters of the infinite-dimensional
unitary group U(∞) = lim
→
U(N). The answer also characterizes totally positive Toeplitz matrices
[19, 47, 48]. Also in the degenerate case t = q2 or t = q1/2 and q → 1, the boundary problem
becomes equivalent to characterizing the space of extreme spherical functions of the infinite-
dimensional Gelfand pairs (U(2∞),Sp(∞)) and (U(∞),O(∞)), respectively. This question, and
in fact a more general one-parameter “Jack”-degeneration, was solved in [37]. A similar degenerate
question in the setting of random matrix theory was studied in [42]. Some of the tools in this
paper can be degenerated easily to these scenarios and they may provide an alternative approach
to their proof as well; for example, the special case of our toolbox in the case t = q → 1 was
used in [24] to study the corresponding boundary problem, and in [15] we also study refine the
asymptotic result that is needed to solve the boundary problem of [37].
Another similar boundary problem in a somewhat different direction is the following. Assume
we consider the Young graph instead of the Gelfand–Tsetlin graph, e.g., see [8]. Assume also
that the cotransition probabilities coming from the branching rule of Macdonald polynomials
are replaced by the cotransition probabilities coming from the Pieri-rule. In this setting, the
Asymptotic Formulas for Macdonald Polynomials 7
boundary problem has not been solved yet, but it is expected that the answer is given by Kerov’s
conjecture, which characterizes Macdonald-positive specializations, see [4, Section 2].
Finally, it was brought to my attention, after I completed the results of this paper, that
Grigori Olshanski has obtained a characterization of the extreme set of (q, t)-central measures
in the extended Gelfand–Tsetlin graph for more general parameters q, t by different methods.
His work follows the setting of the paper [23] of Gorin–Olshanski, which is some sort of analytic
continuation to our proposed boundary problem. Interestingly, new features arise, e.g., two copies
of the space N characterize the boundary in his context, one can define and work with suitable
analogues of zw-measures, etc. Another related work in the t = q case is his recent article [41].
1.4 Organization of the paper
The present work is organized as follows. In Section 2, we briefly recall some important algebraic
properties of Macdonald polynomials that will be used to obtain our main results. We prove
integral representations for Macdonald characters of one variable in Section 3. Next, in Section 4,
we obtain multiplicative formulas for Macdonald characters of a given number of variables
m ∈ N in terms of those of one variable. By making use of our formulas, in Section 5 we obtain
asymptotics of Macdonald characters as the signatures grow to infinity in a specific limit regime.
In Sections 6 and 7, we define and characterize the boundary of the (q, t)-Gelfand–Tsetlin graph
in the case that θ ∈ N and t = qθ. The asymptotic statements of Section 5 play the key role in
the characterization of the boundary.
In Appendix A, we have bundled the necessary language and results of q-theory that are used
throughout the paper. In Appendix B, we make some computations with expressions that appear
in the multiplicative formulas for Macdonald polynomials.
2 Symmetric Laurent polynomials
A canonical reference for symmetric polynomials is [32]. We choose to give a brief overview of the
tools that we need from [32], in order to fix terminology and to introduce lesser known objects,
such as signatures and Macdonald Laurent polynomials.
2.1 Partitions, signatures and symmetric Laurent polynomials
A partition is a finite sequence of weakly decreasing nonnegative integers λ = (λ1 ≥ λ2 ≥ · · · ≥ λk),
λi ∈ Z≥0 ∀ i. We identify partitions that differ by trailing zeroes; for example, (4, 2, 2, 0, 0) and
(4, 2, 2) are the same partition. We define the size of λ to be the sum |λ| def= λ1 + · · ·+ λk, and
its length `(λ) to be the number of strictly positive elements of it. The dominance order for
partitions is a partial order given by letting µ ≤ λ if |µ| = |λ| and µ1 + · · ·+ µi ≤ λ1 + · · ·+ λi
for all i. As usual, we let µ < λ if µ ≤ λ and µ 6= λ.
Partitions can be graphically represented by their Young diagrams. The Young diagram of
partition λ is the array of boxes with coordinates (i, j) with 1 ≤ j ≤ λi, 1 ≤ i ≤ `(λ), where the
coordinates are in matrix notation (row labels increase from top to bottom and column labels
increase from left to right), see Fig. 1.
A signature is a sequence of weakly decreasing integers λ = (λ1 ≥ λ2 ≥ · · · ≥ λk), λi ∈ Z ∀ i.
A positive signature is a signature whose elements are all nonnegative. The length of a signature,
or positive signature, is the number k of elements of it. Positive signatures which differ by trailing
zeroes are not identified, in contrast to partitions; for example, (4, 2, 2, 0, 0) and (4, 2, 2) are
different positive signatures, the first of length 5 and the second of length 3. We shall denote GTN
(resp. GT+
N ) the set of signatures (resp. positive signatures) of length N . Evidently GT+
N can be
identified with the set of all partitions of length ≤ N . Under this identification, we are allowed
8 C. Cuenca
s
Figure 1. Young diagram for the partition λ = (5, 4, 4, 2). Square s = (3, 3) has arm length, arm colength,
leg length and leg colength given by a(s) = 1, a′(s) = 2, l(s) = 0, l′(s) = 2.
to talk about the Young diagram of a positive signature λ ∈ GT+
N , its size, the dominance order,
and other attributes that are typically associated to partitions. Note, however, that length is
defined differently for partitions and for positive signatures.
Let us now switch to notions pertaining to symmetric (Laurent) polynomials. Fix a positive
integer N . Consider the field F = C(q, t) and recall the algebra ΛF [x1, . . . , xN ] of symmetric
polynomials on the variables x1, . . . , xN with coefficients in F . For any m ∈ Z≥0, recall also
the subalgebra ΛmF [x1, . . . , xN ] of symmetric polynomials on x1, . . . , xN that are homogeneous of
degree m; then
ΛF [x1, . . . , xN ] =
⊕
m≥0
ΛmF [x1, . . . , xN ].
We also denote by ΛF [x±1 , . . . , x
±
N ] the algebra of symmetric (with respect to the transpositions
xi ↔ xi+1 for i = 1, . . . , N − 1) Laurent polynomials in the variables x1, . . . , xN .
The connection between partitions/signatures and symmetric polynomials comes from the
observation that dimF (ΛmF [x1, . . . , xN ]) is the number of partitions of size m and length ≤ N , or
equivalently the number of positive signatures of size m and length N . A basis for the space
ΛmF [x1, . . . , xN ] is given by the monomial symmetric polynomials mλ(x1, . . . , xN ), with |λ| = m,
`(λ) ≤ N , defined by
mλ(x1, . . . , xN )
def
=
∑
µ∈SN ·λ
xµ11 · · ·x
µN
N ,
where SN · λ is the orbit of λ under the permutation action of SN , and the sum runs over
distinct elements µ of that orbit. It is implied that {mλ(x1, . . . , xN ) : `(λ) ≤ N} is a basis of
ΛF [x1, . . . , xN ].
2.2 Macdonald polynomials and Macdonald characters
Proposition/Definition 2.1 ([32, Chapter VI, Sections 3, 4, 9]). The Macdonald polynomials
Pλ(x1, . . . , xN ; q, t), for partitions λ with `(λ) ≤ N , are the unique elements of ΛF [x1, . . . , xN ]
satisfying the following two properties
• Triangular decomposition: Pλ(x1, . . . , xN ; q, t) = mλ +
∑
µ : µ<λ
cλ,µmµ, for some cλ,µ ∈ F ,
and the sum is over partitions µ with `(µ) ≤ N , and µ < λ in the dominance order.
• Orthogonality relation: Let [·]0 : F [x±1 , . . . , x
±
N ]→ F be the constant term map ∑
λ=(λ1≥···≥λN )∈ZN
aλx
λ1
1 · · ·x
λN
N
0
= a(0,...,0).
Asymptotic Formulas for Macdonald Polynomials 9
The Macdonald polynomials are orthogonal with respect to the inner product (·, ·)q,t on
ΛF [x±1 , . . . , x
±
N ] given by (f, g)q,t
def
= [f(x1, . . . , xN )g
(
x−1
1 , . . . , x−1
N
)
∆q,t]0, where
∆q,t
def
=
∏
1≤i 6=j≤N
∞∏
k=0
1− qkxix−1
j
1− qktxix−1
j
.
Note that P∅(q, t) = 1. If N < `(λ), we set Pλ(x1, . . . , xN ; q, t)
def
= 0 for convenience.
When we are talking about Macdonald polynomials and some of their properties which hold
regardless of the number N of variables, as long as N is large enough, we simply write Pλ(q, t)
instead of Pλ(x1, . . . , xN ; q, t).
From the triangular decomposition of Macdonald polynomials, Pλ(q, t) is a homogeneous
polynomial of degree |λ|. Moreover {Pλ(x1, . . . , xN ) : `(λ) ≤ N} is a basis of ΛF [x1, . . . , xN ].
Finally, we have the following index stability property:
P(λ1+1,...,λN+1)(x1, . . . , xN ; q, t) = (x1 · · ·xN ) · Pλ(x1, . . . , xN ; q, t). (2.1)
As pointed out before, the set of partitions of length ≤ N is in bijection with GT+
N . Thus
we can index the Macdonald polynomials by positive signatures rather than by partitions: for
any λ ∈ GT+
N , we let Pλ(x1, . . . , xN ; q, t) be the Macdonald polynomial corresponding to the
partition associated to λ. We can slightly extend the definition above and introduce Macdonald
Laurent polynomials Pλ(x1, . . . , xN ; q, t) for any λ ∈ GTN . Let λ ∈ GTN be arbitrary. If λN ≥ 0,
then λ ∈ GT+
N and Pλ(x1, . . . , xN ; q, t) is already defined. If λN < 0, choose m ∈ N such that
λN +m ≥ 0 and so (λ1 +m, . . . , λN +m) ∈ GT+
N . Then define
Pλ(x1, . . . , xN ; q, t)
def
= (x1 · · ·xN )−m · P(λ1+m,...,λN+m)(x1, . . . , xN ; q, t). (2.2)
By virtue of the index stability property, the Macdonald (Laurent) polynomial Pλ(x1, . . . , xN ; q, t)
is well-defined and does not depend on the value of m that we choose. For simplicity, we call
Pλ(x1, . . . , xN ; q, t) a Macdonald polynomial, whether λ ∈ GT+
N or not. In a similar fashion, we
can define monomial symmetric polynomials mλ, for any λ ∈ GTN .
Recall the definitions of the arm-length, arm-colength, leg-length, leg-colength a(s), a′(s), l(s),
l′(s) of the square s = (i, j) of the Young diagram of λ, given by a(s) = λi − j, a′(s) = j − 1,
l(s) = λ′j − i, l′(s) = i− 1; we note that λ′j = |{i : λi ≥ j}| is the length of the jth part of the
conjugate partition λ′, see Fig. 1.
We use terminology from q-analysis, see Appendix A; particularly we use the definition of
q-Pochhammer symbols (z; q)n
def
=
n−1∏
i=0
(1− zqi) and (z; q)∞
def
=
∞∏
i=0
(1− zqi).
For λ ∈
⊔
N≥0
GT+
N , define the dual Macdonald polynomials Qλ(q, t) as the following normaliza-
tion of Macdonald polynomials
Qλ(q, t)
def
= bλ(q, t)Pλ(q, t), bλ(q, t)
def
=
∏
s∈λ
1− qa(s)tl(s)+1
1− qa(s)+1tl(s)
. (2.3)
The complete homogeneous symmetric (Macdonald) polynomials g0 = 1, g1, g2, . . . are the
one-row dual Macdonald polynomials:
gn(q, t)
def
= Q(n)(q, t) =
(q; q)n
(t; q)n
P(n)(q, t). (2.4)
For convenience, we also set gn(q, t)
def
= 0, ∀n < 0.
Now we come to several important theorems on Macdonald polynomials, which will be our
main tools.
10 C. Cuenca
Theorem 2.2 (index-argument symmetry; [32, Chapter VI, Property 6.6]). Let N ∈ N, λ, µ ∈
GT+
N , then
Pλ
(
qµ1tN−1, qµ2tN−2, . . . , qµN ; q, t
)
Pλ
(
tN−1, tN−2, . . . , 1; q, t
) =
Pµ
(
qλ1tN−1, qλ2tN−2, . . . , qλN ; q, t
)
Pµ
(
tN−1, tN−2, . . . , 1; q, t
) .
Theorem 2.3 (evaluation identity; [32, Chapter VI, equations (6.11) and (6.11′)]). Let N ∈ N,
λ ∈ GT+
N , then
Pλ
(
tN−1, tN−2, . . . , 1; q, t
)
= tn(λ)
∏
1≤i<j≤N
(
qλi−λj tj−i; q
)
∞(tj−i+1; q)∞(
qλi−λj tj−i+1; q
)
∞(tj−i; q)∞
= tn(λ)
∏
s∈λ
1− qa′(s)tN−l′(s)
1− qa(s)tl(s)+1
,
where and n(λ)
def
= λ2 + 2λ3 + · · · + (N − 1)λN . The first equality holds, more generally, for
any signature λ ∈ GTN by virtue of the definition (2.2) of Macdonald Laurent polynomials
Pλ(x1, . . . , xN ; q, t).
From the second equality of Theorem 2.3 and the definition of dual Macdonald polynomials,
we obtain
Corollary 2.4. Let N ∈ N, λ ∈ GT+
N , then
Qλ
(
tN−1, tN−2, . . . , 1; q, t
)
= tn(λ)
∏
s∈λ
1− qa′(s)tN−l′(s)
1− qa(s)+1tl(s)
.
Since the Macdonald polynomial Pλ(x1, x2, . . . , xN ; q, t) is symmetric in x1, x2, . . . , xN , it
is also a symmetric polynomial on x2, . . . , xN ; thus it is a linear combination of Macdonald
polynomials Pµ(x2, . . . , xN ; q, t) with coefficients in F [x1]. More precisely, we have the so-called
branching rule for Macdonald polynomials:
Theorem 2.5 (branching rule; [32, Chapter VI, equation (7.13′), Example 2(b) on p. 342]). Let
N ∈ N, λ ∈ GT+
N , then
Pλ(x1, x2, . . . , xN ; q, t) =
∑
µ∈GT+
N−1 : µ≺λ
ψλ/µ(q, t)x
|λ|−|µ|
1 Pµ(x2, . . . , xN ; q, t),
where the branching coefficients are
ψλ/µ(q, t)
def
=
∏
1≤i≤j≤N−1
(
qµi−µj tj−i+1; q
)
∞
(
qλi−λj+1tj−i+1; q
)
∞(
qλi−µj tj−i+1; q
)
∞
(
qµi−λj+1tj−i+1; q
)
∞
×
(
qλi−µj+1tj−i; q
)
∞
(
qµi−λj+1+1tj−i; q
)
∞(
qµi−µj+1tj−i; q
)
∞
(
qλi−λj+1+1tj−i; q
)
∞
,
and the sum is over positive signatures µ ∈ GT+
N−1 that satisfy the interlacing constraint
λN ≤ µN−1 ≤ λN−1 ≤ · · · ≤ µ1 ≤ λ1,
which is written succinctly as µ ≺ λ.
Observe that ψλ/µ(q, t) > 0, whenever q, t ∈ (0, 1). By applying the branching rule several
times, we can deduce the following.
Asymptotic Formulas for Macdonald Polynomials 11
Corollary 2.6. The coefficients cλ,µ = cλ,µ(q, t) in the expansion
Pλ(q, t) =
∑
µ
cλ,µmµ
are such that cλ,λ = 1 and cλ,µ ≥ 0, whenever q, t ∈ (0, 1). Moreover, cλ,µ = 0 unless λ ≥ µ.
We come to our final tool on Macdonald polynomials. It is the main theorem of [30], and is
called the Jacobi–Trudi formula for Macdonald polynomials. For any n ∈ N, nonnegative integers
τ1, . . . , τn, variables u1, . . . , un, define the rational functions C
(q,t)
τ1,...,τn(u1, . . . , un) by
C(q,t)
τ1,...,τn(u1, . . . , un)
def
=
n∏
k=1
tτk
(q/t; q)τk
(q; q)τk
(quk; q)τk
(qtuk; q)τk
∏
1≤i<j≤n
(qui/tuj ; q)τi
(qui/uj ; q)τi
(tui/(q
τiuj); q)τi
(ui/(qτiuj); q)τi
× 1
∆(qτ1u1, . . . , qτnun)
det
1≤i,j≤n
[
(qτiui)
n−j
(
1− tj−1 1− tqτiui
1− qτiui
n∏
k=1
uk − qτiui
tuk − qτiui
)]
, (2.5)
where ∆(z1, . . . , zn)
def
=
∏
1≤i<j≤n
(zi − zj) = det
[
zn−ji
]n
i,j=1
is known as the Vandermonde determi-
nant.
Theorem 2.7 (Jacobi–Trudi formula; [30, Theorem 5.1]). Let N ∈ N, λ ∈ GT+
N , then
Qλ(x1, . . . , xN ; q, t)
=
∑
τ∈M(N)
N−1∏
s=1
C(q,t)
τ1,s+1,...,τs,s+1
ui = q
λi−λs+1+
N∑
j=s+2
(τi,j−τs+1,j)
ts−i : 1 ≤ i ≤ s
×
N∏
s=1
gλs+τ+s −τ−s (x1, . . . , xN ; q, t)
}
,
where M (N) is the set of strictly upper-triangular matrices with nonnegative entries, and for each
1 ≤ s ≤ N , the integers τ+
s , τ−s , depend only on the indexing matrix τ and are defined by
τ+
s
def
=
N∑
i=s+1
τs,i, τ−s
def
=
s−1∑
i=1
τi,s. (2.6)
Remark 2.8. Observe that, even though M (N) is an infinite set, the only nonvanishing terms
in the sum above are those τ ∈ M (N) such that λs + τ+
s − τ−s ≥ 0 ∀ s = 1, . . . , N . In other
words, the sum is indexed by points of the discrete N(N−1)
2 -dimensional simplex with coordinates
{τi,j}1≤i<j≤N satisfying
τi,j ≥ 0, for all 1 ≤ i < j ≤ N,
λn +
N∑
i=n+1
τn,i −
n−1∑
i=1
τi,n ≥ 0, for all n = 1, . . . , N.
Let us introduce the last piece of terminology and main object of study in this paper.
Definition 2.9. For any m,N ∈ N with 1 ≤ m ≤ N , λ ∈ GTN , define
Pλ(x1, . . . , xm;N, q, t)
def
=
Pλ
(
x1, . . . , xm, 1, t, . . . , t
N−m−1; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
) (2.7)
and call Pλ(x1, . . . , xm;N, q, t) the Macdonald unitary character of rank N , number of variables m
and parametrized by λ. For simplicity of terminology, we call Pλ(x1, . . . , xm;N, q, t) a Macdonald
character rather than a Macdonald unitary character. Observe that if q, t ∈ C are such that
|q|, |t| ∈ (0, 1), the evaluation identity for Macdonald polynomials, Theorem 2.3, shows that the
denominator of (2.7) is nonzero.
12 C. Cuenca
3 Integral formulas for Macdonald characters of one variable
In this section, assume q is a real number in the interval (0, 1). There will also be a parameter θ,
typically θ > 0, but we also consider cases when θ is a complex number with <θ > 0. In either
case, the parameter t = qθ satisfies |t| < 1.
3.1 Statements of the theorems
The simplest contour integral representation is the following, which works only when t = qθ,
θ ∈ N, and involves a closed contour around finitely many singularities.
Theorem 3.1. Let θ ∈ N, t = qθ, N ∈ N, λ ∈ GTN and x ∈ C \
{
0, q, q2, . . . , qθN−1
}
. Then
Pλ
(
x, t, t2, . . . , tN−1; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
)
= ln(1/q)
θN−1∏
i=1
1− qi
x− qi
1
2π
√
−1
∮
C0
xz
N∏
i=1
θ−1∏
j=0
(
1− qz−(λi+θ(N−i)+j)
)dz, (3.1)
where C0 is a closed, positively oriented contour enclosing the real poles {λi + θ(N − i) + j : i =
1, . . . , N, j = 0, . . . , θ − 1} of the integrand. For instance, the rectangular contour with vertices
−M − r
√
−1, −M + r
√
−1, M + r
√
−1 and M − r
√
−1, for any − 2π
ln q > r > 0 and any
M > max{0,−λN , λ1 + θN − 1}, is a suitable contour.
The following two theorems are analytic continuations, in the variable θ, of Theorem 3.1
above.
Theorem 3.2. Let θ > 0, t = qθ, N ∈ N, λ ∈ GTN and x ∈ C \ {0}, |x| ≤ 1. The integral below
converges absolutely and the equality holds
Pλ
(
xtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, tN−2, . . . , t, 1; q, t
)
=
ln q
1− q
(
xtN ; q
)
∞
(xq; q)∞
Γq(θN)
2π
√
−1
∫
C+
xz
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
dz. (3.2)
Contour C+ is a positively oriented contour consisting of the segment [M +r
√
−1,M −r
√
−1] and
the horizontal lines [M+r
√
−1,+∞+r
√
−1), [M−r
√
−1,+∞−r
√
−1), for some − π
2 ln q > r > 0
and λN > M , see Fig. 2. Observe that C+ encloses all real poles of the integrand (which accumulate
at +∞) and no other poles.
The reader is referred to Appendix A for a reminder of the definition of the q-Gamma function,
its zeroes and poles.
Theorem 3.3. Let θ > 0, t = qθ, N ∈ N, λ ∈ GTN and x ∈ C, |x| ≥ 1. The integral below
converges absolutely and the equality holds
Pλ
(
x, t, t2, . . . , tN−1; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
)
=
ln q
q − 1
(
x−1tN ; q
)
∞
(x−1q; q)∞
Γq(θN)
2π
√
−1
∫
C−
xz
N∏
i=1
Γq(z − (λi − θi+ θ))
Γq(z − (λi − θi))
dz. (3.3)
Asymptotic Formulas for Macdonald Polynomials 13
<z
=z
Figure 2. Contour C+.
<z
=z
Figure 3. Contour C−.
Contour C− is a positively oriented contour consisting of the segment [M −r
√
−1,M +r
√
−1] and
the horizontal lines [M−r
√
−1,−∞−r
√
−1), [M+r
√
−1,−∞+r
√
−1), for some − π
2 ln q > r > 0
and M > λ1, see Fig. 3. Observe that C− encloses all real poles of the integrand (which accumulate
at −∞) and no other poles.
Remark 3.4. In the formulas above, xz = exp(z lnx). If x /∈ (−∞, 0), we can use the principal
branch of the logarithm to define lnx, and if x ∈ (−∞, 0), then we can define the logarithm in
the complex plane cut along (−
√
−1∞, 0] such that = ln a = 0 for all a ∈ (0,∞).
Remark 3.5. The formulas in Theorems 3.2 and 3.3 probably hold for more general con-
tours C+, C−, but we do not need more generality for our purposes.
Remark 3.6. When θ = 1, Theorem 3.1 recovers [24, Theorem 3.6].
Remark 3.7. The infinite contours in Theorems 3.2 and 3.3 are needed because there are
infinitely many real poles in the integrand and the contour needs to enclose all of them. When
θ ∈ N, the integrands have finitely many poles, and we can therefore close the contours, obtaining
eventually Theorem 3.1. More generally, if θ > 0 is such that θN ∈ N, a similar remark applies.
In fact, we can write the product of q-Gamma function ratios appearing in the integrand (3.2) as
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
=
Γq(λ1 + θ(N − 1)− z)
Γq(λ2 + θ(N − 1)− z)
· · · Γq(λN−1 + θ − z)
Γq(λN + θ − z)
Γq(λN − z)
Γq(λ1 + θN − z)
, (3.4)
and since Γq(t+ 1) = 1−qt
1−q Γq(t), we conclude that the product above is a rational function in q−z
with finitely many real poles. Thus formula (3.2) is true if we replaced contour C+ by a closed
contour C0 containing all finitely many real poles of the integrand. Similarly, we can replace C−
by a closed contour C0 in (3.3).
3.2 An example
Before carrying out the proofs of the theorems above in full generality, we prove some very special
cases, by means of the residue theorem and the q-binomial formula. For simplicity, let |x| < 1 be
a complex number, and consider the empty partition λ = ∅, or equivalently the N -signature
14 C. Cuenca
λ =
(
0N
)
= (0, 0, . . . , 0). As remarked in Section 2.2, we have P(0N )(q, t) = 1, and therefore the
left-hand sides of identities (3.1) and (3.2) are both equal to 1, when λ = (0N ). Let us prove
that the right-hand sides of (3.1) and (3.2) also equal 1, for λ = (0N ).
Let us begin with the case θ /∈ N, i.e., the right-hand side of (3.2). Since the contour C−
encloses all real poles in the integrand in its interior, then the right-hand side of (3.2) equals
ln q
1− q
(
xtN ; q
)
∞
(xq; q)∞
× Γq(θN)×
∞∑
n=0
xn
Resz=n Γq(−z)
Γq(θN − n)
. (3.5)
From the definition of q-Gamma functions, see Appendix A, it is evident that, for any n ∈ Z≥0,
we have
Resz=n Γq(−z) =
(−1)n(1− q)n+1
ln q
q(
n+1
2 )
(q; q)n
.
Furthermore, Γq(t+ 1) = 1−qt
1−q Γq(t) gives
Γq(θN)
Γq(θN−n) = (1− q)−n(qθN−n; q)n, so (3.5) equals
(
xtN ; q
)
∞
(xq; q)∞
×
∞∑
n=0
(−1)nq(
n+1
2 )(qθN−n; q)n
(q; q)n
xn.
The latter indeed equals 1 because of the q-binomial theorem, Theorem A.3, applied to z =
xqθN = xtN , a = q1−θN , and the equality qθNn(q1−θN ; q)n = (−1)nq(
n+1
2 )(qθN−n; q)n ∀n ≥ 0.
Second, let us consider the case θ ∈ N, i.e., the right-hand side of (3.1). Observe that for
λ = (0N ), the integrand in (3.1) can be rewritten as xz ·
θN−1∏
i=0
(1− qz−i)−1, whose set of poles
enclosed in the interior of C0 is {0, 1, 2, . . . , θN − 1}. Since Resz=n (1− qz−n)−1 = −(ln q)−1 =
(ln (1/q))−1, similar considerations as above lead us to conclude that the right side of (3.1) is
equal to the finite sum
θN−1∏
i=1
1− qi
x− qi
×
θN−1∑
n=0
xn∏
0≤i≤θN−1
i 6=n
(1− qn−i)
=
θN−1∏
i=1
1− q−i
1− xq−i
×
θN−1∑
n=0
xn(
q; q)n(q−1; q−1
)
θN−n−1
=
1(
xq−1; q−1
)
θN−1
×
θN−1∑
n=0
(−1)nq−(n+1
2 )(q−1; q−1
)
θN−1(
q−1; q−1
)
n
(q−1; q−1)θN−n−1
xn.
The latter equals 1, because of Corollary A.4 of the q-binomial formula, applied to z = xq−1 and
M = θN − 1.
A simple argument involving the index stability for Macdonald polynomials, see (2.1), shows
that if Theorems 3.1 and 3.2 hold for λ ∈ GTN , then they hold for (λ1 + n ≥ λ2 + n ≥ · · · ≥
λN +n) ∈ GTN , and any n ∈ Z, cf. Step 5 in Section 3.3 below. Thus the present example shows
how to prove Theorems 3.1 and 3.2 for signatures of the form (n, n, . . . , n) ∈ GTN , only by use
of the classical q-binomial theorem.
3.3 Integral formula when t = qθ, θ ∈ N: Proof of Theorem 3.1
Assume θ ∈ N and t = qθ. The proof of Theorem 3.1 is broken down into several steps. In the
first four steps, we prove the statement for positive signatures λ ∈ GT+
N (when all coordinates
are nonnegative: λ1 ≥ · · · ≥ λN ≥ 0), and in step 5 we extend it for all signatures λ ∈ GTN
(when some coordinates of λ could be negative).
Asymptotic Formulas for Macdonald Polynomials 15
Step 1. We derive a contour integral formula for the ratio Pλ(qrtN−1,tN−2,...,t,1;q,t)
Pλ(tN−1,...,t,1;q,t)
of Macdonald
polynomials, and any r ∈ N. The index-argument symmetry, Theorem 2.2, applied to λ =
(λ1, . . . , λN ) and µ = (r) = (r, 0N−1), gives
Pλ
(
qrtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) =
P(r)
(
qλ1tN−1, . . . , qλN−1t, qλN ; q, t
)
P(r)
(
tN−1, . . . , t, 1; q, t
) . (3.6)
The denominator P(r)(t
N−1, . . . , t, 1; q, t) has a simple expression due to the evaluation identity,
Theorem 2.3; it is particularly simple for the row partition (r):
P(r)
(
tN−1, . . . , t, 1; q, t
)
=
N∏
j=2
(
qrtj−1; q
)
∞(tj ; q)∞
(qrtj ; q)∞(tj−1; q)∞
=
(qrt; q)∞(tN ; q)∞
(qrtN ; q)∞(t; q)∞
=
(
tN ; q
)
r
(t; q)r
.
Since we also have P(r)(q, t) = (q;q)r
(t;q)r
gr(q, t), see (2.4), identity (3.6) becomes
Pλ
(
qrtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) =
(q; q)r(
tN ; q
)
r
gr
(
qλ1tN−1, . . . , qλN ; q, t
)
. (3.7)
The symmetric polynomials gr(q, t), in addition to being essentially one-row Macdonald
polynomials, can be defined in terms of their generating function as follows, see [32, Chapter VI]:
N∏
i=1
(txiy; q)∞
(xiy; q)∞
=
∑
r≥0
gr(x1, . . . , xN ; q, t)yr. (3.8)
The relation (3.8) holds formally in the ring ΛF [x1, . . . , xN ][[y]]. If we fix nonzero values
x1, . . . , xN ∈ C \ {0}, the identity above is an equality of real analytic functions in the domain
{y ∈ C : max1≤i≤N |xiy| < 1}. Thus we have the following contour integral representation
gr(x1, . . . , xN ; q, t) =
1
2π
√
−1
∮
C
1
yr+1
N∏
i=1
(txiy; q)∞
(xiy; q)∞
dy,
where C is any circle around the origin and radius smaller than (maxi |xi|)−1. Let xi = qλitN−i
for i = 1, 2, . . . , N in the integral representation of gr(q, t), and replace it into the right-hand side
of (3.7); then
Pλ
(
qrtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) =
(q; q)r(
tN ; q
)
r
1
2π
√
−1
∮
C
1
yr+1
N∏
i=1
(
qλitN−i+1y; q
)
∞(
qλitN−iy; q
)
∞
dy, (3.9)
where C can be taken to be any circle around the origin of radius smaller than 1 (we need here
that λ ∈ GT+
N implies qλitN−i < 1 ∀ i). For t = qθ, θ ∈ N, we can simplify (3.9) to
Pλ
(
qrtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) =
(q; q)r(
qθN ; q
)
r
1
2π
√
−1
∮
C
y−(r+1)dy
N∏
i=1
θ−1∏
j=0
(1− qλi+θ(N−i)+jy)
. (3.10)
Step 2. We obtain a new contour integral representation by modifying (3.10). The resulting
contour integral representation involves an open contour C+, that looks like that of Fig. 2 (but
has a slight difference from that in Theorem 3.2).
Observe that the absolute value of the integrand in (3.10) is of order o(R−r−1) = o(R−2),
if |y| = R is large. An application of Cauchy’s theorem yields that the value of the integral is
16 C. Cuenca
<z
=z
O
Figure 4. Contour C′.
unchanged if the closed contour C is deformed into the “keyhole” contour C′ shown in Fig. 4. Let
us describe the contour C′ in words: it is a positively oriented contour, formed by two lines away
from the origin, of arguments ±3π/4, and the portion of a semicircle of some radius 0 < δ < 1.
Evidently, the straight lines are part of the level lines =(ln(y)) = ±3π/4, while the portion of the
semicircle is part of the level line <(ln(y)) = δ (where ln is defined in C \ (−∞, 0]).
Next, we make the change of variables y = q−z, or z = − ln y
ln q , where ln is defined on its
principal branch. Based on the previous observations about the contour C′ being composed by
level lines, it is clear that the resulting contour for z is a negatively oriented contour formed by
one segment and two straight lines, but we can easily reverse the orientation of the contour at
the cost of switching signs. Let us call the positively oriented contour C+, see Fig. 2; the integral
formula becomes
Pλ
(
qrtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) =
(q; q)r(
qθN ; q
)
r
1
2π
√
−1
∫
C+
q(r+1)zq−z ln qdz
N∏
i=1
θ−1∏
j=0
(
1− qλi+θ(N−i)+j−z
) . (3.11)
Note that points in the horizontal lines of contour C+ in (3.11) have imaginary parts ± 3π
4 ln q , while
points in the vertical segment of C+ have real part − ln δ/ ln q < 0. Thus contour C+ encloses
exactly all the real poles of the integrand in (3.11) and no other poles (it also encloses the origin,
though this not important).
Step 3. We make some final modifications to formula (3.11); the resulting contour integral
representation will include a closed contour C0 as in the statement of the theorem.
Observe that the integrand in (3.11) has finitely many real poles and they are all enclosed
by contour C+; also the integrand is exponentially small as |z| → ∞ along the contour C+.
Therefore, as an application of Cauchy’s theorem, we can replace C+ by a closed contour C0 that
encloses all finitely many real poles of the integrand. Also note that for r > θN , we can write
(q;q)r
(qθN ;q)r
=
θN−1∏
i=1
1−qi
1−qr+i . Thus for r > θN , equation (3.11) can be rewritten as
Pλ
(
qrtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
)
=
ln q
2π
√
−1
θN−1∏
i=1
1− qi
1− qr+i
∮
C0
qrz
N∏
i=1
θ−1∏
j=0
1
1− qλi+θ(N−i)+j−z
dz. (3.12)
Asymptotic Formulas for Macdonald Polynomials 17
We let x = qrtN = qr+θN and replace all instances of qr in (3.12) above by x/tN ; then multiply
both sides of the identity by t|λ|. We claim that the resulting equation is exactly equality (3.1).
In fact, the left-hand side of our equation is
t|λ|
Pλ
(
x/t, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) =
Pλ
(
x, tN−1, tN−2, . . . , t; q, t
)
Pλ
(
tN−1, tN−2, . . . , 1; q, t
) =
Pλ
(
x, t, t2, . . . , tN−1; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
) ,
by homogeneity of the Macdonald polynomials. On the other hand, the right-hand side of our
equation is
t|λ|
ln q
2π
√
−1
θN−1∏
i=1
1− qi
1− xqi−θN
∮
C0
xzt−Nz
N∏
i=1
θ−1∏
j=0
1
1− qλi+θ(N−i)+j−z
dz,
which can be shown to be equal to the right-hand side of (3.1), by simple algebraic manipulations.
The conclusion is that we have proved identity (3.1) for all x = qm with m ∈ N large enough
(to be precise, m is of the form r + θN and r > θN , so the statement was proved for all integers
m > 2θN).
Step 4. Still assuming λ ∈ GT+
N , we prove Theorem 3.1 for all x ∈ C \ ({0} ∪ {qi : i =
1, 2, . . . , θN − 1}). Observe that x 6= 0 is imposed to make sense of the term xz = exp(z lnx)
and x /∈ {q, q2, . . . , qθN−1} is necessary so that the denominator of the right-hand side of (3.1) is
nonzero.
We claim that both sides of (3.1) are rational functions of x. This would prove the desired
result, given that we have shown in step 3 above that (3.1) holds for infinitely many points qm,
where m is large enough. The left-hand side of (3.1) is obviously a polynomial on x. In the
right-hand side of (3.1), we have the product
θN−1∏
i=1
(x− qi)−1 which is a rational function. We
only need to check that the contour integral is also a rational function on x. In fact, this follows
from the residue theorem and the fact that there are finitely many poles in the interior of C0,
all of these being simple and integral. The fact that the poles considered above are simple and
integral can be easily checked and is equivalent to fact that all the values λi + θ(N − i) + j, for
1 ≤ i ≤ N , 0 ≤ j ≤ θ − 1, are pairwise distinct integers.
Step 5. We extend equality (3.1) to all signatures λ ∈ GTN .
Let λ ∈ GTN be arbitrary and we aim to prove (3.1). If λ ∈ GT+
N , the result is already
proved in the first four steps above. Otherwise, choose m ∈ N such that λN +m ≥ 0, and so
λ̃
def
= λ+
(
mN
)
= (λ1 +m,λ2 +m, . . . , λN +m) ∈ GT+
N .
By the index stability (2.1), (x1 · · ·xN )mPλ(x1, . . . , xN ; q, t) = P
λ̃
(x1, . . . , xN ; q, t). Thus
multiplying the left-hand side of the equality (3.1) by xm =
(
x ·t ·t2 · · · tN−1
)m
(1 ·t ·t2 · · · tN−1)−m
gives (
x · t · t2 · · · tN−1
)m · Pλ(x, t, t2, . . . , tN−1; q, t
)(
1 · t · t2 · · · tN−1
)m · Pλ(1, t, t2, . . . , tN−1; q, t
) =
P
λ̃
(
x, t, t2, . . . , tN−1; q, t
)
P
λ̃
(
1, t, t2, . . . , tN−1; q, t
) .
If we multiply the right-hand side of equality (3.1) by xm, and also make the change of variables
z 7→ z −m in the contour integral, we obtain
ln(1/q)
θN−1∏
i=1
1− qi
x− qi
1
2π
√
−1
∮
C0
xz
N∏
i=1
θ−1∏
j=0
(
1− qz−(λ̃i+θ(N−i)+j)
)dz.
Since we have proved equality (3.1) for the positive signature λ̃ already, then (3.1) also holds
for λ, since we have multiplied both sides by xm and obtained equal expressions.
18 C. Cuenca
3.4 Integral formula for general q, t I:
Outline of proof of Theorems 3.2 and 3.3
Theorems 3.2 and 3.3 are analytic continuations of Theorem 3.1, with respect to the variable θ. We
shall need the weak version of Carlson’s lemma below, which is proved in [2, Theorem 2.8.1] (in this
reference, the statement is given for functions that are analytic and bounded on {z ∈ C : <z ≥ 0},
but a simple change of variables z 7→ z +M , for some large positive integer M ∈ N leads us to
the version below).
Lemma 3.8 (Carlson’s lemma). Let f(z) be a holomorphic function on the right half plane
{z ∈ C : <z > 0}, such that f(z) is uniformly bounded on the domain {z ∈ C : <z ≥ M}, for
some M > 0. If f(n) = 0 for all n ∈ N, then f is identically zero.
We outline the steps of the proof of Theorem 3.2 below, and we shall carry out the detailed
proof of each step in the next section. Theorem 3.3 can be proved analogously and we leave the
details to the reader.
• Step 0. Prove that the right-hand side of (3.2) is well-defined, i.e., the contour integral is
absolutely convergent. We prove the absolute convergence for all |x| ≤ 1, x 6= 0, and θ > 0.
• Step 1. Reduce the general statement to the case |x| < 1, x 6= 0, and λ ∈ GT+
N . Assume
the latter conditions are in place for the remaining steps in the proof.
• Step 2. Prove that the equation (3.2) of Theorem 3.2 holds when t = qθ and θ ∈ N.
• Step 3. Prove that both sides of equation (3.2) are holomorphic functions of θ on the right
half plane {θ ∈ C : <θ > 0}.
• Step 4. Prove that, for some (sufficiently large) M > 0, both sides of equation (3.2) are
uniformly bounded functions of θ on {θ ∈ C : <θ ≥M}.
Then we can conclude Theorem 3.2 as follows. From steps 2–4 and Lemma 3.8 above,
identity (3.2) is proved for all x ∈ C with |x| < 1, x 6= 0, λ ∈ GT+
N , θ ∈ {z ∈ C : <z > 0},
and in particular for all θ > 0. Thus step 1 shows the theorem holds for the more general case
x ∈ C \ {0}, |x| ≤ 1, and λ ∈ GTN .
3.5 Integral formula for general q, t II: Proof of Theorem 3.2
In this section, we prove Theorem 3.2; as we already mentioned, Theorem 3.3 can be proved
in an analogous way, and we leave the details to the reader. From the last paragraph of the
previous section, Theorem 3.2 will be proved if we verify steps 0–4 stated there.
Proof of Step 0. Fix θ > 0, x ∈ C \ {0}, |x| ≤ 1, and let
Fq(z; θ)
def
= xz
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
.
From the definition of the q-Gamma function, Fq(z; θ) is a holomorphic function in a neighborhood
of the contour C+, thus
∫
C+ Fq(z; θ)dz is a well-posed integral. To prove its absolute convergence,
it suffices to show that |Fq(z; θ)| ≤ c1 · qc2<z, for some c1, c2 > 0 and all z ∈ C+ with <z large
enough.
Because |x| ≤ 1 and all z in the contour C+ have bounded imaginary parts, we have |xz| =
exp(<z ln |x| − =z arg(x)), z ∈ C+, is upper-bounded by a constant. Thus it will suffice to show∣∣∣∣ Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
∣∣∣∣ ≤ c1 · qc2<z, i = 1, 2, . . . , N,
Asymptotic Formulas for Macdonald Polynomials 19
for some c1, c2 > 0 and all z ∈ C+ with <z large enough. Since each λi + θ(N − i) is real and |=z|
is constant for z ∈ C+, when <z is large enough, the statement reduces to showing∣∣∣∣ Γq(−z)
Γq(θ − z)
∣∣∣∣ ≤ c1 · qc2<z, (3.13)
for some c1, c2 > 0 and all z ∈ C+ with <z large enough.
Let π/2 > d > 0 be the value such that |=z|, for any point z ∈ C+ with <z large enough,
equals −d/ ln q (such d exists by our assumptions on the contour C+). There exists a real number
a > 0 large enough so that the following inequality holds
qa
(
1 + qθ
)
≤ 2 cos d. (3.14)
Now we consider only points z ∈ C+ with <z large enough so that |=z| = −d/ ln q and <z >
θ + a+ 1; let M = M(z) = b<z − θ − ac ∈ N. From the definition of a in (3.14), the restriction
on the values of <z, and q ∈ (0, 1), one can easily obtain∣∣1− qz−θ−j∣∣ ≤ ∣∣1− qz−j∣∣, ∀ j = 0, 1, . . . ,M. (3.15)
As a consequence of inequality (3.15) and the definition of the q-Gamma function, we have∣∣∣∣ Γq(−z)
Γq(θ − z)
∣∣∣∣ =
∣∣∣∣∣(1− q)θ
(
q−z+θ; q
)
∞
(q−z; q)∞
∣∣∣∣∣
=
∣∣∣∣∣(1− q)θ
(
1− qθ−z
)
· · ·
(
1− qθ−z+M
)
(1− q−z) · · ·
(
1− q−z+M
) (q−z+θ+M+1; q
)
∞(
q−z+M+1; q
)
∞
∣∣∣∣∣
=
∣∣∣∣∣(1− q)θ
(
1− qz−θ
)
· · ·
(
1− qz−θ−M
)
(1− qz) · · ·
(
1− qz−M
) qθ(M+1)
(
qθ−z+M+1; q
)
∞(
q−z+M+1; q
)
∞
∣∣∣∣∣
≤ (1− q)θ · qθ(M+1)
∣∣∣∣∣
(
qθ−z+M+1; q
)
∞(
q−z+M+1; q
)
∞
∣∣∣∣∣ .
Since M = b<z − θ − ac, the previous inequality almost shows (3.13). We are left to show
that the absolute value of (qθ−z+M+1; q)∞/(q
−z+M+1; q)∞ is upper bounded by a constant
independent of z ∈ C+ as long as <z is large enough. In fact, we have <z − θ − a ≥
M > <z − θ − a − 1, so |qθ−z+M+1|, |q−z+M+1| ≤ q−θ−a and |=qθ−z+M+1|, |=q−z+M+1| ≥
qθ+M+1−<z sin d ≥ q1−a sin d > 0. Because the q-Pochhammer symbol (x; q)∞ is holomorphic
on x ∈ C, in particular continuous, the absolute value |(x; q)∞| attains a maximum cmax ≥ 0
and a minimum value cmin ≥ 0 on the compact subset {x ∈ C : |x| ≤ q−θ−a, |=x| ≥ q1−a sin d},
and moreover cmin > 0 because all the roots of (x; q)∞ = 0 are real. We have thus proved
|(qθ−z+M+1; q)∞|/|(q−z+M+1; q)∞| ≤ cmax/cmin <∞, as desired.
Proof of Step 1. Assume we proved identity (3.2) when x ∈ C \ {0}, |x| < 1. Then an easy
application of the dominated convergence theorem (we know the integral converges absolutely
when x = 1 because of step 1) shows the equation would also hold for all x ∈ C \ {0}, |x| ≤ 1.
Also, observe that step 5 of the proof of Theorem 3.1 can be repeated almost word-by-word
to extend the theorem to all λ ∈ GTN , assuming that it was proved for all λ ∈ GT+
N . Therefore
we will assume for convenience in the next steps that |x| < 1, x 6= 0, λ ∈ GT+
N , and prove the
theorem only in that case, without loss of generality.
Proof of Step 2. In this step, we consider the case θ ∈ N. Since we are also assuming
|x| < 1, x 6= 0, then clearly xtN /∈
{
0, q, q2, . . . , qθN−1
}
and therefore equality (3.1) holds if x was
20 C. Cuenca
replaced with xtN . After multiplying both sides of the resulting equation by t−|λ|, we claim that
we arrive at the desired (3.2) with C0 in place of C+. In fact, the left-hand side of our equation is
t−|λ|
Pλ
(
xtN , t, t2, . . . , tN−1; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
)
=
Pλ
(
xtN−1, 1, t, . . . , tN−2; q, t
)
Pλ
(
1, t, t2, . . . , tN−1; q, t
) =
Pλ
(
xtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, tN−2, . . . , t, 1; q, t
)
thanks to the homogeneity and symmetry of Macdonald polynomials. On the other hand, the
right-hand side of our equation is
t−|λ| ln(1/q)
θN−1∏
i=1
1− qi
xqθN − qi
1
2π
√
−1
∮
C0
xztNz
N∏
i=1
θ−1∏
j=0
(
1− qz−(λi+θ(N−i)+j)
)dz,
which can be shown equal to the right-hand side of (3.2) (with C+ replaced by C0) after simple
algebraic manipulations. Finally observe that contour C0 can be replaced by C+ by an application
of Cauchy’s theorem.
Proof of Step 3. Let us begin by proving holomorphicity of the left-hand side of (3.2)
with respect to the variable θ; observe that θ only appears inside the variable t = qθ. The
Macdonald polynomials Pλ(x1, . . . , xN ; q, t) are holomorphic functions of θ on {θ ∈ C : <θ >
0} because all the branching coefficients ψµ/ν(q, t) are holomorphic on this domain. Then
Pλ
(
xtN−1, tN−2, . . . , t, 1; q, t
)
and Pλ
(
tN−1, tN−2, . . . , t, 1; q, t
)
are also holomorphic. It follows
that the ratio of these two quantities is holomorphic if we proved that the denominator
Pλ
(
tN−1, tN−2, . . . , t, 1; q, t
)
never vanishes for <θ > 0, or equivalently for |t| < 1; this is
evident from the evaluation identity for Macdonald polynomials, Theorem 2.3.
Next we prove holomorphicity of the right-hand side of (3.2) as a function of θ in the domain
{θ ∈ C : <θ > 0}. Clearly
(
xqθN ; q
)
∞Γq(θN) is holomorphic in the given domain of θ, but it is
less clear that the integral
∫
C+ Fq(z, θ)dz is holomorphic in the right half-plane, where
Fq(z; θ) = xz
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
.
First we claim that Fq(z; θ) is holomorphic on U ×{θ ∈ C : <θ > 0}, for some neighborhood U
of C+. Indeed, the factor xz is clearly entire on z, and does not depend on θ. We can write the
product of ratios of q-Gamma functions in the definition of Fq(z; θ), as we did in Remark 3.7,
see (3.4). For 1 ≤ i ≤ N − 1, we have
Γq(λi + θ(N − i)− z)
Γq(λi+1 + θ(N − i)− z)
=
λi−1∏
n=λi+1
[n+ θ(N − i)− z]q =
λi−1∏
n=λi+1
1− qn+θ(N−i)−z
1− q
,
which is clearly holomorphic on (z, θ) ∈ C2. Finally the remaining factor can be written as
Γq(λN − z)
Γq(λ1 + θN − z)
= (1− q)λ1+θN−λN
(
qλ1+θN−z; q
)
∞(
qλN−z; q
)
∞
.
It is clear that (1− q)λ1+θN−λN
(
qλ1+θN−z; q
)
∞ is holomorphic on (z, θ) ∈ C2. And also there is
a neighborhood U of C+ on which the function
(
qλN−z; q
)−1
∞ of z is holomorphic on U .
Secondly, we claim that
∫
C+ Fq(z; θ)dz is absolutely convergent and moreover
∫
C+ |Fq(z; θ)|dz
is uniformly bounded on compact subsets of {θ ∈ C : <θ > 0}; this will be a consequence of the
stronger statement
Asymptotic Formulas for Macdonald Polynomials 21
Claim 3.9. Consider any compact subset K ⊂ {θ ∈ C : <θ > 0}. There exists a constant M1 > 0,
depending on K, such that
|Fq(z; θ)| < M1|xz|, ∀ z ∈ C+, θ ∈ K. (3.16)
Let us first conclude the proof of step 3 from the claim above.
Since |x| < 1 we have that |xz| decreases exponentially as |z| → ∞, z ∈ C+. Then inequal-
ity (3.16) shows that
∫
C+ Fq(z; θ)dz is absolutely convergent and for all θ belonging to the compact
subset K ⊂ {θ ∈ C : <θ > 0}, we have the bound
∫
C+ |Fq(z; θ)|dz < C(K), for some constant
C(K) > 0 that depends on K.
Let T be any triangular contour belonging to {θ ∈ C : <θ > 0}. Then∫
T
∫
C+
|Fq(z; θ)|dzdθ ≤ C(T )
∫
T
|dθ| <∞.
By Fubini’s and Cauchy’s theorems, we have∫
T
∫
C+
Fq(z; θ)dzdθ =
∫
C+
∫
T
Fq(z; θ)dθdz = 0.
Morera’s theorem implies that
∫
C+ Fq(z; θ)dz is holomorphic on {θ ∈ C : <θ > 0}, concluding
step 3.
Proof of Claim 3.9. Since we have shown before that Fq(z; θ) is holomorphic on U × {θ ∈
C : <θ > 0}, then it suffices to prove inequality (3.16) for all z ∈ C+, <z > M2 and all θ ∈ K,
where M2 is an arbitrarily large positive constant.
We can express the product of q-Gamma function ratios in the definition of Fq(z; θ) as we did
in (3.4). The last ratio is
Γq(λN − z)
Γq(λ1 + θN − z)
=
[−z + λN − 1]q · · · [−z + 1]q[−z]q
[−z + λ1 + θN − 1]q · · · [−z + θN + 1]q[−z + θN ]q
Γq(−z)
Γq(θN − z)
,
because we are assuming λ ∈ GT+
N (and so λ1 ≥ λN ≥ 0). Plugging the equality above into (3.4),
we obtain
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
=
∏
i,j
[θ(N − i) + j − z]q
[λN − z − 1]q · · · [−z]q
[λ1 + θN − z − 1]q · · · [θN − z]q
(3.17)
× Γq(−z)
Γq(θN − z)
, (3.18)
where the product in (3.17) is over 1 ≤ i ≤ N − 1 and λi+1 ≤ j < λi. The term (3.17) is of the
form P (q−z)/Q(q−z), where P,Q ∈ C(q, qθ)[x] are both polynomials of degree λ1. It follows that
there exist constants M ′1,M
′
2 > 0 such that <z > M ′2 implies |P (q−z)|/|Q(q−z)| < M ′1.
Thus we are left to deal with (3.18), which by definition of the q-Gamma function equals
(1− q)θN (qθN−z ;q)∞
(q−z ;q)∞
. For any θ with <θ > 0, we have |(1− q)θ| < 1. Thus we only need to prove
the existence of M ′′1 ,M
′′
2 > 0 such that z ∈ C+, <z > M ′′2 , implies∣∣∣∣∣
(
qθN−z; q
)
∞
(q−z; q)∞
∣∣∣∣∣ < M ′′1 , for all θ ∈ K. (3.19)
22 C. Cuenca
There exists a > 0 large enough such that
q−a
(
1− qN infθ∈K <θ
)
> 2. (3.20)
Note that such a exists because K ⊂ {θ ∈ C : <θ > 0} is compact and so infθ∈K <θ > 0.
Now consider only z ∈ C+ with <z > a + 1, and let M = M(z)
def
= b<z − ac ∈ N. From the
triangle inequality, |1 − qθN−z+j | ≤ 1 + qN<θ−<z+j and |1 − q−z+j | ≥ q−<z+j − 1. Along with
condition (3.20), we deduce that |1− q−z+j | ≥ |1− qθN−z+j |, for all 0 ≤ j ≤M . Since we can
write (
qθN−z; q
)
∞
(q−z; q)∞
=
(
1− qθN−z
)
· · ·
(
1− qθN−z+M
)
(1− q−z) · · ·
(
1− q−z+M
) ·
(
qθN−z+M+1; q
)
∞(
q−z+M+1; q
)
∞
,
it follows that∣∣∣∣∣
(
qθN−z; q
)
∞
(q−z; q)∞
∣∣∣∣∣ ≤
∣∣∣∣∣
(
qθN−z+M+1; q
)
∞(
q−z+M+1; q
)
∞
∣∣∣∣∣ . (3.21)
Thus we only need to show that the right-hand side of (3.21) is bounded by a constant, for all
z ∈ C+ with <z large enough.
Since M = b<z − ac > <z − a− 1, we have |q−z+M+1| = qM+1−<z ≤ q−a and |qθN−z+M+1| ≤
qN<θ−a < q−a. Moreover, for z ∈ C+ with <z large enough, |=q−z+M+1| ≥ q−a+1| sin(ln q ·=z)| def=
m1(z) and since |=z| is a constant between 0 and − π
2 ln q for z ∈ C+ with <z large enough, then
m1(z) = m1 > 0 is a strictly positive constant independent of z ∈ C+ as long as <z is large
enough. Since the function (x; q)∞ is continuous on x ∈ C, we have cmax
def
= sup
|x|≤q−a
|(x; q)∞| <∞,
and cmin
def
= inf |x|≤q−a,|=x|≥m1
|(x; q)∞| ∈ (0,∞). Thus the right-hand side of (3.21) is upper
bounded by the constant cmax/cmin <∞. �
Proof of Step 4. We prove a stronger statement than step 4. Let M > 0 be any positive
number. We show that both sides of (3.2) are uniformly bounded on {θ ∈ C : <θ ≥M}. Let us
begin with the left-hand side of (3.2). Observe that θ appears in the left side only within the
variable t = qθ and |t| = q<θ ≤ qM . Name ε = qM ∈ (0, 1); we have to prove that there exists a
constant C > 0 such that
sup
|t|≤ε
∣∣∣∣∣Pλ
(
xtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, tN−2, . . . , t, 1; q, t
) ∣∣∣∣∣ ≤ C.
Thanks to the branching rule for Macdonald polynomials, Theorem 2.5, and the assumptions
|x| ≤ 1, x 6= 0, λ ∈ GT+
N , we have∣∣∣∣∣Pλ
(
xtN−1, tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, tN−2, . . . , t, 1; q, t
) ∣∣∣∣∣ ≤∑
µ≺λ
|ψλ/µ(q, t)||t|(N−1)(|λ|−|µ|)
∣∣∣∣∣Pµ
(
tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
)∣∣∣∣∣.
Given λ ∈ GT+
N , there are finitely many µ ∈ GT+
N−1 with µ ≺ λ. Thus it suffices to prove that
there exist constants C1, C2 > 0 such that
|ψλ/µ(q, t)| ≤ C1,
∣∣∣∣∣t(N−1)(|λ|−|µ|)Pµ
(
tN−2, . . . , t, 1; q, t
)
Pµ
(
tN−1, . . . , t, 1; q, t
)∣∣∣∣∣ ≤ C2,
where C1, C2 do not depend on t, though they may depend on µ.
Asymptotic Formulas for Macdonald Polynomials 23
The branching coefficient |ψλ/µ(q, t)|, due to the expression in Theorem 2.5, is a finite product
of terms of the form 1−qatb
1−qctd , with a, b, c, d ∈ Z≥0 and (c, d) 6= (0, 0). We have∣∣∣∣1− qatb1− qctd
∣∣∣∣ ≤ 2
(
1− qcεd
)−1
for all |t| ≤ ε and 2(1− qcεd)−1 does not depend on t, so the boundedness of |ψλ/µ(q, t)| follows.
Due to the evaluation identity for Macdonald polynomials, Theorem 2.3, we have
t(N−1)(|λ|−|µ|)Pµ
(
tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
) = t(N−1)(|λ|−|µ|)+n(µ)−n(λ)
×
∏
1≤i<j≤N−1
(qµi−µj tj−i; q)∞(tj−i+1; q)∞
(tj−i; q)∞(qµi−µj tj−i+1; q)∞
∏
1≤i<j≤N
(tj−i; q)∞(qλi−λj tj−i+1; q)∞
(qλi−λj tj−i; q)∞(tj−i+1; q)∞
,
where n(λ)
def
= (N − 1)λN + (N − 2)λN−1 + · · ·+ 2λ3 + λ2 and similarly for n(µ).
The last two terms above are products of a finite number of fractions 1−qatb
1−qctd , with a, b, c, d ∈
Z≥0, (c, d) 6= (0, 0), and as we saw above it is implied that the absolute value of the last two
terms above are upper bounded by a constant independent of t (as long as |t| ≤ ε). Thus our
only goal is to show there is an upper bound for t(N−1)(|λ|−|µ|)+n(µ)−n(λ); this fact follows if the
exponent is nonnegative. In fact, we have
(N − 1)(|λ| − |µ|)− (n(λ)− n(µ))
= (N − 1)(|λ| − |µ|)− ((N − 1)λN + (N − 2)(λN−1 − µN−1) + · · ·+ (λ2 − µ2))
≥ (N − 1)(|λ| − |µ|)− (N − 1)(λN + λN−1 − µN−1 + · · ·+ λ2 − µ2)
= (N − 1)(λ1 − µ1) ≥ 0.
Let us proceed to prove uniform boundedness of the right-hand side of (3.2) on {θ ∈ C : <θ ≥
M}. First of all, the triangular inequality gives
∣∣(xtN ; q
)
∞
∣∣ ≤ ∞∏
i=0
(
1 + |x|qN<θ+i
)
≤
∞∏
i=0
(
1 + |x|qi
)
,
|(xq; q)∞| =
∞∏
i=1
∣∣(1− xqi)∣∣ ≥ ∞∏
i=1
(
1− |x|qi
)
,
so the factor
(
xtN ; q
)
∞/(xq; q)∞ in (3.2) has an upper-bounded absolute value. We are left to
deal with
Γq(θN)
∫
C+
xz
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
dz
=
∫
C+
xzΓq(θN)
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
dz (3.22)
and prove its absolute value is uniformly bounded on {θ ∈ C : <θ ≥M}.
For any M2 > 0, the contribution of the portion C+ ∩ {z ∈ C : <z ≤ M2} of the contour
is bounded by a constant. In fact, <θ ≥ M implies |t| = |qθ| = q<θ ≤ qM and θ appears
in the integrand of (3.2) only as part of the exponent of some q, thus the integrand can be
written as a function of z and t (with q ∈ (0, 1) fixed). Thus for (z, t) in the compact subset
(C+ ∩ {z ∈ C : <z ≤M2})×
[
0, qM
]
, the integrand in (3.22) attains a maximum value L <∞,
24 C. Cuenca
and the contribution of the integral in the portion C+ ∩ {z ∈ C : <z ≤ M2} of the contour is
upper bounded by L times the length of that finite portion.
Since |x| < 1, the term xz decreases exponentially as |z| → ∞, z ∈ C+. Thus to deal with the
infinite portion of the integral C+ ∩ {z ∈ C : <z ≥M2}, it is enough to show that
Γq(θN)
N∏
i=1
Γq(λi + θ(N − i)− z)
Γq(λi + θ(N − i+ 1)− z)
= (1− q) (q; q)∞
(qθN ; q)∞
N∏
i=1
(
qλi+θ(N−i+1)−z; q
)
∞(
qλi+θ(N−i)−z; q
)
∞
has bounded absolute value for all z ∈ C+, <z > M2 and θ ∈ C, <θ ≥M , for a suitable constant
M2 > 0. Clearly
∣∣(qθN ; q
)∣∣−1 ≤
(
qN<θ; q
)−1 ≤
(
qNM ; q
)−1
, thus we only need a bound on the
absolute value of
N∏
i=1
(
qλi+θ(N−i+1)−z; q
)
∞(
qλi+θ(N−i)−z; q
)
∞
, (3.23)
uniformly over all z ∈ C+, <z > M2, and θ ∈ C, <θ ≥ M . We can bound the absolute value
of (3.23) by
N−1∏
i=1
∣∣∣∣∣
(
qλi+1+θ(N−i)−z; q
)
∞(
qλi+θ(N−i)−z; q
)
∞
∣∣∣∣∣×
∣∣∣∣∣
(
qλ1+θN−z; q
)
∞(
qλN−z; q
)
∞
∣∣∣∣∣
=
∏
1≤i≤N−1
λi+1≤j<λi
∣∣1− qθ(N−i)+j−z∣∣× 1∏
λN≤j<λ1
|1− qj−z|
×
∣∣∣∣∣
(
qλ1+θN−z; q
)
∞(
qλ1−z; q
)
∞
∣∣∣∣∣
≤
∏
1≤i≤N−1
λi+1≤j<λi
(
1 + q(N−i)<θ+j−<z)× 1∏
λN≤j<λ1
(
qj−<z − 1
) × ∣∣∣∣∣
(
qλ1+θN−z; q
)
∞(
qλ1−z; q
)
∞
∣∣∣∣∣
≤
∏
λN≤j<λ1
1 + qj−<z
qj−<z − 1
×
∣∣∣∣∣
(
qλ1+θN−z; q
)
∞(
qλ1−z; q
)
∞
∣∣∣∣∣ .
If <z is large enough, the product
∏
λN≤j<λ1
1+qj−<z
qj−<z−1
is clearly upper bounded by a constant. We
still have to bound
∣∣∣ (qλ1+θN−z ;q)∞
(qλ1−z ;q)∞
∣∣∣. Since λ1 is real and =z is constant for z ∈ C+, <z large
enough, it suffices to prove the following statement: there exist constants M1,M2 > 0 such that
z ∈ C+, <z > M2, and <θ ≥M imply∣∣∣∣∣
(
qθN−z; q
)
∞
(q−z; q)∞
∣∣∣∣∣ < M1.
This statement was proved above in step 3, see (3.19). In that case, θ varied over a compact
subset K ⊂ {θ ∈ C : <θ > 0}, but in this case θ varies over a closed infinite domain of the form
{θ ∈ C : <θ ≥M}. However, the expression (qθN−z ;q)∞
(q−z ;q)∞
depends on θ only by means of qθ, so the
proof of (3.19) above can be repeated word-by-word, since we only used |t| ≤ qinfθ∈K <θ < 1 in
that proof.
4 Multiplicative formulas for Macdonald characters
We now come to the multiplicative formulas. All of our results require parameter θ to be a positive
integer. In this section, q is typically a variable (but of course, we can specialize q to a complex
number later).
Asymptotic Formulas for Macdonald Polynomials 25
4.1 Statement of the multiplicative theorem and some consequences
We need some non-standard terminology on q-difference operators. The q-shift operators
{Tq,xi : i = 1, . . . ,m} are linear operators on C(q)[x1, . . . , xm] that act as
(Tq,xif)(x1, . . . , xm)
def
= f(x1, . . . , xi−1, qxi, xi+1, . . . , xm).
The q-degree operators {Dq,xi}i=1,...,m are linear operators on C(q)[x1, . . . , xm] defined by
Dq,xi
def
=
Tq,xi − 1
q − 1
or
(Dq,xif)(x1, . . . , xm) =
f(x1, . . . , xi−1, qxi, xi+1, . . . , xm)− f(x1, . . . , xm)
q − 1
.
The q-difference operators that appear in the multiplicative formulas for Macdonald polynomials
are finite sums of terms
ci1,...,im(x1, . . . , xm; q)T i1q,x1 · · ·T
im
q,xm or ci1,...,im(x1, . . . , xm; q)Di1
q,x1 · · ·D
im
q,xm ,
where (i1, . . . , im) vary over a finite subset of Zm≥0, and ci1,...,im(x1, . . . , xm; q) are rational
functions in the variables q, x1, . . . , xm. Thus the operators that we consider are linear operators
C(q)[x1, . . . , xm] −→ C(q, x1, . . . , xm) that act on polynomials and yield rational functions.
Recall the setting of Jacobi Trudi’s formula for Macdonald polynomials, Theorem 2.7. The
expressions C
(q,qθ)
τ1,...,τn(u1, . . . , un) were defined in (2.5); they are rational functions in q, u1, . . . , un
whose denominators are products of linear factors. We define M
(m)
θ as the set of strictly upper-
triangular m × m matrices whose entries belong to {0, 1, . . . , θ} (the cardinality of M
(m)
θ is
(θ + 1)
(
m
2
)
). For any strictly upper-triangular m×m matrix τ and 1 ≤ i ≤ m, let τ+
i
def
=
m∑
j=i+1
τi,j
(resp. τ−i
def
=
i−1∑
j=1
τj,i) be the sum of the entries of τ to the right of (i, i) (resp. sum of entries of τ
above (i, i)), cf. (2.6). The main theorem of this section is the following.
Theorem 4.1. Let θ ∈ N, t = qθ, N ∈ N, λ ∈ GTN . Then
Pλ
(
x1, . . . , xm;N, q, qθ
)
=
qθ
2(m+1
3 )−{Nθ2−(θ+1
2 )}(m2 )
m∏
i=1
[θ(N − i+ 1)− 1]q!
m∏
i=1
θ(N−m+1)−1∏
j=1
(xi − qj−θ)
× 1∏
1≤i<j≤m
0≤k<θ
(xi − qkxj)
×D(m)
q,θ
m∏
i=1
Pλ
(
xi;N, q, q
θ
) θN−1∏
j=1
(
xi − qj−θ
)
[θN − 1]q!
, (4.1)
where D(m)
q,θ is the q-difference operator C(q)[x1, . . . , xm] −→ C(q, x1, . . . , xm) given by
D(m)
q,θ
def
=
1
(q − 1)θ(
m
2 )
∑
τ∈M(m)
θ
{
C(q,qθ)
τ (x1, . . . , xm)
m∏
i=1
T
(i−1)θ+τ+i −τ
−
i
q,xi
}
, (4.2)
where for any τ ∈M (m)
θ , we denoted
C(q,qθ)
τ (x1, . . . , xm)
26 C. Cuenca
def
=
m−1∏
s=1
C(q,qθ)
τ1,s+1,...,τs,s+1
{ui = x−1
s+1xiq
−θ+
m∑
j=s+2
(τi,j−τs+1,j)
: 1 ≤ i ≤ s}
. (4.3)
The proof of Theorem 4.1 is given in the next subsection. We derive here some conclusions,
namely two special cases when the operator D(m)
q,θ has a simple form. The first simple case is
m = 2.
Corollary 4.2. In the same setting as Theorem 4.1 (for m = 2), we have
Pλ
(
x1, x2;N, q, qθ
)
=
q−(N−1)θ2+(θ+1
2 ) · [θ(N − 1)− 1]q!
θ(N−1)−1∏
j=1
(
x1 − qj−θ
)(
x2 − qj−θ
)
× D̃(2)
q,θ
2∏
i=1
Pλ
(
xi;N, q, q
θ
) θN−1∏
j=1
(
xi − qj−θ
)
[θN − 1]q!
where
D̃(2)
q,θ
def
=
(
1
x1 − x2
◦ (Dq,x2 −Dq,x1)
)θ
=
(
1
x1 − x2
◦ (Dq,x2 −Dq,x1)
)
◦ · · · ◦
(
1
x1 − x2
◦ (Dq,x2 −Dq,x1)
)
︸ ︷︷ ︸
composition of θ operators
.
Proof. For m = 2, we have M
(2)
θ = {[ 0 n
0 0 ] , n ∈ {0, 1, . . . , θ}}, and thus the operator D(2)
q,θ
becomes
D(2)
q,θ =
1
(q − 1)θ
θ∑
n=0
{
C(q,qθ)
n
(
x−1
2 x1q
−θ)Tnq,x1T θ−nq,x2
}
.
We let
a(θ)
n =
C
(q,qθ)
n
(
x−1
2 x1q
−θ)
θ−1∏
i=0
(x1 − qix2)
, 0 ≤ n ≤ θ.
The statement of the theorem can be easily reduced to prove that the coefficient of Tnq,x1T
θ−n
q,x2 in
the product
Dq,θ =
(
1
x1 − x2
◦ (Tq,x2 − Tq,x1)
)θ
=
θ∑
n=0
b(θ)n Tnq,x1T
θ−n
q,x2
is equal to a
(θ)
n , i.e., we prove a
(θ)
n = b
(θ)
n for all θ, n ∈ N, 0 ≤ n ≤ θ, and we do it by induction
on θ. The case θ = 1 can be easily dealt with, using Lemma B.2. Now assume a
(θ−1)
n = b
(θ−1)
n
for all 0 ≤ n ≤ θ − 1, and some θ ≥ 2. We prove a
(θ)
n = b
(θ)
n for all 0 ≤ n ≤ θ. Evidently,
Dq,θ =
(
1
x1−x2 ◦ (Tq,x2 − Tq,x1)
)
◦Dq,θ−1 implies
b(θ)n =
1
x1 − x2
(
Tq,x2b
(θ−1)
n − Tq,x1b
(θ−1)
n−1
)
, n = 1, 2, . . . , θ − 1,
Asymptotic Formulas for Macdonald Polynomials 27
b
(θ)
0 =
1
x1 − x2
Tq,x2b
(θ−1)
0 , b
(θ)
θ =
1
x2 − x1
Tq,x1b
(θ−1)
θ−1 .
From Lemma B.3 in Appendix B, the terms a
(θ)
n satisfy
a(θ)
n =
1
x1 − x2
(
Tq,x2a
(θ−1)
n − Tq,x1a
(θ−1)
n−1
)
, n = 1, 2, . . . , θ − 1,
a
(θ)
0 =
1
θ−1∏
i=0
(x1 − qix2)
, a
(θ)
θ =
1
θ−1∏
i=0
(x2 − qix1)
.
It is not difficult to conclude from these relations, and the inductive hypothesis a
(θ−1)
m = b
(θ−1)
m
∀m = 0, 1, . . . , θ − 1, that a
(θ)
n = b
(θ)
n for all 0 ≤ n ≤ θ, as desired. �
When θ = 1 (equivalently t = q), the result has a compact form as well. Let us recall that when
t = q, the Macdonald polynomials become the well known Schur polynomials sλ(x1, . . . , xN ) =
Pλ(x1, . . . , xN ; q, q). The Schur (Laurent) polynomials sλ(x1, . . . , xN ), λ ∈ GTN , can also be
defined by the simple determinantal formula
sλ(x1, . . . , xN ) =
det
[
xN+1−j
i
]N
i,j=1∏
1≤i<j≤N
(xi − xj)
.
For any m ∈ N with 1 ≤ m ≤ N , we consider
sλ(x1, . . . , xm;N, q)
def
=
sλ
(
x1, . . . , xm, 1, q, . . . , q
N−1−m)
sλ(1, q, . . . , qN−1)
and call it a q-Schur character of rank N , number of variables m and parametrized by λ; it was
defined before in [21]. We recover the following theorem.
Corollary 4.3 ([24, Theorem 3.5]). Let N ∈ N, λ ∈ GTN . Then
sλ(x1, . . . , xm;N, q) =
q(
m+1
3 )−(N−1)(m2 )
m∏
i=1
[N − i]q!
m∏
i=1
N−m∏
j=1
(
xi − qj−1
) 1
∆(x1, . . . , xm)
×D(m)
q
m∏
i=1
sλ(xi;N, q)
N−1∏
j=1
(
xi − qj−1
)
[N − 1]q!
,
where
D(m)
q
def
= det
[
Dj−1
q,xi
]m
i,j=1
=
∏
1≤i<j≤m
(Dq,xj −Dq,xi).
Proof. Letting θ = 1 in Theorem 4.1, we see that the equation above holds if D(m)
q,1 = D(m)
q . In
fact, thanks to Lemma B.2, we have
D(m)
q,1 = (q − 1)−(m2 )
∑
τ∈M(m)
1
m−1∏
s=1
(−1)τ1,s+1+···+τs,s+1
m∏
i=1
T
i−1+τ+i −τ
−
i
q,xi .
28 C. Cuenca
Evidently D(m)
q =
∏
i<j
(Dq,xj −Dq,xi) = (q − 1)−(m2 ) ∏
i<j
(Tq,xj − Tq,xi), thus we need to show
∏
1≤i<j≤m
(Tq,xj − Tq,xi) =
∑
τ∈M(m)
1
(−1)|τ |
m∏
k=1
T
k−1+τ+k −τ
−
k
q,xk , (4.4)
where we denoted by |τ | to the sum of all entries of τ ∈ M (m)
1 . The operators Tq,x1 , . . . , Tq,xm
pairwise commute. When expanding the left-hand side of (4.4), it is clear that the resulting
terms can be parametrized by matrices in M
(m)
1 : the term corresponding to τ ∈ M (m)
1 is the
product of (−1)kT kq,xiT
1−k
q,xj , where k ∈ {0, 1} ranges over the elements of τ strictly above the
main diagonal. Thus the term corresponding to τ is
∏
1≤i<j≤m
(−1)τi,jT
τi,j
q,xiT
1−τi,j
q,xj = (−1)
∑
1≤i<j≤m
τi,j ∏
1≤i<j≤m
T
τi,j
q,xiT
1−τi,j
q,xj .
We are left to show∏
1≤i<j≤m
T
τi,j
q,xiT
1−τi,j
q,xj =
m∏
k=1
T
k−1+τ+k −τ
−
k
q,xk . (4.5)
Both sides of (4.5) are of the form T p1q,x1 · · ·T
pm
q,xm , for some p1, . . . , pm ∈ Z≥0, so we simply need
to check the equality between exponents pk of Tq,xk in both sides, for an arbitrary 1 ≤ k ≤ m. In
the left side, there are k − 1 factors of the form T
τi,k
q,xiT
1−τi,k
q,xk , i ≤ k − 1, which overall contribute
k − 1−
k−1∑
i=1
τi,k = k − 1− τ−k to the exponent of Tq,xk . Moreover there are m− k factors of the
form T
τk,j
q,xkT
1−τk,j
q,xj , k+ 1 ≤ i, which contribute
m∑
i=k+1
τk,i = τ+
k to the exponent of Tq,xk . Therefore
the power of Tq,xk in the left-hand side of (4.5) is T
k−1+τ+k −τ
−
k
q,xk . Evidently, the power of Tq,xk in
the right-hand side of (4.5) if also T
k−1+τ+k −τ
−
k
q,xk , which finishes the proof. �
Example 4.4. We discuss the first nontrivial example of the multiplicative formula for Macdonald
polynomials (an example that is not dealt with in the Corollaries above): θ = 2, m = 3. The
formula in this case is
Pλ
(
x1, x2, x3;N, q, q2
)
=
q25−12N
([2N − 1]q)2([2N − 2]q)2[2N − 3]q[2N − 4]q
× 1
2N−7∏
j=−1
(x1 − qj)(x2 − qj)(x3 − qj)
1∏
1≤i<j≤3
(xi − xj)(xi − qxj)
× D̂(3)
q,2
3∏
i=1
Pλ
(
xi;N, q, q
2
) 2N−3∏
j=−1
(xi − qj)
,
the q-difference operator is
D̂(3)
q,2 =
1
(q − 1)6
∑
a,b,c∈Z
0≤a,b,c≤2
{
C(q,q2)
a
(
x−1
2 x1q
−2+b−c)
× C(q,q2)
b,c
(
x−1
3 x1q
−2, x−1
3 x2q
−2
)
T a+b
q,x1T
2+c−a
q,x2 T 4−b−c
q,x3
}
.
Asymptotic Formulas for Macdonald Polynomials 29
We can also write the q-difference operator in terms of the q-degree operators {Dq,xi : i = 1, 2, 3}
by using 1
q−1Tq,xi = 1
q−1 +Dq,xi . Then we can replace the operator D̂(3)
q,2 with the sum of operators
D(3,top)
q,2 +A(3)
q,2, where
D(3,top)
q,2 =
∑
a,b,c∈Z
0≤a,b,c≤2
{
C(q,q2)
a
(
x−1
2 x1q
−2+b−c)
× C(q,q2)
b,c
(
x−1
3 x1q
−2, x−1
3 x2q
−2
)
Da+b
q,x1D
2+c−a
q,x2 D4−b−c
q,x3
}
,
A(3)
q,2 =
∑
i1,i2,i3≥0
i1+i2+i3≤5
fi1,i2,i3(x1, x2, x3; q, 2)
(q − 1)6−i1−i2−i3 Di1
q,x1D
i2
q,x2D
i3
q,x3 ,
and fi1,i2,i3(x1, x2, x3; q, 2) are certain rational functions. With the help of Sage, we found
fi1,i2,i3(x1, x2, x3; q, 2) = 0, for all i1 + i2 + i3 ≤ 2.
There are nontrivial rational functions fi1,i2,i3(x1, x2, x3; q, 2) as well, for some i1 + i2 + i3 ≥ 3,
e.g.,
f4,1,0(x1, x2, x3; q, 2) = −(q − 1)
(x2 + x3)(x1 − qx3)(x1 − qx2)
(qx2 − x3)(qx1 − x3)(qx1 − x2)
, (4.6)
f2,1,1(x1, x2, x3; q, 2) = −(q − 1)2 (q + 1)(x2 − x3)
(
x2
1 + x2x3 + 2x1x2 + 2x1x3
)
(qx1 − x2)(qx1 − x3)(qx2 − x3)
. (4.7)
Two important observations are in order. First, from Corollaries 4.2 and 4.3, one might believe that
D(m)
q,θ is, in general, homogeneous of degree θ
(
m
2
)
as a functions of the operators {Dq,xi : i = 1, 2, 3}.
However, this example disproves it. Second, the terms fi1,i2,i3(x1, x2, x3; q, 2) above make us
suspect that fi1,i2,i3(x1, x2, x3; q, 2) is divisible by (q − 1)6−i1−i2−i3 ∀ 0 ≤ i1 + i2 + i3 ≤ 5. We
have checked this fact in the computer. In fact, we believe that the analogous statement for
general m, θ ∈ N holds true, but the author could not prove it.
4.2 Proof of Theorem 4.1
Fix a positive signature λ ∈ GT+
N and let us prove equation (4.1); we extend the result for all
signatures λ ∈ GTN at the end.
Let us consider m positive integers n1 > n2 > · · · > nm > θ(N +m). By the index-argument
symmetry, Theorem 2.2, applied to λ ∈ GT+
N and µ = (n1 ≥ · · · ≥ nm ≥ 0 ≥ · · · ≥ 0) ∈ GT+
N , as
well as the definition of the dual Macdonald polynomials Qµ(·; q, t), we obtain
Pλ
(
qn1tN−1, . . . , qnmtN−m, tN−m−1, . . . , t, 1; q, t
)
Pλ
(
tN−1, tN−2, . . . , 1; q, t
)
=
Q(n1,...,nm)
(
qλ1tN−1, qλ2tN−2, . . . , qλN ; q, t
)
Q(n1,...,nm)
(
tN−1, tN−2, . . . , 1; q, t
) . (4.8)
Apply the Jacobi–Trudi formula for Macdonald polynomials, Theorem 2.7, to the numerator
of (4.8), then multiply and divide the term parametrized by τ by the product
m∏
s=1
gns+τ+s −τ−s
(
tN−1, . . . , t, 1; q, t
)
,
30 C. Cuenca
so (4.8) equals
∑
τ∈M(m)
m−1∏
s=1
C(q,t)
τ1,s+1,...,τs,s+1
({
ui = q
ni−ns+1+
m∑
j=s+2
(τi,j−τs+1,j)
ts−i : 1 ≤ i ≤ s
})
×
m∏
s=1
gns+τ+s −τ−s
(
qλ1tN−1, . . . , qλN ; q, t
)
gns+τ+s −τ−s
(
tN−1, . . . , t, 1; q, t
) ×
m∏
s=1
gns+τ+s −τ−s
(
tN−1, . . . , t, 1; q, t
)
Q(n1,...,nm)
(
tN−1, . . . , t, 1; q, t
)
. (4.9)
(In equation (4.9), we are setting gn(q, t) = 0 if n is a nonpositive integer.) Recall that t = qθ,
θ ∈ N. In view of Lemma B.1, the only terms in the sum (4.9) with nonzero contributions are
those parametrized by m×m matrices whose entries belong to the set {0, 1, . . . , θ}; define M
(m)
θ
to be the set of such matrices. Notice that ns + τ+
s − τ−s > 0 for all τ ∈M (m)
θ , because of our
initial assumption on the values of n1, . . . , nm. By another application of the index-argument
symmetry (and of the identity (2.4) above), we have
gns+τ+s −τ+s
(
qλ1tN−1, . . . , qλN ; q, t
)
gns+τ+s −τ−s (tN−1, . . . , t, 1; q, t)
=
Pλ
(
qns+τ
+
s −τ−s tN−1, tN−2, . . . , t, 1; q, t
)
Pλ(tN−1, . . . , t, 1; q, t)
∀ 1 ≤ s ≤ m. (4.10)
Plugging (4.2) into (4.9), we obtain
Pλ
(
qn1+θ(N−1), . . . , qnm+θ(N−m), tN−m−1, . . . , t, 1; q, t
)
Pλ
(
tN−1, tN−2, . . . , 1; q, t
) (4.11)
=
∑
τ∈M(m)
θ
m−1∏
s=1
C(q,qθ)
τ1,s+1,...,τs,s+1
({
ui = q
ni−ns+1+
m∑
j=s+2
(τi,j−τs+1,j)+θ(s−i)
: 1 ≤ i ≤ s
})
×
m∏
s=1
gns+τ+s −τ−s
(
tN−1, . . . , t, 1; q, t
)
Q(n1,...,nm)
(
tN−1, . . . , t, 1; q, t
) × m∏
s=1
Pλ
(
qns+τ
+
s −τ−s +θ(N−1), tN−2, . . . , t, 1; q, t
)
Pλ
(
tN−1, . . . , t, 1; q, t
)
.
Let us make the change of variables
zs
def
= qns+θ(N−s), 1 ≤ s ≤ m, (4.12)
and rewrite some terms from (4.11) in these new variables. Clearly the left-hand side of
equality (4.11) is the Macdonald character Pλ(z1, . . . , zm;N, q, qθ). It is also evident that the
variable ui in the term C
(q,qθ)
τ1,s+1,...,τs,s+1 can be rewritten as ui = z−1
s+1ziq
−θ+
m∑
j=s+2
(τi,j−τs+1,j)
, for
1 ≤ i ≤ s. Additionally we have qns+τ
+
s −τ−s +θ(N−1) = T θs−θ+τ
+
s −τ−s
q,zs (zs), for all s, which implies
that the last product in (4.11) can be written as
m∏
s=1
T θ(s−1)+τ+s −τ−s
q,xs
(
m∏
s=1
Pλ
(
zs;N, q, q
θ
))
.
Next, we need to rewrite
m∏
s=1
gns+τ+s −τ−s
(
tN−1, . . . , t, 1; q, t
)
Q(n1,...,nm)
(
tN−1, . . . , t, 1; q, t
)
Asymptotic Formulas for Macdonald Polynomials 31
in terms of z1, . . . , zm. From Corollary 2.4 of the evaluation identity for Macdonald polynomials,
we obtain
gp
(
tN−1, tN−2, . . . , t, 1; q, t
)
=
∏
s=(1,j)∈(p)
1− qa′(s)+θ(N−l′(s))
1− qa(s)+1+θl(s)
=
p∏
i=1
1− qi−1+θN
1− qp−i+1
=
θN−1∏
j=1
1− qp+j
1− qj
=
1
(1− q)θN−1
θN−1∏
j=1
(1− qp+j)
[θN − 1]q!
(4.13)
for any p ∈ N, p > θN . By similar, but more complicated, computations we find
Q(p1,...,pm)
(
tN−1, . . . , t, 1; q, t
)
=
1
(1− q)m(θN−1)−θm(m−1)/2
×
θ∏
s=1
∏
1≤i<j≤m
qpj − qpi+s+θ(j−i−1)
1− qpi+s+θ(j−i−1)
×
θ(N−m+1)−1∏
j=1
(
1− qpm+j
)
·
θ(N−m+2)−1∏
j=1
(
1− qpm−1+j
)
· · ·
θN−1∏
j=1
(
1− qp1+j
)
[θ(N −m+ 1)− 1]q![θ(N −m+ 2)− 1]q! · · · [θN − 1]q!
, (4.14)
for any partition (p1 > p2 > · · · > pm > 0) with pi > θ(N − i+ 1) for all 1 ≤ i ≤ m. From (4.13)
and (4.14), we obtain
m∏
s=1
gns+τ+s −τ−s
(
tN−1, . . . , t, 1
)
Q(n1,...,nm)
(
tN−1, . . . , t, 1; q, t
) =
1
(1− q)θm(m−1)/2
m∏
i=1
[θ(N − i+ 1)− 1]q!
[θN − 1]q!
(4.15)
×
θN−1∏
j=1
(
1− qn1+τ+1 −τ
−
1 +j
)
θN−1∏
j=1
(
1− qn1+j
) ·
θN−1∏
j=1
(
1− qn2+τ+2 −τ
−
2 +j
)
θ(N−1)−1∏
j=1
(
1− qn2+j
) · · ·
θN−1∏
j=1
(
1− qnm+τ+m−τ−m+j
)
θ(N−m+1)−1∏
j=1
(
1− qnm+j
) (4.16)
×
θ∏
s=1
∏
1≤i<j≤m
1− qni+s+θ(j−i−1)
qnj − qni+s+θ(j−i−1)
. (4.17)
Observe that we used our assumption n1 > n2 > · · · > nm > θ(N +m) to guarantee that (4.13)
and (4.14) are applicable. We have to rewrite both (4.16) and (4.17) in terms of zs. Let us begin
with (4.16), which is a product of m terms; for 1 ≤ r ≤ m, the rth term is
θN−1∏
j=1
(
1− qnr+τ
+
r −τ−r +j
)
θ(N−r+1)−1∏
j=1
(
1− qnr+j
) =
θN−1∏
j=1
(
1− q−θ(N−r)zrqτ
+
r −τ−r +j
)
θ(N−r+1)−1∏
j=1
(
1− q−θ(N−r)zrqj
)
= (−1)θ(r−1)q
−
θN−1∑
j=θ(N−r+1)
(θ(N−r)−j)
·
θN−1∏
j=1
(
zrq
τ+r −τ+r − qθ(N−r)−j
)
θ(N−r+1)−1∏
j=1
(
zr − qθ(N−r)−j
) (4.18)
= (−1)θ(r−1)q
−
θN−1∑
j=θ(N−r+1)
(θ(N−r)−j)
q−(θr−θ)(θN−1)
θN−1∏
j=1
(
zrq
θr−θ+τ+r −τ−r − qθ(N−1)−j)
θ(N−r+1)−1∏
j=1
(
zr − qθ(N−r)−j
) .
32 C. Cuenca
Under the change of indexing j 7→ θ(N − r + 1)− j, the product in the denominator of (4.18)
becomes
θ(N−r+1)−1∏
j=1
(
zr − qθ(N−r)−j
)
=
θ(N−r+1)−1∏
j=1
(
zr − qj−θ
)
,
whereas the numerator of (4.18) can be expressed as
θN−1∏
j=1
(
zrq
θr−θ+τ+r −τ−r − qθ(N−1)−j) = T θr−θ+τ
+
r −τ−r
q,zr
θN−1∏
j=1
(
zr − qθ(N−1)−j).
Define also
c(N,m, θ) = −
m∑
r=1
θN−1∑
j=θ(N−r+1)
(θ(N − r)− j)−
m∑
r=1
(θr − θ)(θN − 1), (4.19)
which is the power of q coming from the terms (4.18), for 1 ≤ r ≤ m. Thus (4.16) equals
(−1)θm(m−1)/2qc(N,m,θ)
m∏
r=1
θ(N−r+1)−1∏
j=1
(
zr − qj−θ
) ·
m∏
r=1
T θr−θ+τ
+
r −τ−r
q,zr
θN−1∏
j=1
(
zr − qθ(N−1)−j) . (4.20)
On the other hand, (4.17) can be expressed in terms of the variables zi as follows:
θ∏
r=1
∏
i<j
1− qni+r+θ(j−i−1)
qnj − qni+r+θ(j−i−1)
=
θ∏
r=1
∏
i<j
qni+r+θ(j−i−1) − 1
qni+r+θ(j−i−1) − qnj
=
θ∏
r=1
∏
i<j
qni+r+θ(N−i−1) − qθ(N−j)
qni+r+θ(N−i−1) − qnj+θ(N−j)
=
θ∏
r=1
∏
i<j
ziq
r−θ − qθ(N−j)
ziqr−θ − zj
=
θ∏
r=1
∏
i<j
zi − qθ(N−j+1)−r
zi − zjqθ−r
. (4.21)
The denominator of (4.21) is
∏
1≤i<j≤m
0≤k<θ
(zi − qkzj), and the numerator is
m∏
r=1
θ(N−i+1)−1∏
j=θ(N−m+1)
(zr − qj−θ).
Then (4.17) equals
m∏
r=1
θ(N−r+1)−1∏
j=θ(N−m+1)
(
zr − qj−θ
)
∏
1≤i<j≤m
0≤k<θ
(zi − qkzj)
. (4.22)
Therefore after the change of variables (4.12), and using (4.15), (4.16), (4.17), (4.20), (4.22)
and the identities after (4.12), equation (4.11) yields the desired (4.1) for any xs = qns+θ(N−s),
1 ≤ s ≤ m, such that n1 > · · · > nm > θ(N +m), provided the equality
c(N,m, θ) = θ2
(
m+ 1
3
)
−
{
Nθ2 −
(
θ + 1
2
)}(
m
2
)
holds, where c(N,m, θ) is defined in (4.19). This is an easy exercise.
Asymptotic Formulas for Macdonald Polynomials 33
We have just proved that the statement of Theorem 4.1 holds for all xs = qns+θ(N−s), for all
n1 > n2 > · · · > nm > θ(N +m). Since both sides of the equality (4.1) are evidently rational
functions in x1, . . . , xm, an easy algebro-geometric argument shows the equality holds for all
x1, . . . , xm ∈ C, as desired.
We still have to extend the theorem to all signatures. Let us prove (4.1) for an arbitrary
λ ∈ GTN . If λ ∈ GT+
N , then we are done. Otherwise, choose any p ∈ N large enough so that
λ̃
def
= (λ1 + p, λ2 + p, . . . , λN + p) ∈ GT+
N . By homogeneity of Macdonald polynomials, we have
P
λ̃
(
x1, . . . , xm;N, q, qθ
)
= Pλ
(
x1, . . . , xm;N, q, qθ
)
(x1 · · ·xm)pq−θpNmqθpm(m+1)/2
and
P
λ̃
(
xi;N, q, q
θ
)
= Pλ
(
xi;N, q, q
θ
)
xpi q
−θpNqθp ∀ 1 ≤ i ≤ m
⇒ T
(i−1)θ+τ+i −τ
−
i
q,xi P
λ̃
(
xi;N, q, q
θ
)
= q−θpNqp(iθ+τ
+
i −τ
−
i )xpi
× T (i−1)θ+τ+i −τ
−
i
q,xi Pλ
(
xi;N, q, q
θ
)
∀ 1 ≤ i ≤ m.
We know that (4.1) holds for λ̃; from the expressions above, it holds also for λ provided that
(the powers of q match)
−θpNm+
θpm(m+ 1)
2
=
m∑
i=1
(−θpN + p(iθ + τ+
i − τ
−
i )).
The latter equation is easy to check, and the proof of Theorem 4.1 is therefore finished.
5 Asymptotics of Macdonald characters
In the remaining of the paper, starting here, we denote the Macdonald polynomial Pλ(x1, . . . , xN ;
q, t) simply by Pλ(x1, . . . , xN ).
In this section, assume that θ ∈ N and let t = qθ. We study the asymptotics of (certain
normalization of) Macdonald characters of a fixed number m of variables, as the rank N tends
to infinity and the signatures λ(N) stabilize in certain way that we define next.
Definition 5.1. Let
N def
= {ν = (ν1, ν2, . . . ) ∈ Z∞ : ν1 ≤ ν2 ≤ . . . }
be the set of weakly increasing integer sequences. We say that the sequence {λ(N)}N≥1 of
signatures, λ(N) ∈ GTN , stabilizes to ν ∈ N if we have the following limits
lim
N→∞
λN−i+1(N) = νi, ∀ i = 1, 2, . . . .
In other words {λ(N)}N≥1 stabilizes to ν if there exists a sequence of integers 0 < N1 < N2 < · · ·
such that λN−i+1(N) = νi, ∀N > Ni, i = 1, 2, . . . .
5.1 Asymptotics of Macdonald characters of one variable
Theorem 5.2. Let t = qθ, θ ∈ N, and {λ(N)}N≥1, λ(N) ∈ GTN , be a sequence of signatures
that stabilizes to ν ∈ N . Then
lim
N→∞
Pλ(N)
(
x, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) = Φν(x; q, t), (5.1)
34 C. Cuenca
where
Φν(x; q, t)
def
=
(q; q)∞
(xq; q)∞
ln q
2π
√
−1
∫
C+
xz
∞∏
i=1
(
q−z+νiti; q
)
∞(
q−z+νiti−1; q
)
∞
dz, (5.2)
and the contour C+ is the infinite positive contour consisting of
[
−M+ π
√
−1
ln q ,−M− π
√
−1
ln q
]
and the
lines
[
−M + π
√
−1
ln q ,+∞+ π
√
−1
ln q
)
,
[
−M − π
√
−1
ln q ,+∞− π
√
−1
ln q
)
for an arbitrary M > max{0,−ν1},
see Fig. 2. The function Φν(x; q, t) is defined by the formula above in the domain
U def
=
⋂
k≥1
{
x 6= q−k
}
∩ {x 6= 0},
and admits an analytic continuation to the domain {x 6= 0} = C \ {0}. Moreover if ν1 ≥ 0, the
function Φν(x; q, t) can be analytically continued to C.
The convergence (5.1) is uniform on compact subsets of C \ {0}, and if ν1 ≥ 0, then it is
uniform on compact subsets of C.
Remark 5.3. The theorem is a direct application of the integral representation in Theorem 3.1.
If we use instead Theorems 3.2 and 3.3, we could possibly extend the theorem for general
q, t ∈ (0, 1). Since our main application does require θ ∈ N and t = qθ, we do not bother to
pursue the more general case q, t ∈ (0, 1).
Proof. Let us first prove that the limit (5.1) holds uniformly for x in compact subsets of U .
We use the integral representation for Macdonald characters of one variable, Theorem 3.1, for
the signature λ(N) ∈ GTN , and with x replaced by xtN , x ∈ U . Since Pλ(N) is a homogeneous
polynomial of degree |λ(N)|, then the left-hand side is
t|λ(N)|Pλ(N)
(
x, t−1, t−2, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) .
After simple algebraic manipulations, the right-hand side becomes
q−(θN2 ) (q; q)θN−1
(xq; q)θN−1
ln q
2π
√
−1
∮
C0
xz
N∏
i=1
θ−1∏
j=0
(
q−(λi(N)+θ(N−i)+j) − q−z
)dz.
Therefore we conclude (make change of variables i 7→ N − i+ 1 in the inner product)
Pλ(N)
(
x, t−1, t−2, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
)
=
(q; q)θN−1
(xq; q)θN−1
ln q
2π
√
−1
∮
C0
xz
N∏
i=1
θ−1∏
j=0
(
1− q−zqλN−i+1(N)+θ(i−1)+j
)dz, (5.3)
where C0 is a finite contour encloses all real poles of the integrand and no other poles. We have
the limit
lim
N→∞
(q; q)θN−1
(xq; q)θN−1
=
(q; q)∞
(xq; q)∞
,
uniformly for x belonging to compact subsets of U .
Next modify the contour C0 into an infinite contour C+ as described in the statement of the
theorem. This is possible to do because all the poles of the integrands of (5.3) belong to the
Asymptotic Formulas for Macdonald Polynomials 35
interior of C+, as N grows. Moreover the resulting integral is well-posed for large enough N since
the integrand is of order
(
|x|qθN
)<z
and for any compact set K ⊂ C, there exists N0 ∈ N such
that sup
N>N0
sup
x∈K
{
|x|qθN
}
< 1.
We now look at the asymptotics of the integral in (5.3), with C0 replaced by C+. The
denominator in the integrand has the following limit
lim
N→∞
N∏
i=1
θ−1∏
j=0
(
1− q−z+λN−i+1(N)+θ(i−1)+j
)
=
∞∏
i=1
θ−1∏
j=0
(
1− q−z+νi+θ(i−1)+j
)
=
∞∏
i=1
(
q−z+νiti−1; q
)
∞(
q−z+νiti; q
)
∞
.
The limit above can be justified properly by using the dominated convergence theorem and the
estimate
N∑
i=1
θ−1∑
j=0
∣∣q−z+λN−i+1(N)+θ(i−1)+j
∣∣
≤
N∑
i=1
θ−1∑
j=0
q−<z+λN (N)+θ(i−1)+j ≤ c
N∑
i=1
θ−1∑
j=0
qθ(i−1)+j <
c
1− q
,
valid for some constant c > 0 such that q−<z+λN (N) < c for all N ≥ 1 (it exists because
lim
N→∞
λN (N) = ν1).
To prove the limit in the statement of the theorem, we are left to show
lim
N→∞
∫
C+
xz
N∏
i=1
θ−1∏
j=0
(
1− q−zqλN−i+1(N)+θ(i−1)+j
)dz =
∫
C+
xz
∞∏
i=1
(
q−z+νiti; q
)
∞(
q−z+νiti−1; q
)
∞
dz.
We already proved the pointwise convergence; to make use of the dominated convergence theorem,
we simply need estimates on the contribution of the tails of C+ that are uniform in N and uniform
for x belonging to compact subsets of C \ {0}. Parametrize the tails of C+ as z = r + π
√
−1
ln q or
z = r− π
√
−1
ln q ; for r ranging from some large R > 1 to +∞, we want to show that the contribution
of each of these lines is small. We have∣∣∣∣∣∣∣∣∣
xz
N∏
i=1
θ−1∏
j=0
(
1− q−zqλN−i+1(N)+θ(i−1)+j
)
∣∣∣∣∣∣∣∣∣ =
∣∣xr±π√−1
ln q
∣∣
N∏
i=1
θ−1∏
j=0
(
1 + q−rqλN−i+1(N)+θ(i−1)+j
)
≤ C1 ×
|x|r
k∏
i=1
θ−1∏
j=0
(
1 + q−rqλN−i+1(N)+θ(i−1)+j
)
≤ C2 ×
|x|r
k∏
i=1
θ−1∏
j=0
(1 + q−r)
≤ C2 ×
|x|r
(q−r)θk
= C2 ×
(
|x|tk
)r
,
for all 1 ≤ k ≤ N , where the constant in the second line is uniform for x over compact subsets
of C \ {0}, and the constant in the third line depends on k, but not on N (note that to go
36 C. Cuenca
from the second to the third line, we needed to use that lim
N→∞
λN−i+1(N) exists for i = 1, . . . , k
and so the sequence {λN−i+1(N)}N≥1 is uniformly bounded for i = 1, . . . , k). Now for any
compact set K ⊂ C \ {0} and any M > 0, there exists k ∈ N such that sup
x∈K
|x| · tk ≤ e−M .
Since
∫∞
R e−Mrdr = e−MR/M
R→∞−−−−→ 0, we have shown that the contribution of the tails of C+ is
uniformly small over x in compact subsets of C \ {0} and for large enough N .
We have proved so far that the limit in the theorem holds uniformly for x in compact subsets
of U . Since the set {q−1, q−2, . . . } is discrete and has no accumulation points, Cauchy’s integral
formula allows us to deduce the uniform convergence in compact subsets of C \ {0} as soon as we
show that Φν(x; q, t) admits an analytic continuation to C \ {0}.
By virtue of Riemann’s theorem of removable singularities, it will suffice to show that Φν(x; q, t)
is uniformly bounded in an open neighborhood of each pole q−k. Let R > 0, R /∈ {qn : n ∈ Z},
be arbitrary. From what we have shown so far, it is clear that
sup
1
R
≤|x|≤R
x∈U
∣∣Φν(x; q, t)
∣∣ ≤ sup
N≥1
sup
1
R
≤|x|≤R
x∈U
∣∣∣∣∣Pλ(N)
(
x, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) ∣∣∣∣∣
≤ sup
N≥1
sup
1
R
≤|x|≤R
∣∣∣∣∣Pλ(N)
(
x, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) ∣∣∣∣∣.
Thanks to the branching rule for Macdonald polynomials, Theorem 2.5, the fact that all the
branching coefficients ψµ/ν(q, t) are nonnegative when q, t ∈ (0, 1), and the fact that each
Pλ(N)
(
x, t−1, . . . , t1−N
)
is a Laurent polynomial in x, we have
sup
1
R
≤|x|≤R
∣∣∣∣∣Pλ(N)
(
x, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) ∣∣∣∣∣ ≤ sup
1
R
≤|x|≤R
Pλ(N)
(
|x|, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
)
≤
Pλ(N)
(
R, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) +
Pλ(N)
(
R−1, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) .
From the pointwise limits
lim
N→∞
Pλ(N)
(
R±1, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) = Φν(R±1; q, t),
we deduce that the sequences
{
Pλ(N)
(
R±1, t−1, . . . , t1−N
)
/Pλ(N)
(
1, t−1, . . . , t1−N
)}
N≥1
are uni-
formly bounded. As a result of the estimates above,
sup
1
R
≤|x|≤R
x∈U
|Φν(x; q, t)| <∞.
Thus Φν(x; q, t) admits an analytic continuation to all the poles in
{
z ∈ C : 1
R ≤ |z| ≤ R
}
∩{
q−1, q−2, . . .
}
. Since R > 0 was arbitrary (outside of a lattice), we conclude that Φν(x; q, t)
admits an analytic continuation to C \ {0}.
If ν1 ≥ 0, then lim
N→∞
λN (N) = ν1 shows that λN (N) ≥ 0 for all N > N0 and N0 ∈ N
large enough. For all N > N0, the functions Pλ(N)
(
x, t−1, . . . , t1−N
)
/Pλ(N)
(
1, t−1, . . . , t1−N
)
are
polynomials in x and therefore holomorphic on C. Similar considerations as above allow us
to analytically continue Φν(x; q, t) to C in this case, and also show that the limit (5.1) holds
uniformly for x belonging to compact subsets of C. �
We end this subsection with a Lemma that will be used in Section 7.
Asymptotic Formulas for Macdonald Polynomials 37
Lemma 5.4. If ν, ν̃ ∈ N are such that
Φν(x; q, t) = Φν̃(x; q, t) ∀x ∈ T, (5.4)
then ν = ν̃.
Proof. In the integral representation of Φν(x; q, t), the set of poles of the integrand is
∞⋃
r=1
θ−1⋃
s=0
{νr + θ(r − 1) + s}
and they are all enclosed by contour C+. We can therefore expand
ln q
2π
√
−1
∫
C+
xz
∞∏
i=1
(q−z+νiti; q)∞
(q−z+νiti−1; q)∞
dz,
at least formally, as the sum of residues
∞∑
r=1
θ−1∑
s=0
xνr+θ(r−1)+s
∏
i≥1
i 6=r
(
q−νr+νi−sti−r+1; q
)
∞(
q−νr+νi−sti−r; q
)
∞
×
∏
θ>j≥0
j 6=s
1
(1− qj−s)
. (5.5)
The series above converges absolutely for all x ∈ C \ {0}, and uniformly for x in compact subsets
of C \ {0}; in fact, we can argue as in the special case θ = 1 of [24, Section 6.2]. We have the
bounds∏
θ>j≥0
j 6=s
∣∣∣∣ 1
(1− qj−s)
∣∣∣∣ ≤ 1
(1− q) · · ·
(
1− qθ
)(
q−1 − 1
)
· · ·
(
q−θ − 1
) ;
∏
i>r
∣∣∣∣∣
(
q−νr+νi−sti−r+1; q
)
∞(
q−νr+νi−sti−r; q
)
∞
∣∣∣∣∣ =
∏
i>r
θ−1∏
j=0
∣∣∣∣∣ 1(
1− q−νr+νi−s+θ(i−r)+j
)∣∣∣∣∣
=
∏
i>r
θ−1∏
j=0
1(
1− q−νr+νi−s+θ(i−r)+j
) ≤
∏
i>r
θ−1∏
j=0
1(
1− q−(θ−1)+θ(i−r)+j
) =
1
(q; q)∞
;
∏
i<r
∣∣∣∣∣
(
q−νr+νi−sti−r+1; q
)
∞(
q−νr+νi−sti−r; q
)
∞
∣∣∣∣∣ =
∏
i<r
θ−1∏
j=0
qνr+θ(r−1)+s−νi−θ(i−1)−j
1− qνr−νi+θ(r−i)−j+s
≤ 1
(q; q)∞
∏
i<r
θ−1∏
j=0
qνr+θ(r−1)+s−νi−θ(i−1)−j
≤ 1
(q; q)∞
m∏
i=1
θ−1∏
j=0
qνr+θ(r−1)+s−νi−θ(i−1)−j , for any 1 ≤ m < r,
= qmθ(νr+θ(r−1)+s) × q−θ(ν1+···+νm)q−(θm2 )
(q; q)∞
.
Choose an arbitrary m ∈ N and fix it. For r > m, the general term in brackets at (5.5) has
modulus upper bounded by(
|x|qmθ
)νr+θ(r−1)+s × c(m, θ; q),
38 C. Cuenca
where c(m, θ; q) > 0 is a constant depending on m, θ, as well as ν1, . . . , νm, but not on r. It
follows that the sum (5.5) converges absolutely for any x ∈ C \ {0} with |x| < q−mθ. Since
m ∈ N was arbitrary, it follows that (5.5) converges absolutely for any x ∈ C \ {0}. The
uniform convergence also follows from the bound above. In particular, (5.5) is the Fourier
expansion of (xq;q)∞
(q;q)∞
Φν(x; q, t). A similar Fourier expansion can be given for (xq;q)∞
(q;q)∞
Φν̃(x; q, t).
After multiplying equality (5.4) by (xq; q)∞/(q; q)∞, we have
ln q
2π
√
−1
∫
C+
xz
∞∏
i=1
(
q−z+νiti; q
)
∞(
q−z+νiti−1; q
)
∞
dz =
ln q
2π
√
−1
∫
C+
xz
∞∏
i=1
(
q−z+ν̃iti; q
)
∞(
q−z+ν̃iti−1; q
)
∞
dz. (5.6)
We can expand both sides of (5.6) as above, to get an equality of the form
∑
k∈Z
ck(ν)xk =∑
k∈Z
ck(ν̃)xk. More precisely, (5.5) gives that the set {k ∈ Z : ck(ν) 6= 0} of indices which appear in
the expansion of the left-hand side of (5.6) is {νr + θ(r− 1) + s : 1 ≤ r, 0 ≤ s ≤ θ− 1}. Similarly,
{k ∈ Z : ck(ν̃) 6= 0} = {ν̃r + θ(r − 1) + s : 1 ≤ r, 0 ≤ s ≤ θ − 1}. The equality of these sets, and
the inequalities ν1 ≤ ν2 ≤ · · · , ν̃1 ≤ ν̃2 ≤ · · · , imply that νr = ν̃r for all r ≥ 1, i.e., ν = ν̃. �
5.2 Asymptotics of Macdonald characters of a fixed number m of variables
The following theorem is the main result for asymptotics of Macdonald characters of any given
rank m ∈ N as N tends to infinity.
Theorem 5.5. Let θ ∈ N, t = qθ, and {λ(N)}N≥1, λ(N) ∈ GTN , be a sequence of signatures
that stabilizes to ν ∈ N . Also let m ∈ N be arbitrary. Then
lim
N→∞
Pλ(N)
(
x1, . . . , xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) = Φν(x1, . . . , xm; q, t), (5.7)
where
Φν(x1, . . . , xm; q, t)
def
=
q−2θ2(m3 )−(θ+1
2 )(m2 )
m∏
i=1
(
xiqtm−1; q
)
∞
1∏
1≤i<j≤m
0≤k<θ
(qkxj − xi)
× D̃(m)
q,θ
{
m∏
i=1
Φν(xi; q, t)(xiq; q)∞
}
,
D̃
(m)
q,θ
def
=
∑
τ∈M(m)
θ
C(q,qθ)
τ (x1, . . . , xm)
m∏
i=1
T
(i−1)θ+τ+i −τ
−
i
q,xi . (5.8)
Above we used the notation of Section 4: M
(m)
θ is the set of all strictly upper triangular matrices
with entries in {0, 1, . . . , θ}, and for τ ∈ M (m)
θ , we let τ+
i
def
=
∑
j>i
τi,j, τ
−
i
def
=
∑
k<i
τk,i. Also for
τ ∈ M
(m)
θ , let C
(q,qθ)
τ (x1, . . . , xm) be the rational functions defined in (4.3). The function
Φν(x1, . . . , xm; q, t) is defined by the formula above in the domain
Um
def
=
⋂
k∈Z
⋂
i<j
{
xi 6= qkxj
}
∩
⋂
k≥1
m⋂
i=1
{
xi 6= q−kt1−m
}
∩
m⋂
i=1
{xi 6= 0},
and admits an analytic continuation to the domain
m⋂
i=1
{xi 6= 0} = (C\{0})m. Moreover if ν1 ≥ 0,
the function Φν(x1, . . . , xm; q, t) can be analytically continued to Cm.
Asymptotic Formulas for Macdonald Polynomials 39
The convergence (5.7) is uniform on compact subsets of (C \ {0})m and if ν1 ≥ 0, then it is
uniform on compact subsets of Cm.
Remark 5.6. For θ = 1, our theorem has a different form than [24, Theorem 6.5]. It is not
immediately clear that the two answers are the same.
Remark 5.7. From their definition, it is clear that the rational functions
{
C
(q,qθ)
τ (x1, . . . , xm) :
τ ∈M (m)
θ
}
are holomorphic in a domain of the form
⋂
−N1≤k≤N1
⋂
i<j
{xi 6= qkxj}, for large enough
N1 ∈ N. In particular, all functions
{
C
(q,qθ)
τ (x1, . . . , xm) : τ ∈M (m)
θ
}
are holomorphic on Um.
Proof. This result is a consequence of Theorem 5.2 and the multiplicative formula for Macdonald
polynomials, Theorem 4.1. Let us give more details. As before we prove first the uniform limit
on compact subsets of Um. Begin by applying Theorem 4.1 for the signature λ(N) ∈ GTN and
tN−1xi instead of xi, for i = 1, . . . ,m. Since Pλ(N) is a homogeneous Laurent polynomial of
degree |λ(N)|, the resulting left-hand side is
Pλ(N)
(
x1, . . . , xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) . (5.9)
As for the right side, the factor
∏
1≤i<j≤m
0≤k<θ
(
tN−1xi − tN−1xjq
k
)−1 × (q − 1)−θ(
m
2 ) ×
m∏
i=1
[θ(N − i+ 1)− 1]q!
[θN − 1]q!
equals
(−1)θ(
m
2 ) q−θ
2(N−1)(m2 )∏
1≤i<j≤m
0≤k<θ
(
xi − qkxj
) m∏
i=1
1(
tN−i+1; q
)
θ(i−1)
. (5.10)
We can also obtain easily the polynomial equality
k∏
j=1
(
tN−1x− qj−θ
)
= (−1)kq(
k+1
2 )−θk
k∏
i=1
(
1− xqθN−i
)
, for any k ∈ N.
Therefore for any k ∈ N, we have
m∏
i=1
k−1∏
j=1
(
xit
N−1 − qj−θ
)
= (−1)m(k−1)qm((k2)−θ(k−1))
m∏
i=1
(
xiq
θN−k+1; q
)
k−1
. (5.11)
Observe also that the rational functions are invariant under the simultaneous transformations
xi 7→ tN−1xi ∀ i = 1, 2, . . . ,m, that is
C(q,qθ)
τ
(
tN−1x1, . . . , t
N−1xm
)
= C(q,qθ)
τ (x1, . . . , xm). (5.12)
By combining (5.9), (5.10), (5.11) for k = θ(N −m+ 1), θN , and (5.12), we obtain
Pλ(N)
(
x1, . . . , xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) = (−1)θ(
m
2 )q−2θ2(m3 )−(θ+1
2 )(m2 ) × 1
m∏
i=1
(
tN−i+1; q
)
θ(i−1)
40 C. Cuenca
× 1∏
1≤i<j≤m
0≤k<θ
(
xi − qkxj
) 1
m∏
i=1
(
xitm−1q; q
)
θ(N−m+1)−1
× D̃(m)
q,θ
{
m∏
i=1
Pλ(N)
(
xi, t
−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) (xiq; q)θN−1
}
.
The following limits hold uniformly for (x1, . . . , xm) belonging to compact subsets of Um
lim
N→∞
m∏
i=1
(
tN−1+1; q
)
θ(i−1)
= 1, lim
N→∞
1
m∏
i=1
(
xitm−1q; q
)
θ(N−m+1)−1
=
1
m∏
i=1
(
xitm−1q; q
)
∞
.
We have moreover the following limit holds uniformly for (x1, . . . , xm) belonging to compact
subsets of (C \ {0})m, because of Theorem 5.2,
lim
N→∞
m∏
i=1
Pλ(N)
(
xi, t
−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) (xiq; q)θN−1 =
m∏
i=1
Φν(xi; q, t)(xiq; q)∞.
It is not difficult to observe that if U ⊂ Cm is a domain preserved by the map of multiplication
by q, and {fn}n≥1, f are sequences of holomorphic functions on U for which lim
n→∞
fn(x) = f(x)
uniformly on compact subsets of U , then
lim
n→∞
T sq,xfn(x) = lim
n→∞
fn(qsx) = f(qsx) = T sq,xf(x)
uniformly for x belonging to compact subsets of U . As an implication, the order of the limit as
N →∞ and the q-difference operator D̃
(m)
q,θ can be interchanged. All the considerations above
immediately imply the desired uniform limit for (x1, . . . , xm) belonging to compact subsets of Um.
As in the proof of Theorem 5.2, the limit in the statement will hold also uniformly for compact
subsets of (C \ {0})m if we show that Φν(x1, . . . , xm; q, t) admits an analytic continuation to this
larger domain. The extension of Riemann’s theorem for removable singularities for several complex
variables, [45, Theorem 8], shows that Φν(x1, . . . , xm; q, t) admits an analytic continuation to
all ({z ∈ C \ {0} : |z| ≤ R})m if we showed only that Φν(x1, . . . , xm; q, t) is bounded on ({z ∈
C \ {0} : |z| ≤ R})m ∩ Um. The latter can be proved by repeating the argument in the proof of
Theorem 5.2 above.
Finally, if ν1 ≥ 0 then lim
N→∞
λN (N) = ν1 implies λ(N)N ≥ 0 for large enough N . It follows
that Pλ(N)
(
x1, . . . , xm, t
−m, . . . , t1−N
)
/Pλ(N)
(
1, t−1, . . . , t1−N
)
is a polynomial in x1, . . . , xm for
large enough N , and therefore holomorphic on Cm. The same argument as in the proof of
Theorem 5.2 again shows Φν(x1, . . . , xm; q, t) admits an analytic continuation to Cm and the
limit holds uniformly for compact subsets of Cm. �
6 Preliminaries on the (q, t)-Gelfand–Tsetlin graph
In this section, assume q, t ∈ (0, 1). We use the notation P − lim
k→∞
Mk = M to indicate that
a sequence of probability measures {Mk}k≥1 converges weakly to M .
6.1 The (q, t)-Gelfand–Tsetlin graph
The (q, t)-Gelfand–Tsetlin graph is an undirected, Z≥0-graded graph with countable vertices,
together with a sequence of cotransition probabilities between the levels of the graph (considered
as discrete spaces).
Asymptotic Formulas for Macdonald Polynomials 41
Begin by defining the set of vertices of the graph as the set of all signatures
GT def
=
⊔
N≥0
GTN ,
where, for convenience, we have also included the singleton GT0
def
= {∅}.
The edges are determined by the interlacing constraints: edges only join vertices associated to
signatures whose lengths differ by 1 and µ ∈ GTN is joined to λ ∈ GTN+1 if and only if µ ≺ λ,
i.e., λN+1 ≤ µN ≤ · · · ≤ λ2 ≤ µ1 ≤ λ1. We also assume ∅ ≺ (k), ∀ (k) ∈ GT1. We call the
graph with vertices and edges just described the Gelfand–Tsetlin graph2. Next we introduce
a (q, t)-deformation of the GT graph by considering certain cotransition probabilities.
Definition 6.1. Consider the numbers ΛN+1
N (λ, µ), µ ≺ λ, defined by the expression
ΛN+1
N (λ, µ) = t|µ|ψλ/µ(q, t)
Pµ
(
1, t, . . . , tN−1
)
Pλ
(
1, t, . . . , tN
) = ψλ/µ(q, t)
Pµ
(
t, t2, . . . , tN
)
Pλ(1, t, . . . , tN
) , (6.1)
for all N ∈ Z≥0, λ ∈ GTN+1, µ ∈ GTN , µ ≺ λ, and where the branching coefficients ψλ/µ(q, t)
are defined in Theorem 2.5. For convenience, also let ΛN+1
N (λ, µ) = 0 if µ 6≺ λ, i.e., if µ is not
adjacent to λ.
The (q, t)-Gelfand–Tsetlin graph3 is the sequence
{
GTN ,ΛN+1
N : N = 0, 1, 2, . . .
}
of data
consisting of the GT graph and the GTN+1×GTN matrices
[
ΛN+1
N (λ, µ)
]
, N ≥ 0, defined above.
In general, the numbers ΛN+1
N (λ, µ) depend on the values q, t, but for simplicity we omit
that dependence from the notation. By virtue of the evaluation identity, Theorem 2.3, and the
assumption q, t ∈ (0, 1), we have
ΛN+1
N (λ, µ) ≥ 0, ∀λ ∈ GTN+1, µ ∈ GTN . (6.2)
Moreover, the branching rule for Macdonald polynomials, Theorem 2.5, shows
Pλ
(
1, t, t2, . . . , tN
)
=
∑
µ : µ≺λ
ψλ/µ(q, t)Pµ
(
t, t2, . . . , tN
)
and then
1 =
∑
µ : µ≺λ
ΛN+1
N (λ, µ). (6.3)
Equations (6.2) and (6.3) show that
[
ΛN+1
N (λ, µ)
]
is a stochastic matrix of format GTN+1×GTN ,
for each N ≥ 0. Thus ΛN+1
N determines a Markov kernel GTN+1 99K GTN . For this reason, we
call ΛN+1
N (λ, µ) the cotransition probabilities. Let us also define the more general Markov kernels
ΛMN : GTM 99K GTN , M > N , by
ΛMN
def
= ΛMM−1ΛM−1
M−2 · · ·Λ
N+1
N ,
or more explicitly
ΛMN (λ, µ)
def
=
∑
λ�λ(M−1)�···�λ(N+1)�µ
ΛMM−1
(
λ, λ(M−1)
)
· · ·ΛN+1
N
(
λ(N+1), µ
)
.
2GT graph, for short.
3(q, t)-GT graph, for short.
42 C. Cuenca
By duality, the kernel ΛM
N also determines a map Mprob(GTM ) → Mprob(GTN ) between
the spaces of probability measures on GTM and GTN , that we denote by the same symbol ΛMN .
For example, if λ ∈ GTM and δλ is the delta mass at λ, then ΛMN δλ is the probability measure
on GTN given by
ΛMN δλ(µ) = ΛMN (λ, µ). (6.4)
Definition 6.2. A sequence {MN}N≥0, such that each MN is a probability measure on GTN , is
called a (q, t)-coherent sequence if the following relations are satisfied
MN (µ) =
∑
λ∈GTN+1
MN+1(λ)ΛN+1
N (λ, µ), ∀N ≥ 0, ∀µ ∈ GTN . (6.5)
Similarly, a finite sequence {MN}N=0,1,...,k is said to be a (q, t)-coherent sequence if the rela-
tions (6.5) hold for N = 0, 1, . . . , k − 1 and all µ ∈ GTN .
It is clear that for any probability measure MN on GTN , there exist probability mea-
sures M0,M1, . . . ,MN−1 on GT0,GT1, . . . ,GTN−1 such that {Mm}m=0,1,...,N is a (q, t)-coherent
sequence, and moreover M0, . . . ,MN−1 are uniquely determined by this condition: in fact,
Mm = ΛNmMN ∀ 0 ≤ m ≤ N − 1.
The set of (infinite) (q, t)-coherent sequences {MN}N≥0 is a convex set. Theorem 1.3 is, in
different terms, a characterization of the extreme points of the set of (q, t)-coherent sequences.
6.2 The path-space T and (q, t)-central measures
The set of (q, t)-coherent sequences defined before is in bijection with a class of probability
measures in the path-space of the GT graph T that we define next.
Definition 6.3. The path-space T is the set of (infinite) paths in the GT graph that begin at
∅ ∈ GT0:
T def
=
{
τ =
(
∅ = τ (0) ≺ τ (1) ≺ τ (2) ≺ · · ·
)
: τ (n) ∈ GTn ∀n ≥ 0
}
.
For any finite path φ = (φ(0) ≺ φ(1) ≺ · · · ≺ φ(n)), define the cylinder set Sφ (or simply S(φ)) by
Sφ
def
=
{
τ ∈ T : τ (i) = φ(i) ∀ i = 0, 1, . . . , n
}
⊂ T .
The set T is equipped with the σ-algebra generated by the cylinder sets Sφ, where φ varies over
all finite paths in the GT graph. We always consider T as a measurable space.
An interesting class of probability measures on T consists of the ones that are coherent with
the sequence of stochastic matrices
{
ΛN+1
N
}
N≥0
. To clarify what such coherence is, define the
natural projection maps
ProjN : T ⊂
∏
n≥0
GTn −→ GTN ,
τ =
(
τ (0) ≺ τ (1) ≺ τ (2) ≺ · · ·
)
7→ τ (N).
It is a standard exercise to show that the σ-algebra of T is the smallest one for which all the maps
ProjN are measurable. Consequently, for any probability measure M on T , we can associate
to it the sequence of its pushforwards under the maps ProjN , namely the sequence {MN}N≥0,
MN = (ProjN )∗M .
Asymptotic Formulas for Macdonald Polynomials 43
Definition 6.4. A probability measure M on T is said to be a (q, t)-central measure if the
following relations hold
M
(
S
(
φ(0) ≺ φ(1) · · · ≺ φ(N−1) ≺ φ(N)
))
= ΛNN−1
(
φ(N), φ(N−1)
)
· · ·Λ1
0
(
φ(1), φ(0)
)
MN
(
φ(N)
)
= t|φ
(N−1)|+···+|φ(0)|ψφ(N)/φ(N−1)(q, t) · · ·ψφ(1)/φ(0)(q, t)
Pφ(N)
(
1, t, . . . , tN−1
) MN
(
φ(N)
)
,
for all N ≥ 0, all finite paths φ =
(
φ(0) ≺ · · · ≺ φ(N)
)
, and for some probability measures MN
on GTN . The branching coefficients ψµ/ν(q, t) are explicit in the statement of Theorem 2.5.
One can verify easily that, if the relations above hold, then the measure MN is the pushforward
(ProjN )∗M , for all N ≥ 0. Moreover, {MN}N≥0 is automatically a (q, t)-coherent sequence.
We denote by Mprob(T ) the set of (q, t)-central (probability) measures. The set of (q, t)-central
measures is a convex set. The set of extreme points of Mprob(T ), equipped with its inherited
topology, is called the boundary of the (q, t)-GT graph4 and is denoted by Ωq,t.
The following proposition implies that the correspondence between the set of (q, t)-central
measures and the set of (q, t)-coherent sequences is a bijection.
Proposition 6.5. Any probability measure M on T has an associated sequence {MN}N≥0 of
probability measures on {GTN}N≥0, as shown above. The map M 7→ {MN}N≥0 is a bijection
between the set Mprob(T ) of (q, t)-central measures on T , and the set of (q, t)-coherent sequences.
The bijection is an isomorphism of convex sets.
Let
{
M (i)
}
i≥1
,M be elements on Mprob(T ) and
{
M
(i)
m
}
i≥1,m≥0
, {Mm}m≥0 be their correspon-
ding (q, t)-coherent sequences. Then the weak limit P− lim
i→∞
M (i) = M holds if and only if the
weak limits P− lim
i→∞
M
(i)
m = Mm hold for all m ∈ N.
Proof. Similar statements are known for other branching graphs, e.g., the case t = q of
our proposition is given in [21, Propositions 4.4 and 4.9], and the case t = q → 1 is in [40,
Proposition 10.3]. In the case t = q, this proposition is given in [21, Propositions 4.4 and 4.9].
The proof at our level of generality can be easily adapted from the proofs in [21]; details are left
to the reader. �
6.3 Macdonald generating functions
We introduce Macdonald generating functions, which will be very helpful in our study of (q, t)-
coherent sequences.
Definition 6.6. Let MN be a probability measure on GTN , then its Macdonald generating
function is the formal sum
PMN
(x1, . . . , xN )
def
=
∑
λ∈GTN
MN (λ)
Pλ
(
x1, x2t, . . . , xN t
N−1
)
Pλ
(
1, t, . . . , tN−1
) .
Note that PMm(x1, . . . , xm) depends on the values q, t, but we omit such dependence for
simplicity.
The sum above is absolutely convergent on the torus (x1, . . . , xm) ∈ Tm. In fact, Theo-
rem 2.5 and the fact that all the branching coefficients ψµ/ν(q, t) are nonnegative imply∣∣Pλ(x1, x2t, . . . , xN t
N−1
)∣∣ ≤ Pλ(|x1|, |x2t|, . . . ,
∣∣xN tN−1
∣∣) = Pλ
(
1, t, . . . , tN−1
)
. Thus not only is
4We may also call it the minimal boundary of the (q, t)-GT graph, to differentiate it from the Martin boundary
of the (q, t)-GT graph, see Section 7.1 below.
44 C. Cuenca
PMN
(x1, . . . , xN ) well-defined as a function on TN , but also sup
x1,...,xN∈T
|PMN
(x1, . . . , xN )| ≤ 1.
Therefore PMN
∈ L∞(Tm).
If MN is supported on the set of positive signatures GT+
N , then each Pλ
(
x1, x2t, . . . , xN t
N−1
)
is a polynomial in x1, . . . , xN and therefore the sum defining PMN
is absolutely convergent on
the closed unit disk (x1, . . . , xm) ∈ Dm. Moreover PMN
∈ L∞(Dm) if MN is supported on GT+
N .
In general, PMN
∈ L∞(Tm) ⊂ L2(Tm). The Fourier series expansion of PMN
can be obtained
by using Corollary 2.6. In fact,
PMN
(x1, . . . , xN ) =
∑
λ∈GTN
MN (λ)
Pλ
(
x1, x2t, . . . , xN t
N−1
)
Pλ
(
1, t, . . . , tN−1
)
=
∑
λ∈GTN
MN (λ)
∑
µ∈GTN
cλ,µmµ
(
x1, . . . , xN t
N−1
)
Pλ
(
1, t, . . . , tN−1
)
=
∑
µ∈GTN
mµ
(
x1, . . . , xN t
N−1
) ∑
λ∈GTN
cλ,µMN (λ)
Pλ
(
1, t, . . . , tN−1
) ,
where the interchange in the order of summation follows from the absolute convergence of all the
sums involved. (For the absolute convergence, the nonnegativity of all coefficients cλ,µ is needed.)
From the expansion above, we can extract the coefficient of xκ11 · · ·x
κN
N in the Fourier series,
for any κ = (κ1 ≥ · · · ≥ κN ) ∈ GTN . In fact, such term appears only in mκ
(
x1, . . . , xN t
N−1
)
with coefficient tn(κ), where n(κ) = κ2 + 2κ3 + · · ·+ (N − 1)κN . Thus the Fourier coefficient of
xκ11 · · ·x
κN
N in the expansion of PMN
(x1, . . . , xN ) is
fκ1,...,κN = tn(κ)
∑
λ∈GTN
cλ,κMN (λ)
Pλ
(
1, t, . . . , tN−1
) . (6.6)
If MN is supported on GT+
N , then the sum defining fκ1,...,κN above is finite. Indeed the only
signatures with a nonzero contribution are λ ∈ GT+
N with |λ| = |κ|. But then |κ| = |λ| ≥ λ1, and
there are finitely many signatures λ ∈ GTN with |κ| ≥ λ1 ≥ · · · ≥ λN ≥ 0. This observation will
be put to use several times.
Lemma 6.7. Let MN , M ′N be probability measures on GTN that are supported on GT+
N . If
PMN
(x1, . . . , xN ) = PM ′N (x1, . . . , xN ) ∀ (x1, . . . , xN ) ∈ TN
then MN = M ′N .
Proof. Both PMN
(x1, . . . , xN ) and PM ′N (x1, . . . , xN ) belong to L2
(
TN
)
. The equality of these
functions implies that their Fourier coefficients agree. From (6.6), this means
∑
µ∈GT+
N
cµ,κMN (µ)
Pµ
(
1, t, . . . , tN−1
) =
∑
µ∈GT+
N
cµ,κM
′
N (µ)
Pµ
(
1, t, . . . , tN−1
) ∀κ ∈ GTN . (6.7)
Observe that we have restricted the sum above to µ ∈ GT+
N , because MN , M ′N are supported
on positive signatures. Let n ∈ Z≥0 be arbitrary. We show that MN (κ) = M ′N (κ) for all κ ∈ GT+
N
with |κ| = n.
Let C be the matrix whose rows and columns are parametrized by λ ∈ GT+
N with |λ| = n,
and such that its entry C(κ, µ) is cµ,κ/Pµ
(
1, t, . . . , tN−1
)
. Observe that C is a finite and square
matrix. Also let M (resp. M ′) be the column vector whose entries are parametrized by λ ∈ GT+
N
with |λ| = n and whose entry µ is MN (µ) (resp. M ′N (µ)). Then (6.7) yields CM = CM ′. The
Asymptotic Formulas for Macdonald Polynomials 45
matrix C is upper-triangular with respect to the order ≥ on signatures because cµ,κ = 0 unless
µ ≥ κ. Moreover the diagonal entries are cµ,µ/Pµ
(
1, t, . . . , tN−1
)
= 1/Pµ
(
1, t, . . . , tN−1
)
6= 0. It
follows that C has an inverse and M = C−1CM = C−1CM ′ = M ′. Therefore MN (κ) = M ′N (κ)
∀κ ∈ GT+
N with |κ| = n. Since n ∈ Z≥0 was arbitrary and both MN , M ′N are supported on GT+
N ,
we conclude MN = M ′N . �
Proposition 6.8. If the sequence {MN}N≥0 (resp. finite sequence {MN}N=0,1,...,k) is a (q, t)-
coherent sequence, then
PMN
(x1, . . . , xN ) = PMN+1
(1, x1, . . . , xN ), ∀ (x1, . . . , xN ) ∈ TN
for all N ≥ 0 (resp. for all N = 0, 1, . . . , k − 1). The converse statement holds if, for each N ≥ 0
(resp. N = 0, 1, . . . , k − 1), MN is supported on GT+
N .
Proof. Let us prove the first part. Let {MN}N≥0 be a (q, t)-coherent sequence. By making use
of the branching rule, Theorem 2.5, the fact that Pµ is homogeneous of degree |µ|, and making
a change in the order of summation, we obtain
PMN+1
(1, x1, . . . , xN ) =
∑
λ∈GTN+1
MN+1(λ)
Pλ
(
1, x1t, . . . , xN t
N
)
Pλ
(
1, t, . . . , tN
)
=
∑
λ∈GTN+1
MN+1(λ)
Pλ
(
1, t, . . . , tN
) ∑
µ∈GTN
ψλ/µ(q, t)Pµ
(
x1t, . . . , xN t
N
)
=
∑
λ∈GTN+1
MN+1(λ)
Pλ
(
1, t, . . . , tN
) ∑
µ∈GTN
t|µ|ψλ/µ(q, t)Pµ
(
x1, . . . , xN t
N−1
)
=
∑
µ∈GTN
Pµ(x1, . . . , xN t
N−1)
Pµ
(
1, t, . . . , tN−1
) ∑
λ∈GTN+1
t|µ|ψλ/µ(q, t)
Pµ
(
1, t, . . . , tN−1
)
Pλ
(
1, t, . . . , tN
) MN+1(λ)
=
∑
µ∈GTN
Pµ
(
x1, . . . , xN t
N−1
)
Pµ
(
1, t, . . . , tN−1
) MN (µ) = PMN
(x1, . . . , xN ).
We can easily show that all sums above are absolutely convergent, so the change in the order of
summations can be justified.
Next we prove the converse statement. Assume that MN , MN+1 are probability measures
on GTN , GTN+1. Assume that they are supported on GT+
N , GT+
N+1, respectively, and also that
PMN
(x1, . . . , xN ) = PMN+1
(1, x1, . . . , xN ) on TN . Let M ′N be the measure on GTN defined by
M ′N (µ) =
∑
λ∈GTN+1
MN+1(λ)ΛN+1
N (λ, µ) ∀µ ∈ GTN .
Since
[
ΛN+1
N (λ, µ)
]
is a stochastic matrix and MN+1 is a probability measure on GTN+1, then M ′N
is a probability measure on GTN . Moreover since MN+1 is supported on GT+
N+1, it follows
that M ′N is supported on GT+
N . In fact, if µ /∈ GTN \ GT+
N (or equivalently µN < 0), then
for any λ ∈ GTN+1, either λN+1 < 0 in which case MN+1(λ) = 0, or λN+1 ≥ 0 in which case
ΛN+1
N (λ, µ) = 0.
We will be done if we showed MN = M ′N . From what we have proved in the first part of the
argument, we have PM ′N (x1, . . . , xN ) = PMN+1
(1, x1, . . . , xN ) on TN . It follows that PMN
= PM ′N
on TN . An application of Lemma 6.7 concludes the proof. �
46 C. Cuenca
Proposition 6.9. Let N ∈ N be arbitrary. If {Mm}m≥1, M are all probability measures on GTN
such that the weak convergence holds P− lim
m→∞
Mm = M , then
lim
m→∞
PMm(x1, . . . , xN ) = PM (x1, . . . , xN ) ∀ (x1, . . . , xN ) ∈ TN . (6.8)
The convergence above is uniform on TN .
Proof. Let ε > 0 be a very small real number. Since M is a probability measure, there exists
c > 0 large enough so that
M ({λ ∈ GTN : c ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ −c}) > 1− ε. (6.9)
From the weak convergence P− lim
m→∞
Mm = M , there exists N1 ∈ N so that m > N1 implies
Mm ({λ ∈ GTN : c ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ −c}) > 1− 2ε. (6.10)
Consider the set GT[−c,c]
N
def
= {λ ∈ GTN : c ≥ λ1 ≥ · · · ≥ λN ≥ −c} ⊂ GTN . It is clear that
GT[−c,c]
N is finite and has cardinality no greater than (2c+ 1)N . Also, since GT[−c,,c]
N is a finite
set, there exists N2 ∈ N such that m > N2 implies
|M(λ)−Mm(λ)| < ε
(2c+ 1)N
∀λ ∈ GT[−c,c]
N . (6.11)
We are ready to make the desired estimate. Use (6.9), (6.10), (6.11) and the triangle inequality
to argue that for any m > max{N1, N2}, we have
sup
(x1,...,xN )∈TN
|PMm(x1, . . . , xN )− PM (x1, . . . , xN )|
≤
∑
λ∈GT[−c,c]
N
|Mm(λ)−M(λ)| sup
(x1,...,xN )∈TN
∣∣∣∣∣Pλ
(
x1, x2t, . . . , xN t
N−1
)
Pλ
(
1, t, . . . , tN−1
) ∣∣∣∣∣
+
∑
λ∈GTN\GT[−c,c]
N
(Mm(λ) +M(λ)) sup
(x1,...,xN )∈TN
∣∣∣∣∣Pλ
(
x1, x2t, . . . , xN t
N−1
)
Pλ
(
1, t, . . . , tN−1
) ∣∣∣∣∣
≤
∑
λ∈GT[−c,c]
N
|Mm(λ)−M(λ)|+
∑
λ∈GTN\GT[−c,c]
N
(Mm(λ) +M(λ))
≤ ε
(2c+ 1)N
∣∣GT[−c,c]
N
∣∣+Mm
(
GTN \GT[−c,c]
N
)
+M
(
GTN \GT[−c,c]
N
)
(6.12)
≤ ε+ 2ε+ ε = 4ε. �
A partial converse to the previous proposition is Proposition 6.10 below.
Proposition 6.10. Let {Mm}m≥1, M , be probability measures on GTN , supported on GT+
N .
Assume the following convergence
lim
m→∞
PMm(x1, . . . , xN ) = PM (x1, . . . , xN )
holds uniformly on TN , then we have the weak convergence P− lim
m→∞
Mm = M .
Asymptotic Formulas for Macdonald Polynomials 47
Proof. All functions {PMm}m≥1, PM belong to L2(Tm). Therefore the limit lim
m→∞
PMm(x1, . . . ,
xN ) = PM (x1, . . . , xN ) implies the convergence of the Fourier coefficients. Due to (6.6), this
implies that, for any κ ∈ GTN , we have
lim
m→∞
∑
λ∈GT+
N
cλ,κM
m(λ)
Pλ
(
1, t, . . . , tN−1
) =
∑
λ∈GT+
N
cλ,κM(λ)
Pλ
(
1, t, . . . , tN−1
) .
We show that lim
m→∞
Mm(λ) = M(λ) for any λ ∈ GT+
N . In fact, let n ∈ Z≥0 be arbitrary and
let us prove lim
m→∞
Mm(λ) = M(λ) for any λ ∈ GT+
N with |λ| = n. Consider the finite, square
matrix C whose rows and columns are parametrized by λ ∈ GT+
N , |λ| = n, and whose entries are
C(κ, λ) = cλ,κ/Pλ
(
1, t, . . . , tN−1
)
. Also let {Mm}m≥0,M be column vectors whose entries are
parametrized by λ ∈ GT+
N with |λ| = n, and whose entries, at λ ∈ GT+
N , are {Mm(λ)}m≥0,M(λ).
From the limit relation above, we have the entrywise limit of column vectors lim
m→∞
CMm = CM .
The matrix C is upper triangular (with respect to the order on signatures given in Corollary 2.6)
and has nonzero diagonal entries, thus it has an inverse C−1. Each entry of the column vectors
Mm =
(
C−1C
)
Mm = C−1(CMm) is a finite linear combination of entries of CMm, and the
same can be said about the entries of C−1CM = M . Thus the entry-wise limit of column vectors
lim
m→∞
Mm = M follows.
By assumption, Mm(λ) = M(λ) = 0 for any λ /∈ GT+
N , so also lim
m→∞
Mm(λ) = M(λ) in this
case. Hence the weak convergence P− lim
m→∞
Mm = M is proved. �
6.4 Automorphisms Ak
Recall the set N of nonincreasing integer sequences, given in Definition 5.1. Equip N with the
topology of pointwise convergence. We denote a generic element of N by ν = (ν1 ≤ ν2 ≤ · · · ). For
each k ∈ Z, we can define the continuous map Ak : N → N by ν 7→ Akν = (ν1 +k ≤ ν2 +k ≤ · · · ).
It is clear that Ak has inverse A−k, so each Ak is a homeomorphism.
Similar automorphisms can be constructed for GT and T . In detail, we can define the map
Ak : GT → GT by λ 7→ Akλ = (λ1 + k ≥ λ2 + k ≥ · · · ), Ak∅ = ∅, whose inverse is A−k, and
moreover it restricts to automorphisms GTN → GTN for each N ∈ Z≥0. It is clear that µ ≺ λ
implies Akµ ≺ Akλ, so the automorphism Ak of GT induces the automorphism of measurable
spaces Ak : T → T , τ =
(
τ (0) ≺ τ (1) ≺ τ (2) ≺ · · ·
)
7→
(
Akτ
(0) ≺ Akτ (1) ≺ Akτ (2) ≺ · · ·
)
.
The same notation Ak is used to define automorphisms of the spaces N , GT and T , but there
should be no confusion each time it is used in the future.
We have introduced the automorphisms Ak because, in Lemma 6.12 below, we will relate
the extreme central probability measures associated to ν and Akν. The starting point is the
following simple statement, which has nothing to do with probability.
Lemma 6.11. Recall the functions Φν(x1, . . . , xm; q, t), defined in the statement of Theorem 5.5.
Let ν ∈ N and k ∈ Z be arbitrary. The following equality holds
ΦAkν(x1, . . . , xm; q, t) = tk(
m
2 )(x1 · · ·xm)kΦν(x1, . . . , xm; q, t),
∀ (x1, . . . , xm) ∈ (C \ {0})m. (6.13)
Proof. As both sides of the identity (6.13) are analytic functions on (C \ {0})m, we only need
to prove the equality for (x1, . . . , xm) ∈ Um, where the domain Um was defined in the statement
of Theorem 5.5. We can now make use of formula (5.8) for Φν(x1, . . . , xm; q, t). Observe that the
only place where ν appears in the right-hand side is inside the univariate functions Φν(xi; q, t).
48 C. Cuenca
The operator D̃
(m)
q,θ satisfies that for any Laurent polynomial f on variables x1, . . . , xm, the
following identity holds
D̃
(m)
q,θ
{
(x1 · · ·xm)kf
}
= tk(
m
2 )(x1 · · ·xm)kD̃
(m)
q,θ {f}.
We deduce that the lemma will be proved for all m ∈ N once we prove it for m = 1, that is,
we need ΦAkν(x; q, t) = xkΦν(x; q, t) ∀x ∈ U =
⋂
k≥1
{x 6= q−k} ∩ {x 6= 0}. The latter statement
easily follows from the integral definition of Φν(x; q, t) in Theorem 5.2 (to be precise, our desired
statement follows after a change of variables z 7→ z + k in the integral). �
Next we introduce new maps {Ak}k∈Z on spaces of probability measures.
For a probability measure Mm on GTm, define AkMm as the pushforward of Mm under the
automorphism Ak of GTm, i.e., AkMm(µ)
def
= Mm(A−kµ) for all µ ∈ GTm. Observe that if we
let δλ be the probability measure on GTm given by the delta mass at λ ∈ GTm, then Akδλ = δAkλ
for any k ∈ Z.
Similarly if M is a probability measure on T , define AkM as the pushforward of M under the
automorphism Ak of T . This can be described concretely as follows. The automorphism Ak of T
induces automorphisms Ak on the set of paths of length m in the GT graph, for any m ∈ N:
φ =
(
φ(0) ≺ φ(1) ≺ · · · ≺ φ(m)
)
7→ Akφ
def
=
(
Akφ
(0) ≺ Akφ(1) ≺ · · · ≺ Akφ(m)
)
.
It is therefore natural to define also an automorphism on the family of cylinder sets by AkSφ
def
=
SAkφ. Then AkM is given by AkM(Sφ)
def
= M(A−kSφ) = M(SA−kφ) for all finite paths φ =(
φ(0) ≺ φ(1) ≺ · · · ≺ φ(m)
)
.
Lemma 6.12. Let M ∈ Mprob(T ) be a (q, t)-central probability measure and {Mm}m≥0 be its
associated (q, t)-coherent sequence. Then also AkM ∈Mprob(T ) and its associated (q, t)-coherent
sequence is {AkMm}m≥0.
Proof. Clearly AkM ∈Mprob(T ) is a consequence of the definition of AkM as a pushforward
of the probability measure M . The second claim can be restated as
(Projm)∗AkM = AkMm.
This equality follows from the definitions of AkM and AkMm as pushforwards of M and Mm,
the fact that (Projm)∗M = Mm, and the evident commutativity of the diagram
T T
GTm GTm.
Ak
Projm Projm
Ak
�
7 The boundary of the (q, t)-Gelfand–Tsetlin graph
In this section we prove Theorem 1.3, which characterizes the (minimal) boundary of the (q, t)-GT
graph. Along the way, we also define and characterize the Martin boundary of the (q, t)-GT
graph.
Assume throughout this section that q ∈ (0, 1), θ ∈ N and set t = qθ. Recall the notation
P− lim
k→∞
Mk = M indicates that a sequence of probability measures {Mk}k≥1 converges weakly
to M .
Asymptotic Formulas for Macdonald Polynomials 49
7.1 The Martin boundary: definition and preliminaries
For any λ ∈ GTN , let δλ be the delta mass at λ. As remarked in Section 6.1, there exists a unique
(q, t)-coherent sequence {Mλ
m}m=0,1,...,N such that each Mλ
m is a probability measure on GTm and
Mλ
N = δλ. Such a sequence is given explicitly by Mλ
N = δλ and Mλ
m = ΛN
mδλ ∀ 0 ≤ m ≤ N − 1,
where the probability measures ΛN
mδλ on GTm are given explicitly in (6.4). Moreover recall
that for a (q, t)-central probability measure M on T , we can associate a (q, t)-coherent sequence
{Mm}m≥0 as in Definition 6.4.
Definition 7.1. The Martin boundary of the (q, t)-Gelfand–Tsetlin graph is the subset of
(q, t)-central probability measures M ∈Mprob(T ) for which there exists a sequence {λ(N)}N≥1,
λ(N) ∈ GTN , such that the following weak limits hold
P− lim
N→∞
ΛNmδλ(N) = Mm ∀m = 0, 1, 2, . . . .
Let us denote the Martin boundary by ΩMartin
q,t ⊂ Mprob(T ) and equip it with its subspace
topology, namely with the topology of weak convergence.
To characterize the Martin boundary ΩMartin
q,t , we begin by giving necessary conditions on
sequences of signatures {λ(N)}N≥1 which yield a weak convergence as above. In this section,
we sometimes denote Mλ
N
def
= δλ the delta mass at λ ∈ GTN , and {Mλ
m = ΛN
mδλ}m=0,1,...,N the
corresponding (q, t)-coherent sequence.
Lemma 7.2. Assume that {λ(N)}N≥1, λ(N) ∈ GTN , is a sequence of signatures such that
{ΛN1 δλ(N)}N≥1 converges weakly, as N →∞, to some probability measure m on GT1 = Z. Then
the sequence {λ(N)N}N≥1 is bounded.
Proof. We use the Macdonald generating functions of Section 6.3 above. By Proposition 6.8,
we have
Pδλ(N)
(
1N−1, z
)
= P
M
λ(N)
1
(z)
and then
Pλ(N)
(
1, t, . . . , tN−2, tN−1z
)
Pλ(N)
(
1, t, . . . , tN−2, tN−1
) =
Pλ(N)
(
z, t−1, t−2, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) =
∑
n∈Z
M
λ(N)
1 (n)zn. (7.1)
In the sum of the right-hand side above, n ranges from λ(N)N to λ(N)1 because of the branching
rule for Macdonald polynomials. We multiply the equality by z−λ(N)N and then set z = 0, so in
the right-hand side one clearly picks up the coefficient of zλ(N)N , namely M
λ(N)
1 (λ(N)N ). By
the index stability of Macdonald polynomials, 2.2), the left-hand side is
z−λ(N)N
Pλ(N)
(
z, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
)∣∣∣∣∣
z=0
=
(
z · t−1 · · · t1−N
)−λ(N)N(
1 · t−1 · · · t1−N
)−λ(N)N
Pλ(N)
(
z, t−1, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
)∣∣∣∣∣
z=0
=
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N ,0)
(
z, t−1, . . . , t1−N
)
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N ,0)
(
1, t−1, . . . , t1−N
)∣∣∣∣∣
z=0
=
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N )
(
t−1, . . . , t1−N
)
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N ,0)
(
1, t−1, . . . , t1−N
) .
50 C. Cuenca
Thanks to Theorem 2.3, we can then obtain a lower bound for M
λ(N)
1 (λ(N)N ) as follows:
M
λ(N)
1 (λ(N)N ) =
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N )
(
t−1, . . . , t1−N
)
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N ,0)
(
1, t−1, . . . , t1−N
)
=
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N )
(
tN−2, . . . , t, 1
)
P(λ(N)1−λ(N)N ,...,λ(N)N−1−λ(N)N ,0)
(
tN−1, . . . , t, 1
)
=
∏
1≤i<j≤N−1
(
qλ(N)i−λ(N)j tj−i; q
)
∞(tj−i+1; q)∞(
qλ(N)i−λ(N)j tj−i+1; q
)
∞(tj−i; q)∞
×
∏
1≤i<j≤N
(
qλ(N)i−λ(N)j tj−i+1; q
)
∞(tj−i; q)∞(
qλ(N)i−λ(N)j tj−i; q
)
∞(tj−i+1; q)∞
=
N−1∏
i=1
(
qλ(N)i−λ(N)N tN−i+1; q
)
∞(tN−i; q)∞(
qλ(N)i−λ(N)N tN−i; q
)
∞(tN−i+1; q)∞
=
N−1∏
i=1
(
1− tN−i
)(
1− qtN−i
)
· · ·
(
1− qθ−1tN−i
)(
1− qλ(N)i−λ(N)N tN−i
)
· · ·
(
1− qλ(N)i−λ(N)N+θ−1tN−i
)
≥
N−1∏
i=1
(
1− tN−i
)(
1− qtN−i
)
· · ·
(
1− qθ−1tN−i
)
≥
N−1∏
i=1
(
1− tN−i
)θ ≥ ((t; t)∞)θ.
On the other hand, since (t; t)∞ ∈ (0, 1) and P− lim
N→∞
M
λ(N)
1 = m, then there exist N1, N2 ∈ N
large enough such that
M
λ(N)
1 ({(n) ∈ GT1 : −N1 ≤ n ≤ N1}) > 1− ((t; t)∞)θ ∀N > N2.
We conclude that −N1 ≤ λ(N)N ≤ N1, for all N > N2. Therefore the sequence {λ(N)N}N≥1 is
bounded. �
One actually has the following more general statement.
Lemma 7.3. Let k ∈ N. Assume that {λ(N)}N≥1, λ(N) ∈ GTN , is a sequence of signatures
such that the sequence ΛN
k δλ(N) converges weakly, as N → ∞, to some probability measure m
on GTk. Then for any i = 1, . . . , k, the sequence {λ(N)N−i+1}N≥1, is bounded.
Proof. The argument here is very similar to that of the previous proof for k = 1. As before, by
making use of Macdonald generating functions, we can derive a more general equation than (7.1),
which is
Pλ(N)
(
z1t
1−k, . . . , zk−1t
−1, zk, t
−k, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
)
=
∑
µ∈GTk
M
λ(N)
k (µ)
Pµ
(
z1t
1−k, . . . , zk−1t
−1, zk
)
Pµ
(
1, t−1, . . . , t1−k
) . (7.2)
In the sum of the right-hand side above, note that µ ranges over signatures in GTk such that
µk ≥ λ(N)N , µk−1 ≥ λ(N)N−1, . . . , µ1 ≥ λ(N)N−k+1. (7.3)
This is a consequence of the branching rule for Macdonald polynomials. Another relevant
observation is that for any µ ∈ GTk satisfying (7.3), any monomial cm1,...,mkz
m1
1 · · · zmkk with
cm1,...,mk 6= 0 in the expansion of Pµ(z1t
1−k, . . . , zk−1t
−1, zk) satisfies: mk ≥ λ(N)N ; if mk =
Asymptotic Formulas for Macdonald Polynomials 51
λ(N)N then mk−1 ≥ λ(N)N−1; and so on until, if m2 = λ(N)N−k+2, . . . ,mk ≥ λ(N)N , then
m1 ≥ λ(N)N−k+1. This is a consequence of the triangularity property of the Macdonald
polynomials, see Definition/Proposition 2.1.
In equation (7.2) above, multiply both sides by z
−λ(N)N
k and then set zk = 0. From the
branching rule for Macdonald polynomials and the fact that the branching coefficients satisfy
ψλ(N)/(λ(N)1,...,λ(N)N−1)(q, t) = 1,
the resulting left-hand side is
P(λ(N)1,...,λ(N)N−1)
(
z1t
1−k, . . . , zk−1t
−1, t−k, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
) .
(Note that the argument zk is no longer present in the numerator.) Similarly, from the property
ψµ/(µ1,...,µk−1)(q, t) = 1, for any µ ∈ GTk,
the resulting right-hand side is∑
µ∈GTk : µk=λ(N)N
M
λ(N)
k (µ)
P(µ1,...,µk−1)
(
z1t
1−k, . . . , zk−1t
−1
)
Pµ
(
1, t−1, . . . , t1−k
) .
After that, multiply both sides by (zk−1t
−1)−λ(N)N−1 and then set zk−1 = 0; this gives
P(λ(N)1,...,λ(N)N−2)
(
z1t
1−k, . . . , zk−2t
−2, t−k, . . . , t1−N
)
Pλ(N)
(
1, t−1, . . . , t1−N
)
=
∑
µ∈GTk :
µk−1=λ(N)N−1, µk=λ(N)N
M
λ(N)
k (µ)
P(µ1,...,µk−2)
(
z1t
1−k, . . . , zk−2t
−2
)
Pµ
(
1, t−1, . . . , t1−k
) .
Repeat the same procedure k times, until we have multiplied both sides by
(
z1t
1−k)−λ(N)N−k+1
and set z1 = 0. The end result is
M
λ(N)
k (λ(N)N−k+1, . . . , λ(N)N )
=
P(λ(N)1,...,λ(N)N−k)
(
t−k, . . . , t1−N
)
P(λ(N)N−k+1,...,λ(N)N )
(
1, t−1, . . . , t1−k
)
Pλ(N)
(
1, t−1, . . . , t1−N
)
= t(N−k)(λ(N)N−k+1+λ(N)N−k+2+···+λ(N)N )
×
P(λ(N)1,...,λ(N)N−k)
(
1, t, . . . , tN−k−1
)
P(λ(N)N−k+1,...,λ(N)N )
(
1, t, . . . , tk−1
)
Pλ(N)
(
1, t, t2, . . . , tN−1
)
=
N−k∏
i=1
N∏
j=N−k+1
(
1− tj−i
)(
1− qtj−i
)
· · ·
(
1− qθ−1tj−i
)(
1−qλ(N)i−λ(N)j tj−i
)(
1−qλ(N)i−λ(N)j+1tj−i
)
· · ·
(
1−qλ(N)i−λ(N)j+θ−1tj−i
)
≥
N−k∏
i=1
N∏
j=N−k+1
(
1− tj−i
)(
1− qtj−i
)
· · ·
(
1− qθ−1tj−i
)
≥ ((t; t)∞)θk,
where we used the homogeneity of Macdonald polynomials for the second equality, as well as three
applications of Theorem 2.3 for the third equality. Then M
λ(N)
k (λ(N)N−k+1, . . . , λ(N)N ) ≥ ck =
((t; t)∞)θk > 0. On the other hand, since M
λ(N)
k converges weakly to m, there exist N1, N2 ∈ N
large enough such that
M
λ(N)
k ({(n1, . . . , nk) ∈ GTk : −N1 ≤ n1, . . . , nk ≤ N1}) > 1− ck ∀N > N2.
52 C. Cuenca
Therefore we conclude that −N1 ≤ λ(N)N−k+1, . . . , λ(N)N ≤ N1 for all N > N2. We conclude
that each sequence {λ(N)N−i+1}N≥1, i = 1, . . . , k, is uniformly bounded by a constant. �
Next we show that ΩMartin
q,t is in bijection with the set N . We do so by first constructing
a map ΩMartin
q,t → N that will later be shown to be bijective.
Proposition 7.4. Let M be a (q, t)-central probability measure on T and {Mm}m=0,1,2,... be its
associated (q, t)-coherent system. If M belongs to ΩMartin
q,t , then there exists a unique ν ∈ N such
that
Φν
(
x1t
1−m, . . . , xm−1t
−1, xm; q, t
)
= PMm(x1, . . . , xm),
∀ (x1, . . . , xm) ∈ Tm, ∀m ≥ 1. (7.4)
The function Φν(z1, . . . , zm; q, t) was defined in Theorem 5.2 for m = 1, and in Theorem 5.5 for
general m.
Proof. Let M ∈ ΩMartin
q,t . By definition, there exists a sequence {λ(N)}N≥1 such that λ(N) ∈
GTN , N ≥ 1, and P− lim
m→∞
ΛNmδλ(N) = Mm holds weakly, for any m ∈ N.
By Lemma 7.3, it follows that for any i = 1, 2, . . . , the sequence {λ(N)N−i+1}N≥1 is uniformly
bounded. In particular, since {λ(N)N}N≥1 is uniformly bounded, there exists a subsequence{
N1
1 < N1
2 < N1
3 < · · ·
}
⊂ N such that λ
(
N1
1
)
N1
1
= λ
(
N1
2
)
N1
2
= · · · . Analogously, using that
{λ(N)N−1}N≥1 is bounded, there exists a subsequence
{
N2
1 < N2
2 < N2
3 < · · ·
}
⊂
{
N1
1 <
N1
2 < N1
3 < · · ·
}
such that λ
(
N2
1
)
N2
1−1
= λ
(
N2
2
)
N2
2−1
= · · · . In a similar fashion, we can define
subsequences
{
Nk
1 < Nk
2 < · · ·
}
inductively.
Now consider the subsequence
{
N1 = N1
1 < N2 = N2
2 < N3 = N3
3 < · · ·
}
⊂ N. By
construction, the sequence {λ(Nk)}k≥1 is such that the limits lim
k→∞
λ(Nk)Nk−i+1 exist for all
i = 1, 2, . . . . Consequently, there exists ν ∈ N such that {λ(Nk)}k≥1 stabilizes to ν in the sense
of Definition 5.1. From Theorem 5.5, the following limit
lim
k→∞
Pλ(Nk)
(
x1t
1−m, x2t
2−m, . . . , xm, t
−m, . . . , t1−Nk
)
Pλ(Nk)
(
1, t−1, . . . , t1−Nk
)
= Φν
(
x1t
1−m, x2t
2−m, . . . , xm; q, t
)
(7.5)
holds uniformly for (x1, . . . , xm) in compact subsets of the domain (C \ {0})m, in particular the
convergence is uniform on Tm.
On the other hand, since we also have the weak convergence P− lim
k→∞
ΛNkm δλ(Nk) = Mm, then
by Proposition 6.9:
lim
k→∞
P
Λ
Nk
m δλ(Nk)
(x1, . . . , xm) = PMm(x1, . . . , xm) (7.6)
uniformly on Tm. By Proposition 6.8 and the homogeneity of the Macdonald polynomials, the
Macdonald generating function of ΛNkm δλ(Nk) equals
P
Λ
Nk
m δλ(Nk)
(x1, . . . , xm) = Pδλ(Nk)
(
1Nk−m, x1, . . . , xm
)
=
Pλ(Nk)
(
1, t, . . . , tNk−m−1, tNk−mx1, . . . , t
Nk−1xm
)
Pλ(Nk)
(
1, t, t2, . . . , tNk−1
)
=
Pλ(Nk)
(
x1t
1−m, x2t
2−m, . . . , xm, t
−m, . . . , t1−Nk
)
Pλ(Nk)
(
1, t−1, t−2, . . . , t1−Nk
) . (7.7)
Asymptotic Formulas for Macdonald Polynomials 53
By combining (7.5), (7.6) and (7.7), we conclude
Φν
(
x1t
1−m, x2t
2−m, . . . , xm; q, t
)
= PMm(x1, . . . , xm) for all (x1, . . . , xm) ∈ Tm
and all m ≥ 1, as desired. The statement about the uniqueness of ν ∈ N follows from
Lemma 5.4. �
Because of Proposition 7.4, for each M ∈ ΩMartin
q,t , there exists a unique ν ∈ N such that
equation (7.4) is satisfied for all m ∈ N. Thus there is a well-defined map of sets
N : ΩMartin
q,t → N ,
M 7→ ν, (7.8)
which is determined by setting ν = N(M) be the unique element of N such that (7.4) is satisfied.
We prove below that N is bijective, but first we show the convenient fact that N commutes with
the automorphisms Ak, k ∈ Z, of Section 6.4.
Lemma 7.5. Let M ∈ ΩMartin
q,t , k ∈ Z. Then AkM ∈ ΩMartin
q,t and N(AkM) = AkN(M).
Proof. As M ∈ ΩMartin
q,t ⊂MProb(T ), Lemma 6.12 shows AkM ∈MProb(T ) and {AkMm}m≥0 is
its corresponding (q, t)-coherent sequence. By definition, there exists a sequence {λ(N)}N≥1,
λ(N) ∈ GTN , such that the weak convergence holds P − lim
N→∞
ΛNmδλ(N) = Mm ∀m ≥ 0. We
claim that P− lim
N→∞
ΛNmδAkλ(N) = AkMm ∀m ≥ 0, which would show AkM ∈ ΩMartin
q,t indeed.
Take any m ∈ Z≥0, µ ∈ GTm, then(
ΛNmδAkλ(N)
)
(µ) = ΛNm(Akλ(N), µ) = ΛNm(λ(N), A−kµ)
=
(
ΛNmδλ(N)
)
(A−kµ)
N→∞−−−−→Mm(A−kµ) = AkMm(µ),
where we have used ΛNm(Akν,Akκ) = ΛNm(ν, κ), previously stated in the proof of Lemma 6.12.
Let us move on to the second part of the lemma. Let N(M) = ν; to show N(AkM) = Akν,
we need
PAkMm(x1, . . . , xm) = ΦAkν
(
x1t
1−m, . . . , xm−1t
−1, xm; q, t
)
∀ (x1, . . . , xm) ∈ Tm, ∀m ∈ N.
For any m ∈ N, (x1, . . . , xm) ∈ Tm, we have
PAkMm(x1, . . . , xm) =
∑
λ∈GTm
AkMm(λ)
Pλ
(
x1, x2t, . . . , xmt
m−1
)
Pλ
(
1, t, t2, . . . , tm−1
)
=
∑
λ∈GTm
Mm(A−kλ)
Pλ
(
x1, x2t, . . . , xmt
m−1
)
Pλ
(
1, t, t2, . . . , tm−1
)
=
∑
λ∈GTm
Mm(λ)
PAkλ
(
x1, x2t, . . . , xmt
m−1
)
PAkλ
(
1, t, t2, . . . , tm−1
)
= (x1x2 · · ·xm)k ·
∑
λ∈GTm
Mm(λ)
Pλ
(
x1, x2t, . . . , xmt
m−1
)
Pλ
(
1, t, t2, . . . , tm−1
)
= (x1x2 · · ·xm)k · PMm(x1, . . . , xm)
= (x1 · · ·xm)k · Φν
(
x1t
1−m, . . . , xm; q, t
)
= tk(
m
2 )((x1t
1−m) · · · (xm)
)k · Φν
(
x1t
1−m, . . . , xm; q, t
)
= ΦAkν
(
x1t
1−m, . . . , xm; q, t
)
,
where the last equality follows from Lemma 6.11. �
54 C. Cuenca
Proposition 7.6. The Martin boundary ΩMartin
q,t of the (q, t)-GT graph is bijective to N under
the map N defined in (7.8) above.
Proof. Step 1. We prove N is surjective.
Let ν ∈ N be arbitrary; we want to show it belongs to the range of N. From Lemma 7.5,
we may assume ν1 = 0 without any loss of generality. Consider the sequence {λ(N) = (νN ≥
νN−1 ≥ · · · ≥ ν1)}N≥1 of signatures that stabilizes to ν. Let m ∈ N be arbitrary. The first claim
is that the limit lim
N→∞
M
λ(N)
m (λ) exists for any λ ∈ GTm.
By definition (6.4), M
λ(N)
m (λ) = ΛN
m(λ(N), λ) = 0 unless λm ≥ λ(N)N = ν1 = 0. Thus
the previous claim is clear if λm < ν1 = 0, in which case lim
N→∞
M
λ(N)
m (λ) = 0. Assume now
λm ≥ ν1 = 0, i.e., λ ∈ GT+
m.
From Theorem 5.5, we have the uniform limit
lim
N→∞
Pλ(N)
(
x1t
1−m, . . . , xm−1t
−1, xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
)
= Φν
(
x1t
1−m, . . . , xm−1t
−1, xm; q, t
)
(7.9)
on Tm. Correspondingly, the Fourier coefficients of the normalized Macdonald characters converge
to those of Φν(x1t
1−m, . . . , xm−1t
−1, xm; q, t). Proposition refprop:coherentsequences and the
expansion of Corollary 2.6 give
Pλ(N)
(
x1t
1−m, . . . , xm−1t
−1, xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
)
=
∑
µ∈GTm
Mλ(N)
m (µ)
Pµ
(
x1, x2t, . . . , xmt
m−1
)
Pµ
(
1, t, . . . , tm−1
)
=
∑
κ∈GTm
mκ
(
x1, x2t, . . . , xmt
m−1
) ∑
µ∈GTm
cµ,κM
λ(N)
m (µ)
Pµ
(
1, t, . . . , tm−1
) . (7.10)
Let κ ∈ GTm be arbitrary, and denote n(κ)
def
= κ2 + 2κ3 + · · · + (m − 1)κm. Observe that
xκ11 · · ·xκm appears only in the monomial symmetric polynomial mκ(x1, x2t, . . . , xmt
m−1) and
the corresponding term is tn(κ)xκ11 · · ·xκmm , so (7.10) is essentially the Fourier expansion of the
prelimit functions in (7.9). Then we have that, for any κ ∈ GTm, the following limit
lim
N→∞
tn(κ)
∑
µ∈GTm
cµ,κM
λ(N)
m (µ)
Pµ
(
1, t, . . . , tm−1
) (7.11)
exists and equals the Fourier coefficient of xκ11 · · ·xκmm in the function Φν(x1t
m−1, . . . , xm; q, t).
As mentioned before, M
λ(N)
m (µ) = 0 unless µm ≥ ν1 ≥ 0. Thus we can restrict the sum in (7.11)
to µ ∈ GT+
m. From the same analysis as in Proposition 6.10, we can obtain that the limit
lim
N→∞
M
λ(N)
m (µ) exists for any µ ∈ GT+
m with |λ| = n; let us denote Mm(µ)
def
= lim
N→∞
M
λ(N)
m (µ).
Immediately from the definition (and Fatou’s lemma) it follows that
Mm(λ) ≥ 0 ∀λ ∈ GTm,∑
λ∈GTm
Mm(λ) ≤ 1.
Recall that we observed M
λ(N)
m (λ) = 0 if λ /∈ GT+
m, therefore Mm(λ) = 0 if λ /∈ GT+
m.
Asymptotic Formulas for Macdonald Polynomials 55
Next consider the following function
F (x1, . . . , xm) =
∑
µ∈GTm
Mm(µ)
Pµ
(
x1, x2t, . . . , xmt
m−1
)
Pµ
(
1, t, . . . , tm−1
) .
Clearly F defines a function on Tm, as it is defined by an absolutely convergent series on
the m-dimensional torus. Moreover its absolute value is upper bounded by 1, therefore F ∈
L∞(Tm) ⊂ L2(Tm). By the same argument preceding (6.6), one shows, for any κ ∈ GTm, that
the Fourier coefficient of xκ11 · · ·xκmm in F is
tn(κ)
∑
µ∈GTm
cµ,κMm(µ)
Pµ
(
1, t, . . . , tm−1
) . (7.12)
Observe that, if κ ∈ GTm is fixed, the sums in both (7.11) and (7.12) are finite because
cµ,κMm(µ) = 0 unless µ ≥ κ and µ ∈ GT+
m. Therefore lim
N→∞
M
λ(N)
m (µ) = Mm(µ) for all µ ∈ GT+
m
implies the equality between (7.11) and (7.12). In other words, the following convergence holds
Pλ(N)
(
x1t
1−m, . . . , xm−1t
−1, xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) Fourier−−−−→ F (x1, . . . , xm) (7.13)
in the sense that all Fourier coefficients of the left side of (7.13) converge to the corresponding
Fourier coefficients of F (x1, . . . , xm), as N tends to infinity. But we already knew that the
Fourier coefficients of the left side of (7.13) converge to the corresponding Fourier coefficients of
Φν(x1t
1−m, . . . , xm; q, t). Therefore all the Fourier coefficients of the difference F (x1, . . . , xm)−
Φν(x1t
1−m, . . . , xm; q, t) of square-integrable functions on Tm must be zero. It follows that
F (x1, . . . , xm) = Φν(x1t
1−m, . . . , xm; q, t). In particular, the equality holds for x1 = · · · = xm = 1,
resulting in∑
µ∈GTm
Mm(µ) = F (1, . . . , 1) = Φν
(
t1−m, t2−m, . . . , 1; q, t
)
= lim
N→∞
Pλ(N)
(
x1t
1−m, . . . , xm−1t
−1, xm, t
−m, . . . , t1−N
)
Pλ(N)
(
1, t−1, t−2, . . . , t1−N
) ∣∣∣∣∣
x1=···=xm=1
= lim
N→∞
1 = 1.
Moreover Φν(x1t
1−m, . . . , xm; q, t) = F (x1, . . . , xm) is the Macdonald generating function of Mm.
We are almost done. For each m ∈ N, we have constructed probability measures Mm as limits
of M
λ(N)
m and shown that Φν(x1t
1−m, . . . , xm; q, t) is the generating function of Mm. Complete
the sequence with M0 = δ∅, the delta mass at ∅. We claim that {Mm}m≥0 is a (q, t)-coherent
sequence. In fact, since {Mλ(N)
m }m=0,1,...,N is a (q, t)-coherent sequence, then
Mλ(N)
m (µ) =
∑
λ∈GTm+1
M
λ(N)
m+1 (λ)Λm+1
m (λ, µ) (7.14)
for any 0 ≤ m < N , µ ∈ GTm. As N goes to infinity, the left side of (7.14) tends to Mm(µ).
By an argument similar to that in the proof of Proposition 6.9, one shows that the right side
of (7.14) converges to
∑
λ∈GTm+1
Mm+1(λ)Λm+1
m (λ, µ). Indeed, the argument simply relies on the
weak convergence M
λ(N)
m+1 →Mm+1 and the uniform (on λ) bound |Λm+1
m (λ, µ)| = Λm+1
m (λ, µ) ≤ 1.
Therefore the limit of (7.14) as N →∞ is
Mm(µ) =
∑
λ∈GTm
Mm+1(λ)Λm+1
m (λ, µ),
56 C. Cuenca
for any m ∈ Z≥0, µ ∈ GTm. Thus {Mm}m≥0 is a (q, t)-coherent sequence and has an associated
probability measure M on T , as given by Proposition 6.5. By the definition of N, we conclude
N(M) = ν.
Step 2. Next we show that N is injective.
Let M,M ′ ∈ ΩMartin
q,t have the same image ν under the map N. The goal is to prove M = M ′.
From Lemma 7.5, we may assume ν1 = 0 without any loss of generality. Furthermore, we can
assume that M is the element of ΩMartin
q,t such that N(M) = ν and that was contructed in the
first step.
Let {Mm}m≥0 and {M ′m}m≥0 be the (q, t)-coherent sequences associated to M,M ′, then
PM1(x) = Φν(x; q, t) = PM ′1(x) ∀x ∈ T. (7.15)
As it was mentioned in Section 6.3, Macdonald generating functions are uniformly bounded on
the torus, in particular, Pm ∈ L∞(T) ⊂ L2(T) for any probability measure m on GT1 = Z. Write
the first equality of (7.15) for x = eiθ as follows:∑
n∈Z
M1(n)einθ =
∑
n∈Z
M ′1(n)einθ.
Both sums above are expansions of a square integrable function on T in terms of the basis
{einθ} ⊂ L2(T). Thus the (Fourier) coefficients in both sums must agree, i.e., M1(n) = M ′1(n)
∀n ∈ Z, and so M1 = M ′1.
We aim to apply a similar argument to show that Mm = M ′m for any m ∈ N. We will be done
once this is proved, as Proposition 6.5 would then show M = M ′. For general m, we make use of
the fact that M ∈ ΩMartin
q,t is the probability measure constructed in step 1: we use that each Mm
is supported on GT+
m.
As above, the definition of the map N implies
PMm(x1, . . . , xm) = Φν
(
x1t
1−m, . . . , xm−1t
−1, xm; q, t
)
= PM ′m(x1, . . . , xm) ∀ (x1, . . . , xm) ∈ Tm.
The equality of the functions above implies the equality of corresponding Fourier coefficients. It
follows that for any κ ∈ GTm we have, see (6.6),
∑
µ∈GTm
cµ,κMm(µ)
Pµ
(
1, t, . . . , tm−1
) =
∑
µ∈GTm
cµ,κM
′
m(µ)
Pµ
(
1, t, . . . , tm−1
) . (7.16)
When κ /∈ GT+
m (or equivalently κm < 0), we claim that the left side of (7.16) is zero. In fact,
for any µ ∈ GTm, either cµ,κ = 0 when µm ≥ 0 or Mm(µ) = 0 when µm < 0, because Mm is
supported on GT+
m. Then also the right side of (7.16) is zero if κ /∈ GT+
m. By using also the
properties of the coefficients cµ,κ stated in Corollary 2.6, we have
0 =
∑
µ∈GTm
cµ,κM
′
m(µ)
Pµ
(
1, t, . . . , tm−1
) ≥ cκ,κM
′
m(κ)
Pκ
(
1, t, . . . , tm−1
) =
M ′m(κ)
Pκ
(
1, t, . . . , tm−1
) ≥ 0.
Then we must have M ′m(κ) = 0 if κ /∈ GT+
m, i.e., M ′m is supported on GT+
m. Finally an application
of Lemma 6.7 says that if PMm = PM ′m on Tm, and Mm, M ′m are both supported on GT+
m,
implies Mm = M ′m, thus finishing the proof. �
Asymptotic Formulas for Macdonald Polynomials 57
7.2 Characterization of the Martin boundary
Our next goal is to characterize the topological space ΩMartin
q,t completely. Recall the map
N : ΩMartin
q,t → N , defined above in (7.8), by letting N(M) = ν be the unique element of N such
that
Φν
(
x1t
1−m, . . . , xm−1t
−1, xm; q, t
)
= PMm(x1, . . . , xm) ∀ (x1, . . . , xm) ∈ Tm, ∀m ≥ 1.
Proposition 7.6 shows that N is a bijection, so the inverse map N−1 is well-defined.
Definition 7.7. For any ν ∈ N , we denote N−1(ν) by Mν and the corresponding (q, t)-coherent
sequence by {Mν
m}m≥0.
From step 1 of the proof of Proposition 7.6, and Lemma 7.5, we have that for the sequence of
signatures {λ(N) = (νN ≥ · · · ≥ ν1)}N≥1 which stabilizes to ν, the following weak convergence
holds
P− lim
N→∞
ΛNmδλ(N) = Mν
m ∀m ∈ N,
i.e.,
lim
N→∞
ΛNmδλ(N)(µ) = lim
N→∞
ΛNm(λ(N), µ) = Mν
m(µ) ∀m ∈ N, ∀µ ∈ GTm.
In fact, we note the same analysis as in step 1 of the proof of Proposition 7.6 shows that the
weak convergence above holds for any sequence of signatures {λ(N)}N≥1 stabilizing to ν.
We need the following lemma to prove that N is a homeomorphism.
Lemma 7.8.
1. For any ν ∈ N , m ∈ N, Mν
m is supported on {µ = (µ1 ≥ · · · ≥ µm−1 ≥ µm) ∈ GTm : µm ≥
ν1, µm−1 ≥ ν2, . . . , µ1 ≥ νm}.
2. For any m ∈ N, there exists cm ∈ (0, 1) such that for any integers n1 ≤ n2 ≤ · · · ≤ nm and
ν ∈ N with ν1 = n1, ν2 = n2, . . . , νm = nm, we have Mν
m(nm ≥ · · · ≥ n2 ≥ n1) ≥ cm.
3. Let m ∈ N and let ν̃, ν ∈ N be such that ν̃i ≥ νi ∀ i > m and ν̃i = νi ∀ 1 ≤ i ≤ m, then
M ν̃
m(νm ≥ · · · ≥ ν2 ≥ ν1) ≤Mν
m(νm ≥ · · · ≥ ν2 ≥ ν1).
Proof. Let us prove (1). We use the weak convergence of probabilities P− lim
N→∞
ΛNmδλ(N) = Mν
m
∀m ∈ N, for the sequence of signatures {λ(N) = (νN ≥ · · · ≥ ν2 ≥ ν1)}N≥1. Because of the
definition (6.4) for the maps ΛN
m, it follows that ΛN
mδλ(N)(µ) = ΛN
m(λ(N), µ) = 0, unless µm ≥
λ(N)N = ν1, . . . , µ1 ≥ λ(N)N−m+1 = νm. Thus also Mν
m(µ) = 0 unless µm ≥ ν1, . . . , µ1 ≥ νm.
Next we show (2); to get started fix n1 ∈ N and let us show that Mν
1 (n1) ≥ ((t; t)∞)θ for
any ν ∈ N with ν1 = n1 (so we can set c1 = ((t; t)∞)θ). The weak convergence mentioned
above gives Mν
1 (n1) = lim
N→∞
ΛN1 δλ(N)(n1) = lim
N→∞
ΛN1 δλ(N)(λ(N)N ), for any ν ∈ N with ν1 = n1
and {λ(N)}N≥1 as constructed above. The calculations in the proof of Lemma 7.2 show
ΛN1 δλ(N)(λ(N)N ) ≥ ((t; t)∞)θ for all N ∈ N, and thus also Mν
1 (n1) ≥ c1
def
= ((t; t)∞)θ > 0, for any
ν ∈ N with ν1 = n1.
For a general m ∈ N, we have
Mν
m(nm ≥ · · · ≥ n2 ≥ n1) = lim
N→∞
ΛNmδλ(N)(λ(N)N−m+1 ≥ · · · ≥ λ(N)N ),
for {λ(N)}N≥1 as constructed above. In the proof of Lemma 7.3, we showed ΛNmδλ(N)(λ(N)N−m+1
≥ · · · ≥ λ(N)N ) ≥ cm
def
= ((t; t)∞)θm holds for all N ≥ 1. Therefore Mν
m(nm ≥ · · · ≥ n2 ≥ n1) ≥
cm > 0, for any ν ∈ N with ν1 = n1, ν2 = n2, . . . , νm = nm.
58 C. Cuenca
Finally we move on to (3). To get started, we prove it for m = 1, so let ν̃, ν ∈ N be
such that ν̃i ≥ νi ∀ i ≥ 2 and ν̃1 = ν1. Consider the following pair of sequences of signatures
{λ̃(N) = (ν̃N ≥ · · · ≥ ν̃2 ≥ ν̃1)}N≥1 and {λ(N) = (νN ≥ · · · ≥ ν2 ≥ ν1)}N≥1. Then we have the
weak limits
P− lim
N→∞
ΛNmδλ̃(N)
= M ν̃
m, P− lim
N→∞
ΛNmδλ(N) = Mν
m ∀m ∈ N. (7.17)
By the calculations in Lemma 7.2, we have
ΛN1 δλ(N)(ν1) = ΛN1 δλ(N)(λ(N)N )
=
N−1∏
i=1
(1− tN−i)(1− qtN−i) · · · (1− qθ−1tN−i)
(1− qλ(N)i−λ(N)N tN−i) · · · (1− qλ(N)i−λ(N)N+θ−1tN−i)
=
N−1∏
i=1
(1− tN−i)(1− qtN−i) · · · (1− qθ−1tN−i)
(1− qνN−i+1−ν1tN−i) · · · (1− qνN−i+1−ν1+θ−1tN−i)
≥
N−1∏
i=1
(1− tN−i)(1− qtN−i) · · · (1− qθ−1tN−i)
(1− qν̃N−i+1−ν̃1tN−i) · · · (1− qν̃N−i+1−ν̃1+θ−1tN−i)
=
N−1∏
i=1
(1− tN−i)(1− qtN−i) · · · (1− qθ−1tN−i)
(1− qλ̃(N)i−λ̃(N)N−θtN−i+1) · · · (1− qλ̃(N)i−λ̃(N)N−1tN−i+1)
= ΛN1 δλ̃(N)
(λ̃(N)N ) = ΛN1 δλ̃(N)
(ν1).
Thus taking into account the limits (7.17) for m = 1, we deduce Mν
1 (ν1) ≥M ν̃
1 (ν1).
The proof of the third item for a general m ∈ N follows from similar calculations that are
used to prove Lemma 7.3. We leave the details to the reader. �
Theorem 7.9. The bijective map N : ΩMartin
q,t → N is a homeomorphism.
Proof. The first step shows that N−1 is continuous and the second one shows that N is
continuous.
Step 1. Let
{
Mν(i)
}
i≥1
⊂ ΩMartin
q,t and
{
ν(i)
}
i≥1
be the corresponding images under the
map N. If lim
i→∞
ν(i) = ν ∈ N pointwise, then we prove the weak limit P− lim
i→∞
Mν(i) = Mν .
By applying some automorphism Ak with large k, if necessary, and invoking Lemma 7.5, we
can assume ν1 ≥ 0. Observe that lim
i→∞
ν
(i)
1 = ν1 ≥ 0 implies ν
(i)
1 ≥ 0 for large enough i. Thus let
us also assume ν
(i)
1 ≥ 0 ∀ i ≥ 1, for simplicity.
We reduce our desired statement to simpler ones. We claim that lim
i→∞
ν(i) = ν implies
lim
i→∞
Φν(i)(x1, . . . , xm; q, t) = Φν(x1, . . . , xm; q, t) ∀m ∈ N (7.18)
uniformly on Tm.
First let us deduce our desired weak convergence from the limit above. Recall that, due to
Theorem 5.5, all functions
{
Φν(i)(x1, . . . , xm; q, t)
}
i≥1
and Φν(x1, . . . , xm; q, t) are entire functions
and, in particular, they are continuous on the torus Tm. So the convergence (7.18) implies the
convergence of Fourier coefficients. Thus for any κ ∈ GTm we obtain, see (6.6),
lim
i→∞
∑
µ∈GT+
m
cµ,κM
ν(i)
m (µ)
Pµ(1, t, . . . , tm−1)
=
∑
µ∈GT+
m
cµ,κM
ν
m(µ)
Pµ(1, t, . . . , tm−1)
. (7.19)
Asymptotic Formulas for Macdonald Polynomials 59
Note that in both sides of (7.19), the sums have been restricted to GT+
m. In fact, since we are
assuming ν
(i)
1 , ν1 ≥ 0 ∀ i ≥ 1, Lemma 7.8(1) implies that all probability measures
{
Mν(i)
m
}
i≥1
,Mν
m
are supported on GT+
m.
From Proposition 6.10, the limit (7.19) yields lim
i→∞
Mν(i)
m (µ) = Mν
m(µ) ∀µ ∈ GT+
m. But also
Mν(i)
m (µ) = Mν
m(µ) = 0 for any i ≥ 1 and any µ /∈ GT+
m. Therefore lim
i→∞
Mν(i)
m (µ) = Mν
m(µ) holds
for all µ ∈ GTm, and so P− lim
i→∞
Mν(i)
m = Mm. Note that we proved the weak convergence for
any m ∈ N. By Proposition 6.5, we conclude P− lim
i→∞
Mν(i) = M .
We are left with the task of proving (7.18). We show the uniform convergence in a neighborhood
of the torus Tm. But it suffices to prove the limit on compact subsets of Um. From the definition
of Φν(x1, . . . , xm; q, t) on Um, see (5.8), and observing that Tm ⊂ Um, it follows that the result
holds for any m ∈ N provided it holds for m = 1. We prove (7.18) for m = 1 on an open
neighborhood of T.
Clearly a small enough open neighborhood of T is a subset of U . By the definition of Φν(x; q, t)
on U , see (5.2), the desired uniform convergence will hold if we verify
lim
i→∞
∫
C+
xz
∞∏
j=1
(q−z+ν
(i)
j tj ; q)∞
(q−z+ν
(i)
j tj−1; q)∞
=
∫
C+
xz
∞∏
j=1
(q−z+νj tj ; q)∞
(q−z+νj tj−1; q)∞
(7.20)
uniformly on compact subsets of x ∈ C \ {0}. Note that, since all
{
ν
(i)
1
}
i≥1
, ν1, are nonnegative,
we can take the same contour C+ for both of the integrals in (7.20).
The pointwise convergence of integrands in (7.20) is clear. We still need some uniform
estimates for the contribution of the tails of the left side in (7.20). This is similar to the proof of
Theorem 5.2.
Let K ⊂ C \ {0} be any compact set. Parametrize the tails of C+ as z = r ± π
√
−1
ln q ; for r
ranging from some large R > 0 to +∞, we want to show that the contribution of each of these
lines is small. We have
sup
x∈K
∣∣∣∣∣∣xz ·
∞∏
j=1
(
q−z+ν
(i)
j tj ; q
)
∞(
q−z+ν
(i)
j tj−1; q
)
∞
∣∣∣∣∣∣
≤ const× |x|r ×
∞∏
j=1
1(
1 + q−r+ν
(i)
j +θ(j−1)) · · · (1 + q−r+ν
(i)
j +θj−1)
≤ const · |x|r
q−krqν
(i)
1 +ν
(i)
2 +···+ν(i)k
= const ·
(
|x|qk
)r
qν
(i)
1 +···+ν(i)k
,
for any z = r ± π
√
−1
ln q , and any k ∈ N. The constant above depends on K but not on i. Choose
k ∈ N large enough so that a
def
= sup
x∈K
|x| · qk ∈ (0, 1). Since lim
i→∞
ν
(i)
j = νj for all j = 1, 2, . . . , k,
then sup
i≥1
q−ν
(i)
1 −ν
(i)
2 −···−ν
(i)
k ≤ const <∞. Then there exists a constant c = cK > 0, independent
of i, such that
sup
i≥1
sup
x∈K
∣∣∣∣∣∣xz ·
∞∏
j=1
(
q−z+ν
(i)
j tj ; q
)
∞(
q−z+ν
(i)
j tj−1; q
)
∞
∣∣∣∣∣∣ ≤ c · ar, if z = r ± π
√
−1
ln q
.
Since
∫∞
R ardr = −eR ln a/ ln a
R→∞−−−−→ 0, we have just shown that the contribution of the tails
of C+ is uniformly small. We can then apply dominated convergence theorem to conclude (7.20),
as desired.
60 C. Cuenca
Step 2. As in the step above, let
{
Mν(i)
}
i≥1
be a sequence in ΩMartin
q,t , whose images under N
are
{
ν(i)
}
i≥1
. If the weak limit P− lim
i→∞
Mν(i) = M holds, then we show that M ∈ ΩMartin
q,t , that
there a pointwise limit lim
i→∞
ν(i) = ν ∈ N and moreover M = Mν .
Let m ∈ N be arbitrary. Thanks to Proposition 6.5, the limit P − lim
i→∞
Mν(i) = M implies
P − lim
i→∞
Mν(i)
m = Mm. By Lemma 7.8(2), there exists cm ∈ (0, 1) such that Mν(i)
m
(
ν
(i)
m ≥
· · · ≥ ν
(i)
1
)
≥ cm > 0 for all i ≥ 1. Since Mm is a probability measure, there exists N0 ∈ N
large enough such that Mm((λ1, . . . , λm) ∈ GTm : N0 ≥ λ1, . . . , λm ≥ −N0) > 1 − cm
2 . Then
P− lim
i→∞
Mν(i)
m = Mm implies Mν(i)
m ((λ1, . . . , λm) ∈ GTm : N0 ≥ λ1, . . . , λm ≥ −N0) > 1− cm for
all i large enough. It follows that N0 ≥ ν(i)
m , . . . , ν
(i)
1 ≥ −N0 for all i large enough. In particular,
the sequence
{
ν
(i)
m
}
i≥1
is bounded.
By using the boundedness of the sequences
{
ν
(i)
m
}
i≥1
and the “diagonal argument” of Proposi-
tion 7.4, there exists a subsequence 1 ≤ i1 < i2 < i3 < · · · such that the limits lim
k→∞
ν
(ik)
m exist,
for all m ∈ N. In other words, we have the pointwise limit lim
k→∞
ν(ik) = ν, for some ν ∈ N .
We have the obvious implication
P− lim
i→∞
Mν(i) = M =⇒ P− lim
k→∞
Mν(ik) = M,
But step 1 in this proof shows that lim
k→∞
ν(ik) = ν implies P− lim
k→∞
Mν(ik) = Mν . By uniqueness
of weak limits, we have M = Mν .
We are left to show that lim
i→∞
ν(i) = ν. Above we only showed the existence of a subsequence
{ik}k≥1 such that lim
k→∞
ν(ik) = ν. Nevertheless the same argument gives us more. In fact, it
shows that any subsequence
{
ν(ir)
}
r≥1
⊂
{
ν(i)
}
i≥1
must have a subsubsequence
{
ν(ir(s))
}
s≥1
⊂{
ν(ir)
}
r≥1
such that a pointwise limit lim
s→∞
ν(ir(s)) = ν ′ exists. Then M = Mν′ = Mν , but since
the map N is bijective, we have ν = ν ′. We therefore conclude that the sequence
{
ν(i)
}
i≥1
itself
must converge to ν, finishing the proof. �
7.3 Relation between the Martin boundary and the (minimal) boundary
The basic relation between the Martin and minimal boundary of the (q, t)-GT graph is the
following statement, which actually holds in a much greater generality.
Proposition 7.10 (consequence of [37, Theorem 6.1]). The following inclusion holds Ωq,t ⊆
ΩMartin
q,t . In other words, for any M ∈ Ωq,t, there exists a sequence {λ(N)}N≥1, λ(N) ∈ GTN ,
such that
P− lim
N→∞
Mλ(N)
m = Mm ∀m = 0, 1, 2, . . . .
In many examples, especially in the context of asymptotic representation theory, it is known
that the Martin boundary of a branching graph is equal to its minimal boundary, e.g., [21, 37]. In
our case, we will also prove that this is the case by following the ideas in [21, 41]. The following
statement, which also holds in greater generality, will be useful.
Asymptotic Formulas for Macdonald Polynomials 61
Proposition 7.11 (consequence of [37, Theorem 9.2]). Let M ′ be any probability measure
on Mprob(T ). There exists a unique Borel probability measure π belonging to Ωq,t such that
M ′(Sφ) =
∫
M∈Ωq,t
M(Sφ)π(dM),
for any finite path φ =
(
φ(0) ≺ φ(1) ≺ · · · ≺ φ(n)
)
in the (q, t)-GT graph.
7.4 Characterization of the boundary of the (q, t)-Gelfand–Tsetlin graph
Proof of Theorem 1.3. Let us now prove the items (1), (2) stated in the theorem.
(1) Thanks to Proposition 7.10, we have Ωq,t ⊆ ΩMartin
q,t ⊆Mprob(T ). We claim that ΩMartin
q,t ⊆
Ωq,t. Because of Theorem 7.9, the Martin boundary ΩMartin
q,t (with its induced topology from
Mprob(T )) is homeomorphic to N , under the map N. The claim would then show that the
minimal boundary Ωq,t of the (q, t)-GT graph is equal to the Martin boundary ΩMartin
q,t and
therefore homeomorphic to N , under the map N.
Let us prove the claim above. Let ν ∈ N be arbitrary and let Mν ∈ ΩMartin
q,t be the
corresponding element of ΩMartin
q,t . Let i : Ωq,t ↪→ ΩMartin
q,t be the natural inclusion, considered as
a measurable map. By Proposition 7.11, there exists a unique probability measure π on Ωq,t such
that
Mν =
∫
M∈Ωq,t
Mπ(dM) =
∫
M∈ΩMartin
q,t
M(i∗π)(dM). (7.21)
Note that i∗π is a probability measure on ΩMartin
q,t with (i∗π)(Ωq,t) = 1. Let π̃ be the pushforward
of i∗π under the homeomorphism N : ΩMartin
q,t → N , so π̃ is a Borel probability measure on N .
Equation (7.21) can be rewritten as
Mν =
∫
ν̃∈N
M ν̃ π̃(dν̃). (7.22)
We make a subclaim: π̃ is the delta mass at ν ∈ N . Let us first deduce ΩMartin
q,t ⊆ Ωq,t from
this latter claim. In fact, if π̃ is the delta mass at ν ∈ N , then i∗π is the delta mass at Mν . But
since we had (i∗π)(Ωq,t) = 1, then Mν ∈ Ωq,t. Since Mν was an arbitrary element of ΩMartin
q,t ,
then we conclude ΩMartin
q,t ⊆ Ωq,t.
Let us now prove the subclaim that the probability measure π̃ on N satisfying (7.22) must be
the delta mass at ν ∈ N . We show first that π̃ is supported on {ν̃ ∈ N : ν̃ ≥ ν} def
= {ν̃ ∈ N : ν̃1 ≥
ν1, ν̃2 ≥ ν2, ν̃3 ≥ ν3, . . . }. Since π̃ is a Borel measure, the opposite would mean the existence of
m ∈ N and κ1 ≤ κ2 ≤ · · · ≤ κm, such that κi < νi for some 1 ≤ i ≤ m, and
π̃
({
ν̃ ∈ N : ν̃1 = κ1, . . . , ν̃m = κm
})
> 0.
As a consequence of (7.22) we have, for all m ∈ N,
Mν
m =
∫
ν̃∈N
M ν̃
mπ̃(dν̃). (7.23)
We can now apply (7.23) to κ = (κm ≥ · · · ≥ κ1) ∈ GTm:
Mν
m(κ) =
∫
ν̃∈N
M ν̃
m(κ)π̃(dν̃).
From Lemma 7.8(1), the left-hand side of the equality above vanishes, while Lemma 7.8(2) shows
that the right-hand side is at least cm · π̃({ν̃ ∈ N : ν̃1 = κ1, . . . , ν̃m = κm}) > 0, thus there is
a contradiction.
62 C. Cuenca
From the fact that π̃ is supported on {ν̃ ∈ N : ν̃ ≥ ν} def
= {ν̃ ∈ N : ν̃1 ≥ ν1, ν̃2 ≥ ν2, . . . }, and
Lemma 7.8, parts (1) (3), we have that (7.23) evaluated at (νm ≥ · · · ≥ ν1) is
Mν
m(νm ≥ · · · ≥ ν1) =
∫
ν̃∈N
M ν̃
m(νm ≥ · · · ≥ ν1)π̃(dν̃) =
∫
ν̃∈N
ν̃≥ν
M ν̃
m(νm ≥ · · · ≥ ν1)π̃(dν̃)
=
∫
ν̃∈N , ν̃≥ν
ν̃i=νi ∀i=1,...,m
M ν̃
m(νm ≥ · · · ≥ ν1)π̃(dν̃)
+
∫
ν̃∈N , ν̃≥ν
ν̃i 6=νi for some i∈{1,...,m}
M ν̃
m(νm ≥ · · · ≥ ν1)π̃(dν̃)
=
∫
ν̃∈N ,ν̃≥ν
ν̃i=νi ∀ i=1,...,m
M ν̃
m(νm ≥ · · · ≥ ν1)π̃(dν̃)
≤Mν
m(νm ≥ · · · ≥ ν1) · π̃({ν̃ ∈ N : ν̃1 = ν1, . . . , ν̃m = νm}).
Next Lemma 7.8(2) says that Mν
m(νm ≥ · · · ≥ ν1) ≥ cm > 0, so we must have π̃({ν̃ ∈ N : ν̃1 =
ν1, . . . , ν̃m = νm}) = 1. Since this is true for any m ∈ N, it follows that π̃ must be the delta mass
at ν, thus proving our second claim and the full characterization of Ωq,t.
Let us return to the statement of item (1) in Theorem 1.3. By definition of the map N, the
relations (1.2) hold. We have already observed that the Macdonald generating function defining
PMν
m
(x1, . . . , xm) is absolutely convergent on Tm. The last statement that says Mν is determined
by the relations (1.2) follows from the uniqueness statement in Proposition 7.4.
(2) Let {Mν
m}m≥0, {MAkν
m }m≥0, be the (q, t)-coherent sequences associated to Mν and MAkν ,
respectively. By Lemma 7.5, we have the first statement MAkν(SAkφ) = Mν(Sφ), for any finite
path φ. Next by virtue of Lemma 6.12, MAkν
m = AkM
ν
m for all m ≥ 0. Thus by following
the definitions, MAkν
m (Akλ) = AkM
ν
m(Akλ) = Mν
m(A−kAkλ) = Mν
m(λ), for any λ ∈ GTm and
m ∈ Z≥0. �
A Basics on q-analysis
A good reference for the material on q-analysis is [2, Chapter 10]. Assume |q| < 1 is an arbitrary
complex number. Most statements work if q is an indeterminate too.
The q-numbers and the q-factorial are defined by
[n]q
def
=
1− qn
1− q
, n ∈ N,
[n]q!
def
= [n]q · · · [2]q[1]q, n ∈ N, [0]q!
def
= 1.
It is evident that [n]q → n and [n]q!→ n!, as q → 1, for any n ∈ Z≥0. Observe that we can also
define [x]q for any x ∈ C, as before, and it also holds that [x]q → x as q → 1. The q-Gamma
function is defined by
Γq(z)
def
= (1− q)1−z (q; q)∞
(qz; q)∞
.
From the definition, the q-functional equation
Γq(z + 1) = [z]qΓq(z), z /∈ {. . . ,−2,−1, 0},
is evident. The q-Gamma function is a meromorphic function with simple poles at z =
0,−1,−2, . . . and all their shifts by an integral multiple of 2π
√
−1/ ln q. The q-Gamma function
has no zeroes in C. Moreover, we have the following convergence to the Gamma function.
Asymptotic Formulas for Macdonald Polynomials 63
Theorem A.1 ([2, Corollary 10.3.4]). For any z ∈ C \ {. . . ,−2,−1, 0}, we have
lim
q→1
Γq(z) = Γ(z).
Remark A.2. As a consequence of Stieltjes–Vitali theorem, the convergence lim
q→1
Γq(z) = Γ(z)
holds uniformly on compact subsets of C \ {. . . ,−2,−1, 0}.
Other important identities we use in our paper are the q-binomial theorems. To state them,
we need to define the q-Pochhammer symbols (x; q)n and (x; q)∞, for any x ∈ C and n ∈ Z≥0 by
(x; q)n
def
=
1 if n = 0,
n∏
i=1
(
1− xqi−1
)
if n ≥ 1,
∞∏
i=1
(
1− xqi−1
)
if n =∞.
Note that |q| < 1 implies that the product defining (x; q)∞ is uniformly convergent for x ∈ C,
and thus (x; q)∞ is an entire function.
The q-binomial formula is the following
Theorem A.3 ([2, Theorem 10.2.1]). For |z| < 1,
∞∑
n=0
(a; q)n
(q; q)n
zn =
(az; q)∞
(z; q)∞
Corollary A.4. For z ∈ C, m ∈ N,
M∑
n=0
(q−1; q−1)M
(q−1; q−1)n(q−1; q−1)M−n
(−1)nq−(n2)zn =
(
z; q−1
)
M
.
Proof. Let a = q−M in Theorem A.3, and use (q; q)n = (−1)nq(
n+1
2 )(q−1; q−1)n. The statement
is then proved for any |z| < 1; therefore it also holds for any z ∈ C because both sides are
polynomials on z. �
Another application of the q-binomial theorem is the following limit.
Theorem A.5 ([2, Theorem 10.2.4]). For any a, b ∈ R such that b− a /∈ Z, the following limit
lim
q→1
(xqa; q)∞
(xqb; q)∞
= (1− x)b−a
holds uniformly on compact subsets of {x ∈ C : |x| ≤ 1, x 6= 1}.
Remark A.6. If b − a ∈ Z, the limit in Theorem A.5 holds uniformly on compact subsets
of C \ {1}.
B Some properties of the rational functions C(q,t)
τ1,...,τn
(u1, . . . , un)
Lemma B.1. Assume t = qθ, for some θ ∈ N. Let τ1, . . . , τn ∈ Z≥0 and u1, . . . , un be n variables,
then C
(q,t)
τ1,...,τn(u1, . . . , un) = 0 if some of the integers τ1, . . . , τn is strictly larger than θ.
64 C. Cuenca
Proof. Let us first rewrite the expression for C
(q,t)
τ1,...,τn(u1, . . . , un), as it was done in [30, Section 6].
Let vi = qτiui for i = 1, . . . , n. Also, given τ = (τ1, . . . , τn), let T = Tτ
def
= {k ∈ {1, 2, . . . , n} : τk 6=
0}. Then
C(q,t)
τ (u1, . . . , un) =
∏
k∈T
tτk−1 (q/t; q)τk−1
(q; q)τk−1
(quk; q)τk
(qtuk; q)τk
×
∏
1≤i<j≤n
(qui/tuj ; q)τi
(qui/uj ; q)τi
(tui/vj ; q)τi
(ui/vj ; q)τi
Fτ (u; q, t), (B.1)
where
Fτ (u; q, t)
def
=
∑
K⊂T
(−1)|K|(1/t)(
|K|
2 )
∏
j∈T−K
t− qτj
1− qτj
×
∏
k∈K
j∈T−K
vj − vk/t
vj − vk
∏
k∈K
1− tvk
1− vk
∏
i∈T
i 6=k
ui − vk
ui − vk/t
.
Due to the factor
∏
k∈T
(q/t; q)τk−1 in (B.1), it follows that τk − θ ∈ N = {1, 2, . . . }, for some k,
implies C
(q,t)
τ1,...,τn(u1, . . . , un) = 0, as desired. �
Lemma B.2 ([30, Lemma 6.1]). Assume t = q. Let τ1, . . . , τn ∈ Z≥0 and u1, . . . , un be n
variables, then C
(q,t)
τ1,...,τn(u1, . . . , un) = 0 if some of the integers τ1, . . . , τn is strictly larger than 1.
If all τ1, . . . , τn ∈ {0, 1}, then C
(q,t)
τ1,...,τn(u1, . . . , un) does not depend on the variables u1, . . . , un
and
C(q,t)
τ1,...,τn = (−1)τ1+···+τn .
Lemma B.3. Assume θ ∈ N. If we let a
(θ)
n =
C
(q,qθ)
n (x−1
2 x1q−θ)
θ−1∏
i=0
(x1−qix2)
, for all 0 ≤ n ≤ θ, then these
expressions satisfy the relations
a(θ)
n =
1
x1 − x2
(
Tq,x2a
(θ−1)
n − Tq,x1a
(θ−1)
n−1
)
, 1 ≤ n ≤ θ − 1,
a
(θ)
0 =
1
θ−1∏
i=0
(x1 − qix2)
, a
(θ)
θ =
1
θ−1∏
i=0
(x2 − qix1)
.
Proof. The expression C
(q,t)
n (u), for n ∈ Z≥0, is much simpler than the general expression (2.5).
It was first found by Jing and Joźefiak in [26] and it reads C
(q,t)
n (u) = tn (1/t;q)n
(q;q)n
(u;q)n
(qtu;q)n
1−q2nu
1−u .
Then, for 0 ≤ n ≤ θ:
C(q,qθ)
n
(
x1/
(
qθx2
))
=
qθx2 − q2nx1
qθx2 − x1
1
tn
n∏
i=1
{
qθ − qi−1
1− qi
qθx2 − qi−1x1
x2 − qix1
}
. (B.2)
From (B.2), it is only a matter of tedious computation to check the three identities given in the
lemma. �
Asymptotic Formulas for Macdonald Polynomials 65
Acknowledgements
It is my pleasure to thank Alexei Borodin for his generous sharing of time and ideas. I am equally
indebted to Vadim Gorin, for his interest in my work, many helpful discussions and for sharing
some of his notes on the q-GT graph. This work would not exist without them. I would also like
to thank Jiaoyang Huang, for being an excellent sounding board at the beginning stage of this
project, Konstantin Matveev for his expert help with the software Mathematica, and Grigori
Olshanski for comments in a previous draft of this paper and for asking a question that led to my
proof of Theorem 1.3. The suggestions of the referees helped improved this text greatly; many
thanks are due to them.
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1 Introduction
1.1 Description of the formulas
1.2 The boundary of the (q, t)-Gelfand–Tsetlin graph
1.3 Comments on Theorem 1.3 and connections to existing literature
1.4 Organization of the paper
2 Symmetric Laurent polynomials
2.1 Partitions, signatures and symmetric Laurent polynomials
2.2 Macdonald polynomials and Macdonald characters
3 Integral formulas for Macdonald characters of one variable
3.1 Statements of the theorems
3.2 An example
3.3 Integral formula when t = q, N: Proof of Theorem 3.1
3.4 Integral formula for general q, t I: Outline of proof of Theorems 3.2 and 3.3
3.5 Integral formula for general q, t II: Proof of Theorem 3.2
4 Multiplicative formulas for Macdonald characters
4.1 Statement of the multiplicative theorem and some consequences
4.2 Proof of Theorem 4.1
5 Asymptotics of Macdonald characters
5.1 Asymptotics of Macdonald characters of one variable
5.2 Asymptotics of Macdonald characters of a fixed number m of variables
6 Preliminaries on the (q, t)-Gelfand–Tsetlin graph
6.1 The (q, t)-Gelfand–Tsetlin graph
6.2 The path-space T and (q, t)-central measures
6.3 Macdonald generating functions
6.4 Automorphisms Ak
7 The boundary of the (q, t)-Gelfand–Tsetlin graph
7.1 The Martin boundary: definition and preliminaries
7.2 Characterization of the Martin boundary
7.3 Relation between the Martin boundary and the (minimal) boundary
7.4 Characterization of the boundary of the (q, t)-Gelfand–Tsetlin graph
A Basics on q-analysis
B Some properties of the rational functions C(q, t)1, …, n(u1, …, un)
References
|
| id | nasplib_isofts_kiev_ua-123456789-209463 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:57:56Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cuenca, C. 2025-11-21T19:15:48Z 2018 Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph / C. Cuenca // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D52; 33D90; 60B15; 60C05 arXiv: 1704.02429 https://nasplib.isofts.kiev.ua/handle/123456789/209463 https://doi.org/10.3842/SIGMA.2018.001 We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052-3132], and are expected to provide tools for the study of statistical mechanical models, representation theory, and random matrices. As the first application of our formulas, we characterize the boundary of the (q,t)-deformation of the Gelfand-Tsetlin graph when t=qθ and θ is a positive integer. It is my pleasure to thank Alexei Borodin for his generous sharing of time and ideas. I am equally indebted to Vadim Gorin for his interest in my work, for many helpful discussions, and for sharing some of his notes on the q-GT graph. This work would not exist without them. I would also like to thank Jiaoyang Huang for being an excellent sounding board at the beginning stage of this project, Konstantin Matveev for his expert help with the software Mathematica, and Grigori Olshanski for comments in a previous draft of this paper and for asking a question that led to my proof of Theorem 1.3. The suggestions of the referees helped improve this text greatly; many thanks are due to them. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph Article published earlier |
| spellingShingle | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph Cuenca, C. |
| title | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph |
| title_full | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph |
| title_fullStr | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph |
| title_full_unstemmed | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph |
| title_short | Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph |
| title_sort | asymptotic formulas for macdonald polynomials and the boundary of the (q,t)-gelfand-tsetlin graph |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209463 |
| work_keys_str_mv | AT cuencac asymptoticformulasformacdonaldpolynomialsandtheboundaryoftheqtgelfandtsetlingraph |