Matrix Bailey Lemma and the Star-Triangle Relation
We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus, we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a conseque...
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| citation_txt | Matrix Bailey Lemma and the Star-Triangle Relation / K.Yu. Magadov, V.P. Spiridonov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. |
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| description | We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus, we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.
|
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 121, 13 pages
Matrix Bailey Lemma and the Star-Triangle Relation
Kamil Yu. MAGADOV † and Vyacheslav P. SPIRIDONOV ‡§
† Deceased; Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
‡ Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region, 141980 Russia
E-mail: spiridon@theor.jinr.ru
§ National Research University Higher School of Economics, Moscow, Russia
Received August 10, 2018, in final form October 30, 2018; Published online November 10, 2018
https://doi.org/10.3842/SIGMA.2018.121
Abstract. We compare previously found finite-dimensional matrix and integral operator re-
alizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With
the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced
to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can
be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.
Key words: elliptic hypergeometric functions; Bailey lemma; star-triangle relation
2010 Mathematics Subject Classification: 33D60; 33E20
1 Introduction
Quantum integrable systems can be realized as statistical mechanics models [5] and solved by
the quantum version of the inverse scattering method [22]. One of the important examples is
given by the hard hexagons model solved in [4]. During its investigation Baxter met a particular
type of q-hypergeometric series identities called the Rogers–Ramanujan identities [2] without
knowing how to prove them. It appeared that Bailey [3] has found already a systematic way of
proving such relations. The key ingredient in his method is the so-called Bailey lemma. Taking
appropriate entries for this lemma, called Bailey pairs, Rogers–Ramanujan identities can be
easily proved. An iterative mechanism for building new Bailey pairs from a given one has been
found by Andrews [1] (see also [13]). Sequences of these pairs form the Bailey chain. A survey
of the results derived from the Bailey lemma in the first 50 years was given by Warnaar in [23].
Shortly after appearance of the summary [23], an extension of the Bailey chains technique to
elliptic hypergeometric series was proposed by the second author in [18]. Some further analysis
of the elliptic Bailey chains for such series was performed by Warnaar in [24]. A principal
generalization of the whole Bailey formalism to the level of integrals and its application to
derivation of infinite sequences of symmetry transformations for elliptic hypergeometric integrals
was discovered by the second author in [19]. It is curious to note that Bailey himself was
interested in the generalization of his techniques from q-series to integrals, see the personal
letters from Bailey to Dyson [25]. However, he was not able to find any use of such an idea,
which was realized on the full scale only in [19].
In [10] the integral Bailey pairs were used for building an integral operator (an R-matrix)
satisfying the Yang–Baxter equation [5, 22]. The ingredients of the Bailey lemma serve as
building blocks of this R-matrix. In this case the key operator identity generating an infinite
sequence of Bailey pairs, known as the star-triangle relation, can be interpreted as an infinite-
dimensional space realization of the cubic Coxeter relations for a permutation group. Such
This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.
The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html
mailto:spiridon@theor.jinr.ru
https://doi.org/10.3842/SIGMA.2018.121
https://www.emis.de/journals/SIGMA/EHF2017.html
2 K.Yu. Magadov and V.P. Spiridonov
integral operator realizations of the generating relations of permutation groups were proposed
earlier in [8, 9] at the level of simpler special functions.
Constructions of the matrix and integral Bailey lemmas in [18] and [19] look similar in their
structure. Nevertheless, the direct connection between them has not been described in detail
yet. A residue calculus for a quite sophisticate type of (multivariate) integral Bailey lemmas
was discussed by Rains in [15], but the exposition is not as explicit as one would wish to see.
Our goal is to give an elementary description how an appropriate residue calculus applied to
the integral Bailey lemma of [19] generates the matrix Bailey lemma of [18]. This gives an
interpretation of the corresponding key matrix identity as a star-triangle relation which was not
considered in [18]. Consequently, we also interpret this relation as a realization of the Coxeter
relations of a permutation group, similar to the integral case.
2 Integral Bailey lemma and the star-triangle relation
We start our consideration from the integral Bailey lemma derived in [19]. For this we recall
that two functions α(z, t) and β(z, t) depending on two complex variables z and t are called the
Bailey pair, if they are related by the following (univariate) integral transformation
β(w, t) = M(t)wzα(z, t) := κ
∫
T
Γ
(
tw±1z±1; p, q
)
Γ
(
t2, z±2; p, q
) α(z, t)
dz
z
, (2.1)
κ :=
(p; p)∞(q; q)∞
4πi
, (z; q)∞ :=
∞∏
j=0
(
1− zqj
)
,
where T is the positive oriented unit circle, |tw|, |t/w| < 1, and we assume that α(z, t) is
analytical near z ∈ T.
The kernel of the integral operator M(t)wz contains a combination of seven elliptic gamma
functions
Γ(z; p, q) :=
∞∏
j,k=0
1− z−1pj+1qk+1
1− zpjqk
, |p|, |q| < 1,
with the convention
Γ(a, b; p, q) := Γ(a; p, q)Γ(b; p, q), Γ
(
tz±1; p, q
)
:= Γ(tz; p, q)Γ
(
tz−1; p, q
)
.
The integration contour T in (2.1) can be replaced by any contour C encircling the same set of
singularities of the kernel function which allows analytic continuation of the action of M(t)wz to
a wider range of values of parameters t and w, provided C does not cross singularities of α(z, t).
Let us list briefly the key properties of the elliptic gamma function. These are the symmetry
in bases
Γ(z; p, q) = Γ(z; q, p)
and the finite-difference equations
Γ(qz; p, q) = θ(z; p)Γ(z; p, q), Γ(pz; p, q) = θ(z; q)Γ(z; p, q),
where θ(z; p) is the “short” Jacobi theta function
θ(z; p) := (z; p)∞
(
pz−1; p
)
∞ =
1
(p; p)∞
∑
n∈Z
pn(n−1)/2(−z)n.
Matrix Bailey Lemma and the Star-Triangle Relation 3
Other properties we use include the inversion relation
Γ(z; p, q) =
1
Γ
(pq
z ; p, q
) , (2.2)
the quadratic transformation (note its appearance in (2.1))
Γ
(
z2; p, q
)
= Γ
(
± z,±q1/2z,±p1/2z,±(pq)1/2z; p, q
)
,
and the limiting relation
lim
z→1
(1− z)Γ(z; p, q) =
1
(p; p)∞(q; q)∞
,
which is needed in the residue calculus.
The integral transformation (2.1) is also called the elliptic Fourier transformation. One of
the justification for such a name comes from the fact established in [21] that for actions on the
A1-symmetric functions, f
(
z−1
)
= f(z), and under some constraints on the parameters, the
inversion of this transform is obtained by the reflection t→ t−1, i.e.,
M(t)−1wz = M
(
t−1
)
wz
, or M(t)wzM
(
t−1
)
zx
f(x) = f(w),
where we assume an analytic continuation of the action of the M -operators by an appropriate
deformation of the contours of integration and test functions f(x) with relevant analytical prop-
erties. For a discussion of this procedure and partial description of the null space of the integral
operator M(t)wz, see [11].
Let us define now the following combination of four elliptic gamma functions
D(s; y, w) := Γ
(√
pqs−1y±1w±1; p, q
)
.
From the inversion relation (2.2) we find that
D
(
s−1; y, w
)
=
1
D(s; y, w)
.
The integral Bailey lemma established in [19] states that from a given Bailey pair α(w, t) and
β(w, t) one can obtain a new Bailey pair with the replacement of the parameter t by another
arbitrary variable. Namely, the functions
α′(w, st) := D(s; y, w)α(w, t), (2.3)
β′(w, st) := D
(
t−1; y, w
)
M(s)wxD(st; y, x)β(x, t), (2.4)
with the assumption |x| = 1 and the constraints |t|, |s|,
∣∣√pqs−1t−1y±1∣∣ < 1, form a new Bailey
pair, i.e.,
β′(w, st) = M(st)wzα
′(z, st).
The transformed functions α′ and β′ depend now on two new complex variables s and y. Evi-
dently, this procedure has an iterative character, i.e., from a given Bailey pair it generates in-
finitely many such pairs containing as many free variables as many times we apply the maps (2.3)
and (2.4). Note that for taken restrictions on parameters one automatically has
∣∣√pqs−1y±1∣∣ < 1
and the unit circle separates sequences of poles of D(st; y, x) and D(s; y, x) converging to x = 0
from their reciprocals.
4 K.Yu. Magadov and V.P. Spiridonov
It is easy to see that this statement leads to the following operator identity
M(s)wxD(st; y, x)M(t)xz = D(t; y, w)M(st)wzD(s; y, z), (2.5)
which is called the star-triangle relation. The proof of (2.5) is based on the explicit elliptic beta
integral evaluation formula derived in [17]. Namely, for six complex parameters t1, t2, . . . , t6
subject to the constraints
6∏
j=1
tj = pq and |tj | < 1 one has
κ
∫
T
6∏
j=1
Γ
(
tjz
±1; p, q
)
Γ
(
z±2; p, q
) dz
z
=
∏
1≤j<k≤6
Γ(tjtk; p, q). (2.6)
In a special limit p→ 0 formula (2.6) is reduced to the Rahman q-beta integral [14].
In [10] the star-triangle relation (2.5) was used for a rigorous construction of an integral
operator solving the Yang–Baxter equation, which is currently the most complicated known
rank 1 solution of this important equation of mathematical physics. Earlier the functional form
of relation (2.5) was applied for building new integrable two-dimensional lattice spin systems,
see survey [6] and references therein.
Following [10], let us recall how equality (2.5) can be interpreted as the cubic Coxeter relation
for the permutation group S3. Consider three complex variables u = (u1, u2, u3) and define the
elementary generators of the permutation group
s1u = (u2, u1, u3), s2u = (u1, u3, u2).
Let us denote
s = e2πia, t = e2πib, a = u2 − u3, b = u1 − u2.
Now one can introduce the integral operators S1 and S2, acting as
[S1(u)f ](w) := M(t)wxf(x), [S2(u)f ](w) := D(s; y, w)f(w).
Here one can write symbolically S2 as an integral operator with a singular (delta-function) kernel,
S2(u)wx = D(s; y, x)1lwx, where 1l := 1lwx is the unit operator acting in the infinite-dimensional
space of complex functions, 1lwxf(x) = f(w).
The operators S1(u) and S2(u) depend not on all variables uj , but on their particular com-
binations, namely:
S1(u) = S1(u1 − u2), S2(u) = S2(u2 − u3).
One takes the following multiplication rule for these operators [9, 10]:
SjSk := Sj(sku)Sk(u), j, k = 1, 2.
Then it is easy to see that
S2
2 := S2(−a)S2(a) = 1l, a = u2 − u3.
As mentioned above, for functions satisfying the restriction f
(
z−1
)
= f(z) and under some
constraints on the values of a and the contours of integration [11, 21], one has also the inversion
relation
S2
1 := S1(−b)S1(b) = 1l, b = u1 − u2.
Matrix Bailey Lemma and the Star-Triangle Relation 5
Using the accepted multiplication rule, the relation (2.5) can be written as follows [10]
S1S2S1 := S1(s2s1u)S2(s1u)S1(u) = S1(a)S2(a+ b)S1(b)
= S2(b)S1(a+ b)S2(a) = S2(s1s2u)S1(s2u)S2(u) =: S2S1S2.
So, the integral Bailey lemma operators D and M determine the generators of elementary
permutations of parameters acting as integral operators in the infinite-dimensional space of
complex functions, and the basic relation (2.5) is equivalent to the cubic Coxeter relation
S1S2S1 = S2S1S2.
3 Matrix reduction of the integral Bailey pairs
We consider possible reductions of the integral operator M(t)xz to simpler matrix forms. For
that we discuss first analytic continuation of the action of M(t)xz. The kernel of this integral
operator has poles in the integration variable at the following points
z =
{
tx±1qapb, t−1x±1q−ap−b
}
a,b∈Z≥0
.
Using the inversion formula for the Γ-function, one finds
1
Γ
(
z±2; p, q
) = θ
(
z2; q
)
θ
(
z−2; p
)
.
Therefore zeroes of the kernel are located at the points:
z =
{
± qa/2,±pa/2
}
a∈Z ∪
{
t−1x±1qa+1pb+1, tx±1q−a−1p−b−1
}
a,b∈Z≥0
.
Evidently, at the point z = 0 there is an essential singularity. In the definition (2.1) we assumed
that
∣∣tw±1∣∣ < 1, in which case the contour T separates all geometric sequences of poles accumu-
lating to zero from their reciprocals that go to infinity. We can deform T to different contours as
long as we do not cross poles of the kernel and integrated function and this provides the desired
analytic continuation of the action of M(t)xz to wider ranges of the variables t and x.
Now we assume that the function α(z) in (2.1) satisfies the condition α(z) = α
(
z−1
)
. The
reason for such a restriction comes from the fact that the image of operator M(t)xz obeys this
symmetry. Moreover, the inversion relation M−1(t)xz = M
(
t−1
)
xz
holds only for such functions.
Let us suppose that the poles of α(z) are simple and that they are located at the points{
z = z0q
m
}
m=0,1,...,N
with |z0| < 1 and their z → 1/z reciprocals. Now we deform the contour
of integration T to a contour C that crosses these poles of α(z) without touching any other pole.
From the Cauchy theorem we obtain the equality
β(x) = M(t)xzα(z) = κ
∫
C
Γ
(
tx±1z±1; p, q
)
Γ
(
t2, z±2; p, q
) α(z)
dz
z
+ 4πiκ
N∑
m=0
lim
z→z0qm
(
1− z0qm/z
)Γ
(
tx±1z±1; p, q
)
Γ
(
t2, z±2; p, q
) α(z)
= κ
∫
C
Γ
(
tx±1z±1; p, q
)
Γ
(
t2, z±2; p, q
) α(z)
dz
z
+ 4πiκ
N∑
m=0
Γ
(
tx±1(z0q
m)±1; p, q
)
Γ
(
t2, (z0qm)±2; p, q
) α̃m, (3.1)
where α̃m = lim
z→z0qm
(
1− z0qm/z
)
α(z) are the residues of poles of α(z)/z.
In order to match with the notation of [18] we set
z0 = a1/2, t =
k1/2
a1/2
, x = ε−1k1/2qN ,
6 K.Yu. Magadov and V.P. Spiridonov
Re(z)
Im(z)
n
0
tx
txq
n
C
C
Pinching of the contour
takes place here ...
... and here
z0
0
z q
t x-1 -1
t x q-1 -1 -n
n tx -1
tx
-1
q
t x-1
z
0
-1
-1
0z q
-n
qt x
-1 -n
Figure 1. Pinching of the integration contour C by sequences of poles
{
z = tx−1qn, z−1
0 q−m
}
and{
z = t−1xq−n, z0q
m
}
in the limit ε→ 1.
where N is the same integer number as in (3.1). The sum of residues takes the following form
4πiκ
N∑
m=0
Γ
(
εqm−N , kε−1qm+N , a−1εq−m−N , ka−1ε−1qN−m; p, q
)
Γ
(
k/a,
(
aq2m
)±1
; p, q
) α̃m.
For ε sufficiently close to 1 the condition |tx−1| < 1 will be broken and in the definition of our
M -operator we have to assume that the contour C does not cross corresponding poles lying
outside T. However, in the limit ε → 1 we have the following geometrical picture: before we
force the contour C to cross poles of α(z) it gets pinched by 2(N + 1) poles of α(z) from one
side (the doubling comes from the symmetry z → 1/z) and 2(N + 1) poles of the kernel of
M -operator from the other side, as described on the Fig. 1 for the choice 0 < q < 1.
The multiplier Γ
(
tx−1z0q
m
)
= Γ
(
εqm−N
)
in the residue sum is diverging in the limit ε→ 1.
To remove this divergency we multiply the above expression by 1− ε and take the needed limit.
Denote
β̃N (a, k) := lim
ε→1
(1− ε)β(x)
and use the relation
lim
ε→1
(1− ε)Γ
(
εqm−N ; p, q
)
=
1
θ
(
qm−N , qm−N+1, . . . , q−1; p
)
(q; q)∞(p; p)∞
,
with the convention θ(a, b; p) := θ(a; p)θ(b; p). Since the integral part in (3.1) is finite for ε→ 1,
multiplication by 1− ε and subsequent limit ε→ 1 kills it completely. As a result we obtain
β̃N (a, k) =
N∑
m=0
Γ
(
kqN+m; p, q
)
Γ
(
k
aq
N−m; p, q
)
Γ
(
a−1q−N−m; p, q
)
θ
(
qm−N , qm−N+1, . . . , q−1; p
)
Γ(k/a; p, q)Γ
((
aq2m
)±1
; p, q
) α̃m(a, k). (3.2)
Matrix Bailey Lemma and the Star-Triangle Relation 7
Now we apply the inversion relation (2.2), the equation
Γ
(
zqn; p, q
)
= θ(z; p)nΓ(z; p, q),
where
θ(z; p)n = θ(z)n =
n−1∏
j=0
θ
(
zqj ; p
)
, n > 0,
−n∏
j=1
1
θ
(
zq−j ; p
) , n < 0
is the elliptic Pochhammer symbol, and use the quasiperiodicity properties of θ(z; q)-function.
After corresponding simplifications, relation (3.2) reduces to the following form
β̃N (a, k) =
Γ(k; p, q)
Γ(a; p, q)
N∑
m=0
MNm(a, k)qN(N+1)−m(m−1)α̃m(a, k),
where
MNm(a, k) =
θ(k)N+mθ
(
k
a
)
N−m
θ(qa)N+mθ(q)N−m
θ
(
aq2m; p
)
θ(a; p)
aN−m. (3.3)
Let us renormalize the sequences α̃n and β̃n,
βN (a, k) := β̃N (a, k)q−N(N+1), αm(a, k) := α̃m(a, k)q−m(m+1)Γ(k; p, q)
Γ(a; p, q)
.
Finally, we obtain the matrix relation
βN (a, k) =
N∑
m=0
MNm(a, k)αm(a, k). (3.4)
This is precisely the definition of the discrete Bailey pairs (αn, βn) introduced in [18], where
the matrix MNm(a, k) has appeared for the first time. For p → 0 it reduces to the Bressoud
matrix [7]. Note that due to the presence of the factor 1/θ(q)N−m matrix elements (3.3) vanish
for m > N , which reduces the matrix to a triangular form.
Thus we have shown how in the appropriate limit the elliptic Fourier transformation operator
is reduced to a matrix action (a discrete finite-difference operator). A different type of reduction
of the M -operator has been considered in [10]. Namely, in the limit t2 → q−N , N ∈ Z≥0,
sequences of poles z = txqn and z = t−1xq−m and their reciprocals start to pinch the contour of
integration and there emerges also the vanishing factor 1/Γ(t2; p, q). Forcing the latter contour
to cross half of these poles and taking the limit one finds
[M(t)f ](x) =
Γ
(
x−2; p, q
)
Γ
(
t−2x−2; p, q
) N∑
k=0
θ
(
(tx)2q2k; p
)
θ
(
(tx)2; p
) θ
(
t2, (tx)2; p
)
k
θ
(
q, qx2; p
)
k
f
(
tqkx
)
t4kx2kqk2
,
where t = ±q−N/2. Note that this is an analytical finite-difference operator acting on functions
of complex variable x, whereas in our present case we have a matrix acting on sequences of
numbers.
In [10] an even more complicated reduction associated with the limit t2 → q−Np−M , N,M ∈
Z≥0, was considered. It results in the finite-difference operator involving scalings by two inde-
pendent variables q and p which we do not present here. We only note that a similar complicated
two-base reduction can be applied to the matrix reductions considered in this note. This should
lead to a “doubling” of our matrices, but we skip such an opportunity for now.
8 K.Yu. Magadov and V.P. Spiridonov
4 Matrix reduction of the integral Bailey lemma
Denoting as α and β columns formed by αn and βn, n = 0, . . . , N , we can rewrite relation (3.4)
in the matrix form β(a, k) = M(a, k)α(a, k). Define the diagonal matrix
Dnm(a; b, c) = Dm(a; b, c)δnm, Dm(a; b, c) =
θ(b, c)m
θ(aq/b, aq/c)m
(aq
bc
)m
.
The discrete Bailey lemma of [18] states that from a given Bailey pair α(a, t) and β(a, t) one
can form infinitely many such pairs using the following transformation rules
α′(a, k) = D(a; b, c)α(a, t),
β′(a, k) = D(k; qt/b, qt/c)M(t, k)D(t; b, c)β(a, t),
where k, b, c are arbitrary new parameters satisfying the constraint kbc = qat. Note that the
parameter t in the Bailey pairs is replaced by a new variable k. This means validity of the
relations β′(a, k) = M(a, k)α′(a, k), which leads to the following matrix identity
M(a, k)D(a; b, c)M(t, a) = D(k; qt/b, qt/c)M(t, k)D(t; b, c). (4.1)
After substitution of the explicit expressions for matrices, one can see that it holds true due to
the Frenkel–Turaev summation formula [12].
Let us reduce now relation (2.5) and show how it generates identity (4.1). We consider the
left-hand side of equality (2.5) and deform the contour of integration for z-variable T to Cz in
the same way as we did before by crossing the poles of α(z) at z = z0q
m, m = 0, . . . , N, and
their reciprocals. For simplicity of the analysis of the structure of pole sequences we will assume
that 0 < q < 1. In the final results it will be easy to make analytic continuation to arbitrary
complex q. After the residue calculus we obtain
M(s)wxD(st; y, x)M(t)xzα(z) = 4πiκ2
∫
T
dx
x
Γ
(
sw±1x±1; p, q
)
Γ
(
s2, x±2; p, q
) Γ
(√
pqs−1t−1y±1x±1; p, q
)
×
N∑
m=0
α̃m
Γ
(
tx±1
(
z0q
m
)±1
; p, q
)
Γ
(
t2,
(
z0qm
)±2
; p, q
) + κ2
∫
T
dx
x
Γ
(
sw±1x±1; p, q
)
Γ
(
s2, x±2; p, q
)
× Γ
(√
pqs−1t−1y±1x±1; p, q
) ∫
Cz
dz
z
Γ
(
tx±1z±1; p, q
)
Γ
(
t2, z±2; p, q
) α(z), (4.2)
where we denoted as before α̃m = lim
z→z0qm
(
1− z0qm/z
)
α(z).
Consider the first term of this expression. Its integrand has poles at the points
x = sw±1pjqk,
√
pqs−1t−1y±1pjqk, t
(
z0q
m
)±1
pjqk, j, k ∈ Z≥0,
and their x → 1/x reciprocals. Now we deform the x-integration contour from T to Cx which
crosses the poles x = tz0q
n, n = m,m+1, . . . , N , and their x→ 1/x reciprocals without touching
any other singularity. Again applying the residue calculus we come to the expression
M(s)wxD(st; y, x)M(t)xzα(z)
= (4πiκ)2
N∑
m=0
N∑
n=m
α̃m lim
x→tz0qn
(
1− tz0q
n
x
)
Γ
(
tz0q
m
x
; p, q
)
Γ
(
sw±1x±1; p, q
)
Γ
(
s2, x±2; p, q
)
× Γ
(√
pqs−1t−1y±1x±1; p, q
)Γ
(
tx
(
z0q
m
)±1
; p, q
)
Γ
(
tx−1
(
z0q
m
)−1
; p, q
)
Γ
(
t2,
(
z0qm
)±2
; p, q
) (4.3)
Matrix Bailey Lemma and the Star-Triangle Relation 9
+ 4πiκ2
∫
Cx
dx
x
Γ
(
sw±1x±1; p, q
)
Γ
(
s2, x±2; p, q
) Γ
(√
pqs−1t−1y±1x±1; p, q
) N∑
m=0
α̃m
Γ
(
tx±1
(
z0q
m
)±1
; p, q
)
Γ
(
t2,
(
z0qm
)±2
; p, q
)
+κ2
∫
T
dx
x
Γ
(
sw±1x±1; p, q
)
Γ
(
s2, x±2; p, q
) Γ
(√
pqs−1t−1y±1x±1; p, q
) ∫
Cz
dz
z
Γ
(
tx±1z±1; p, q
)
Γ
(
t2, z±2; p, q
) α(z).
Using the relations
lim
x→tz0qn
(
1− tz0q
n
x
)
Γ
(
tz0q
m
x
; p, q
)
=
1
(p; p)∞(q; q)∞θ
(
qm−n
)
n−m
,
θ
(
qm−n
)
n−m = (−1)n−mq−
(n−m)(n−m+1)
2 θ(q)n−m,
one can see that there is the multiplier 1/θ(q)n−m in the sum which vanishes for n < m.
Therefore the summation over n actually starts from zero and we can interchange summations
over m and n.
In the reduction of Bailey pairs considered in the previous section we introduced an ε-
parametrization of the external coordinate of the M -operator with the subsequent limit ε→ 1,
which converted the integral operator to a matrix. In the present case it works as follows.
Namely, we fix
w = ε−1stz0q
N , i.e., Γ
(
sw−1tz0q
n; p, q
)
= Γ
(
εq−N+n; p, q
)
,
multiply the whole expression by 1− ε, and take the limit ε→ 1. It can be seen that during the
deformation of x-variable integration contour T to Cx in passing from (4.2) to (4.3) the limit
ε → 1 leads to pinching of this contour precisely as it took place in the previous section (see
Fig. 1), i.e., computation of the residues at x = tz0q
n is inevitable. In the same way as before it
shows that the second term in (4.3) with the integral over x remains finite for ε→ 1, because Cx
crossed already dangerous poles without touching other singularities. Similar situation holds
with the third term in (4.3) containing integrals over x and z. Namely, since
∣∣sw−1∣∣ becomes
bigger than 1, we have to deform x-integration contour appropriately not to cross over any pole.
This is possible due to the choice of the z-variable integration contour Cz, which escapes the
dangerous z-values z =
(
z0q
m
)±1
, and a special choice of the function α(z). The constraint
0 < q < 1 helps to make such a statement transparent. Therefore after multiplication by 1 − ε
both integral terms in (4.3) vanish in the limit ε→ 1.
As a result, we find the following replacements of the integral operator factors
M(t)xz →
Γ
(
t2z20 ; p, q
)
Γ
(
z20 ; p, q
) N∑
m=0
Mnm
(
z20 , t
2z20
)
qn(n+1)−m(m+1),
M(s)wx →
Γ
(
s2x20; p, q
)
Γ
(
x20; p, q
) N∑
n=0
MNn
(
x20, s
2x20
)
qN(N+1)−n(n+1),
where x0 = tz0. For reducing the D-function multiplier, we have
D(st; y, x)→ Γ
(√
pqs−1t−1y±1
(
x0q
n
)±1
; p, q
)
.
After denoting a := x20, k := s2x20, t̃ := z20 we obtain
D(st; y, x)→ Γ(b, pc; p, q)
Γ(qa/c, pqa/b; p, q)
Dn(a; b, c),
Dn(a; b, c) :=
θ(b)nθ(c)n
θ(qa/c)nθ(qa/b)n
(qa
bc
)n
,
10 K.Yu. Magadov and V.P. Spiridonov
where we have introduced two parameters b and c,
b :=
√
pqt̃a
k
y, c :=
√
qt̃a
pk
y−1, (4.4)
satisfying the important product rule
kbc = qat̃.
The total expression for the left-hand side takes the following form
M(s)wxD(st; y, x)M(t)xzα(z)
→ Γ(k, b, pc; p, q)qN(N+1)
Γ
(
t̃, qa/c, pqa/b; p, q
) N∑
n,m=0
MNn(a, k)Dn(a; b, c)Mnm
(
t̃, a
)
q−m(m+1)α̃m. (4.5)
Now we consider the right-hand side of equality (2.5). Similarly to the previous case, we
deform the integration contour of M(st)wz from T to such a contour C that crosses the poles
z = z0q
m, m = 0, . . . , N, of α(z) and does not touch other singularities. As a result, we obtain
D(t; y, w)M(st)wzD(s; y, z)α(z) = κ
∫
T
Γ
(√
pqt−1y±1w±1; p, q
)
×
Γ
(
stw±1z±1; p, q
)
Γ
(
s2t2, z±2; p, q
) Γ
(√
pqs−1y±1z±1; p, q
)
α(z)
dz
z
= κ
∫
C
Γ
(√
pqt−1y±1w±1; p, q
)
×
Γ
(
stw±1z±1; p, q
)
Γ
(
s2t2, z±2; p, q
) Γ
(√
pqs−1y±1z±1; p, q
)
α(z)
dz
z
+ Γ
(√
pqt−1y±1w±1; p, q
)
× 4πiκ
N∑
m=0
α̃m
Γ
(
stw±1
(
z0q
m
)±1
; p, q
)
Γ
(
s2t2,
(
z0qm
)±2
; p, q
) Γ
(√
pqs−1y±1
(
z0q
m
)±1
; p, q
)
. (4.6)
We set as before w = ε−1stz0q
N , multiply the whole expression by 1 − ε and take the limit
ε → 1. Again the integral part in (4.6) is finite and disappears in this limit. After detailed
computations, we find the following replacements of the operator factors
D(t; y, w)→
Γ
(
qt̃/c, pqt̃/b; p, q
)
Γ
(
kb/t̃, pkc/t̃; p, q
)DN
(
k; qt̃/c, qt̃/b
)
,
M(st)wz →
Γ(k; p, q)
Γ
(
t̃; p, q
) N∑
m=0
MNm
(
t̃, k
)
qN(N+1)−m(m+1),
D(s; y, z)→ Γ(b, pc; p, q)
Γ
(
qt̃/c, pqt̃/b; p, q
)Dm(t̃; b, c).
The whole right-hand side expression takes the form
lim
ε→1
(1− ε) r.h.s. expression =
N∑
m=0
Γ(k, b, pc; p, q)
Γ
(
t̃, qac ,
pqa
b ; p, q
)qN(N+1)−m(m+1)
×DN
(
k; qt̃/c, qt̃/b
)
MNm(t̃, k)Dm
(
t̃; b, c
)
α̃m. (4.7)
Equating expressions (4.5) and (4.7), cancelling common prefactors independent of the summa-
tion indices, and denoting αm := q−m(m+1)α̃m we obtain the identity
MNn(a, k)Dn(a; b, c)Mnm
(
t̃, a
)
αm = DN
(
k; qt̃/c, qt̃/b
)
MNm
(
t̃, k
)
Dm
(
t̃; b, c
)
αm,
Matrix Bailey Lemma and the Star-Triangle Relation 11
where we assume that there is a summation over repeated matrix indices. After removing
arbitrary numbers αm we see that this is precisely the key matrix Bailey lemma identity of [18]
as described above (4.1). The matrix relation (4.1) was explicitly written in such a form in [20].
Our result consists in the demonstration that it is nothing else than the direct reduction of the
integral star-triangle relation to the discrete form.
5 Matrix realization of the Coxeter relations
Let us rewrite now the matrix Bailey lemma identity (4.1) as a Coxeter relation of the per-
mutation group S3. We take the three parameters set t =
(
t̃, a, k
)
and introduce generators of
elementary permutations sj :
s1
(
t̃, a, k
)
=
(
a, t̃, k
)
, s2
(
t̃, a, k
)
=
(
t̃, k, a
)
.
They satisfy simple quadratic Coxeter relations s2i = 1 and the cubic one s1s2s1 = s2s1s2.
Now we define two operators S1 and S2 as the following matrices
S1
(
t̃, a, k
)
:= M
(
t̃, a
)
, S2
(
t̃, a, k
)
:= D
(
t̃; b, c
)
,
where b and c are fixed in (4.4). These operators are assumed to use the twisted multiplication
rule
SiSj := Si(sjt)Sj(t), i, j = 1, 2.
Then one can easily check that
S2
1 = M
(
a, t̃
)
M
(
t̃, a
)
= 1,
which follows from the result of [18]. Permutation of a and k leads to the changes b→ qt̃/c and
c→ qt̃/b. As a result, we have
S2
2 = D
(
t̃; qt̃/c, qt̃/b
)
D
(
t̃; b, c
)
= 1.
Consider now the following cubic combination of Sj-operators
S1S2S1 := S1(s2s1t)S2(s1t)S1(t) = S1
(
a, k, t̃
)
S2
(
a, t̃, k
)
S1
(
t̃, a, k
)
.
The far right S1-factor coincides with M(t̃, a). As the permutation of a with t̃ does not change b
and c, we have S2
(
a, t̃, k
)
= D(a; b, c). By the definition we have also S1
(
a, k, t̃
)
= M(a, k). So,
S1S2S1 coincides with the left-hand side of the key Bailey lemma equality (4.1):
S1S2S1 = M(a, k)D(a; b, c)M
(
t̃, a
)
.
Now we consider another cubic combination of Sj-operators
S2S1S2 := S2(s1s2t)S1(s2t)S2(t) = S2
(
k, t̃, a
)
S1
(
t̃, k, a
)
S2
(
t̃, a, k
)
.
By definition, S2
(
t̃, a, k
)
= D
(
t̃; b, c
)
and S1
(
t̃, k, a
)
= M
(
t̃, k
)
. The far left S2-operator can be
written in the form
S2
(
k, t̃, a
)
= D
k; y
√
pqkt̃
a
, y−1
√
qkt̃
pa
= D
(
k;
qt̃
c
,
qt̃
b
)
.
12 K.Yu. Magadov and V.P. Spiridonov
Therefore we have
S2S1S2 = D
(
k; qt̃/c, qt̃/b
)
M
(
t̃, k
)
D
(
t̃; b, c
)
.
Finally, we see that the cubic Coxeter relation
S1S2S1 = S2S1S2.
is nothing else than the key Bailey lemma identity (4.1).
To conclude, in this paper we have described direct reduction of the integral Bailey lemma
to the matrix form. We interpreted also the key matrix identity of the latter lemma as the
star-triangle relation, or the cubic Coxeter relation for permutation group S3.
As a next step of the development of derived results it is necessary to apply them to the
problem of reducing the Yang–Baxter equation built with the help of integral Bailey lemma
in [10]. For that it is necessary to understand whether it is possible to extend the construction
of S3-group generators given above to the group S4 with appropriate interpretation of the cor-
responding Coxeter relations. An investigation of the relation of matrix Bailey lemma with the
Sklyanin algebra is another problem for future considerations.
Another relevant subject concerns the supersymmetric field theories. Elliptic hypergeometric
integrals are known to describe superconformal indices of such theories in four dimensions.
The residue calculus corresponds in this picture to giving vacuum expectation values to some
fields or insertion of surface defects and the results of the present paper may be useful for that
interpretation as well. A comprehensive survey of this topic can be found in [16].
Acknowledgements
This work is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government
grant, ag. no. 14.641.31.0001.
References
[1] Andrews G.E., Multiple series Rogers–Ramanujan type identities, Pacific J. Math. 114 (1984), 267–283.
[2] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications,
Vol. 71, Cambridge University Press, Cambridge, 1999.
[3] Bailey W.N., Identities of the Rogers–Ramanujan type, Proc. London Math. Soc. 50 (1948), 1–10.
[4] Baxter R.J., Hard hexagons: exact solution, J. Phys. A: Math. Gen. 13 (1980), L61–L70.
[5] Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
[6] Bazhanov V.V., Kels A.P., Sergeev S.M., Quasi-classical expansion of the star-triangle relation and integrable
systems on quad-graphs, J. Phys. A: Math. Theor. 49 (2016), 464001, 44 pages, arXiv:1602.07076.
[7] Bressoud D.M., A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), 446–448.
[8] Derkachov S.E., Factorization of the R-matrix. I, J. Math. Sci. 143 (2007), 2773–2790, math.QA/0503396.
[9] Derkachov S.E., Karakhanyan D., Kirschner R., Yang–Baxter R-operators and parameter permutations,
Nuclear Phys. B 785 (2007), 263–285, hep-th/0703076.
[10] Derkachov S.E., Spiridonov V.P., Yang–Baxter equation, parameter permutations, and the elliptic beta
integral, Russian Math. Surveys 68 (2013), 1027–1072, arXiv:1205.3520.
[11] Derkachov S.E., Spiridonov V.P., Finite-dimensional representations of the elliptic modular double, Theoret.
and Math. Phys. 183 (2015), 597–618, arXiv:1310.7570.
[12] Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang–Baxter equation and modular hypergeometric
functions, in The Arnold–Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, 171–204.
[13] Paule P., On identities of the Rogers–Ramanujan type, J. Math. Anal. Appl. 107 (1985), 255–284.
https://doi.org/10.2140/pjm.1984.114.267
https://doi.org/10.1017/CBO9781107325937
https://doi.org/10.1112/plms/s2-50.1.1
https://doi.org/10.1088/0305-4470/13/3/007
https://doi.org/10.1088/1751-8113/49/46/464001
https://arxiv.org/abs/1602.07076
https://doi.org/10.2307/2044991
https://doi.org/10.1007/s10958-007-0164-8
https://arxiv.org/abs/math.QA/0503396
https://doi.org/10.1016/j.nuclphysb.2007.05.022
https://arxiv.org/abs/hep-th/0703076
https://doi.org/10.1070/rm2013v068n06abeh004869
https://arxiv.org/abs/1205.3520
https://doi.org/10.1007/s11232-015-0284-0
https://doi.org/10.1007/s11232-015-0284-0
https://arxiv.org/abs/1310.7570
https://doi.org/10.1007/978-1-4612-4122-5_9
https://doi.org/10.1016/0022-247X(85)90368-3
Matrix Bailey Lemma and the Star-Triangle Relation 13
[14] Rahman M., An integral representation of a 10ϕ9 and continuous bi-orthogonal 10ϕ9 rational functions,
Canad. J. Math. 38 (1986), 605–618.
[15] Rains E.M., Multivariate quadratic transformations and the interpolation kernel, SIGMA 14 (2018), 019,
69 pages, arXiv:1408.0305.
[16] Rastelli L., Razamat S.S., The supersymmetric index in four dimensions, J. Phys. A: Math. Theor. 50
(2017), 443013, 34 pages, arXiv:1608.02965.
[17] Spiridonov V.P., On the elliptic beta function, Russian Math. Surveys 56 (2001), 185–186.
[18] Spiridonov V.P., An elliptic incarnation of the Bailey chain, Int. Math. Res. Not. 2002 (2002), 1945–1977.
[19] Spiridonov V.P., A Bailey tree for integrals, Theoret. and Math. Phys. 139 (2004), 536–541,
math.CA/0312502.
[20] Spiridonov V.P., Essays on the theory of elliptic hypergeometric functions, Russian Math. Surveys 63 (2008),
405–472, arXiv:0805.3135.
[21] Spiridonov V.P., Warnaar S.O., Inversions of integral operators and elliptic beta integrals on root systems,
Adv. Math. 207 (2006), 91–132, math.CA/0411044.
[22] Takhtadzhan L.A., Faddeev L.D., The quantum method of inverse problem and the Heisenberg XYZ model,
Russian Math. Surveys 34 (1979), no. 5, 11–68.
[23] Warnaar S.O., 50 years of Bailey’s lemma, in Algebraic Combinatorics and Applications (Gößweinstein,
1999), Springer, Berlin, 2001, 333–347, arXiv:0910.2062.
[24] Warnaar S.O., Extensions of the well-poised and elliptic well-poised Bailey lemma, Indag. Math. (N.S.) 14
(2003), 571–588, math.CA/0309241.
[25] Zudilin W., Hypergeometric heritage of W.N. Bailey. With an appendix: Bailey’s letters to F. Dyson,
arXiv:1611.08806.
https://doi.org/10.4153/CJM-1986-030-6
https://doi.org/10.3842/SIGMA.2018.019
https://arxiv.org/abs/1408.0305
https://doi.org/10.1088/1751-8121/aa76a6
https://arxiv.org/abs/1608.02965
https://doi.org/10.1070/rm2001v056n01ABEH000374
https://doi.org/10.1155/S1073792802205127
https://doi.org/10.1023/B:TAMP.0000022745.45082.18
https://arxiv.org/abs/math.CA/0312502
https://doi.org/10.1070/RM2008v063n03ABEH004533
https://arxiv.org/abs/0805.3135
https://doi.org/10.1016/j.aim.2005.11.007
https://arxiv.org/abs/math.CA/0411044
https://doi.org/10.1070/RM1979v034n05ABEH003909
https://doi.org/10.1007/978-3-642-59448-9_23
https://arxiv.org/abs/0910.2062
https://doi.org/10.1016/S0019-3577(03)90061-9
https://arxiv.org/abs/math.CA/0309241
https://arxiv.org/abs/1611.08806
1 Introduction
2 Integral Bailey lemma and the star-triangle relation
3 Matrix reduction of the integral Bailey pairs
4 Matrix reduction of the integral Bailey lemma
5 Matrix realization of the Coxeter relations
References
|
| id | nasplib_isofts_kiev_ua-123456789-209883 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T12:55:39Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Magadov, K.Yu. Spiridonov, V.P. 2025-11-28T09:42:06Z 2018 Matrix Bailey Lemma and the Star-Triangle Relation / K.Yu. Magadov, V.P. Spiridonov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D60; 33E20 arXiv: 1810.10806 https://nasplib.isofts.kiev.ua/handle/123456789/209883 https://doi.org/10.3842/SIGMA.2018.121 We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus, we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group. This work is partially supported by the Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. no. 14.641.31.0001. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Matrix Bailey Lemma and the Star-Triangle Relation Article published earlier |
| spellingShingle | Matrix Bailey Lemma and the Star-Triangle Relation Magadov, K.Yu. Spiridonov, V.P. |
| title | Matrix Bailey Lemma and the Star-Triangle Relation |
| title_full | Matrix Bailey Lemma and the Star-Triangle Relation |
| title_fullStr | Matrix Bailey Lemma and the Star-Triangle Relation |
| title_full_unstemmed | Matrix Bailey Lemma and the Star-Triangle Relation |
| title_short | Matrix Bailey Lemma and the Star-Triangle Relation |
| title_sort | matrix bailey lemma and the star-triangle relation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209883 |
| work_keys_str_mv | AT magadovkyu matrixbaileylemmaandthestartrianglerelation AT spiridonovvp matrixbaileylemmaandthestartrianglerelation |