Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW
We obtain orthogonal polynomial self-duality functions for the multi-species version of the symmetric exclusion process (SEP(2)) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have > 1 species of particles. In addition, we allow up to 2 particle...
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| description | We obtain orthogonal polynomial self-duality functions for the multi-species version of the symmetric exclusion process (SEP(2)) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have > 1 species of particles. In addition, we allow up to 2 particles to occupy each site in the multi-species SEP(2). The duality functions for the multi-species SEP(2) and the multi-species IRW come from unitary intertwiners between different ∗-representations of the special linear Lie algebra ₙ₊₁ and the Heisenberg Lie algebra ₙ, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP(2) and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.
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| first_indexed | 2026-03-21T04:33:38Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 113, 11 pages
Orthogonal Polynomial Stochastic Duality Functions
for Multi-Species SEP(2j) and Multi-Species IRW
Zhengye ZHOU
Department of Mathematics, Texas A&M University, College Station, TX 77840, USA
E-mail: zyzhou@tamu.edu
Received October 16, 2021, in final form December 24, 2021; Published online December 26, 2021
https://doi.org/10.3842/SIGMA.2021.113
Abstract. We obtain orthogonal polynomial self-duality functions for multi-species version
of the symmetric exclusion process (SEP(2j)) and the independent random walker process
(IRW) on a finite undirected graph. In each process, we have n > 1 species of particles. In
addition, we allow up to 2j particles to occupy each site in the multi-species SEP(2j). The
duality functions for the multi-species SEP(2j) and the multi-species IRW come from unitary
intertwiners between different ∗-representations of the special linear Lie algebra sln+1 and
the Heisenberg Lie algebra hn, respectively. The analysis leads to multivariate Krawtchouk
polynomials as orthogonal duality functions for the multi-species SEP(2j) and homogeneous
products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.
Key words: orthogonal duality; multi-species SEP(2j); multi-species IRW
2020 Mathematics Subject Classification: 60K35
1 Introduction
In recent years, stochastic duality has been used as a powerful tool in the study of stochastic
processes (see, e.g., [3, 7, 15, 17]). More recently, orthogonal stochastic dualities were derived for
some classical interacting particle systems. For instance, the independent random walker process
(IRW) is self-dual with respect to Charlier polynomials [5, 12], and the symmetric exclusion
process (SEP(2j)) is self-dual with respect to Krawtchouk polynomials [5, 12]. Orthogonal
polynomial duality functions turn out to be useful in applications as they form a convenient
orthogonal basis in a suitable space of the systems’ observables. There are many applications of
orthogonal duality functions (see, e.g., [1, 2, 6]). In a series of previous works, several ways to find
orthogonal dualities were introduced. In [9], the two-terms recurrence relations of orthogonal
polynomials were used, method via generating functions was used in [16], while Lie algebra
representations and unitary intertwiners were used in [12]. In addition, two more approaches
were described in [5]. The first approach is based on unitary symmetries, another one is based
on scalar products of classical duality functions. In this paper, we make use of the method
introduced in [12].
We first study a model of interacting particle systems with symmetric jump rates. The
multi-species symmetric exclusion process is a generalization of the SEP(2j) to multi-species
systems, where we have up to 2j ∈ N particles allowed for each site and we have n > 1 species
particles in the system. It is worth mentioning that the multi-species SEP(2j) we consider
is closely related to other multi-species (multi-color) exclusion processes studied over the past
decades. For example, when j = 1
2 , it degenerates to a special case of the multi-color exclusion
processes studied in [4, 8]. This model also arises naturally as a special case of the multi-species
ASEP(q, j) studied in [14] when q = 1. Given the fact that the single-species SEP(2j) is self-dual
with respect to Krawtchouk polynomials, it’s expected that similar results could be found for the
mailto:zyzhou@tamu.edu
https://doi.org/10.3842/SIGMA.2021.113
2 Z. Zhou
multi-species SEP(2j). We prove that the multi-species SEP(2j) is self-dual with multivariate
Krawtchouk polynomials as duality functions.
Another process we study is the multi-species independent random walker, which can be
thought of as n > 1 independent copies of IRW evolving simultaneously. Although it is straight-
forward to obtain the duality functions using the independence property, it is still interesting
to show how the duality functions arise from representations of the nilpotent Heisenberg Lie
algebra hn.
The organization of this paper is as follows. In Section 2 we give an overview of the method
that we use to construct orthogonal duality functions. In Section 3 we obtain the orthogonal
duality functions for the multi-species SEP(2j) and in Section 4 for the multi-species IRW.
2 Background
In this section we describe the method to obtain the orthogonal dualities which was introduced
in [12]. We start by recalling the definition of stochastic duality.
Definition 2.1. Two Markov processes st and s′t on state spaces S and S′ are dual with respect
to duality function D(·, ·) on S×S′ if
Es[D(st, s
′)] = E′
s′ [D(s, s′t)] for all s ∈ S, s′ ∈ S′, and t > 0,
where Es denotes expectation with respect to the law of st with s0 = s and similarly for E′
s′ .
If s′t is a copy of st, we say that the process st is self-dual.
In most relevant examples, duality could also be stated at the level of Markov generators.
We say that generator L1 is dual to L2 with respect to duality function D(·, ·) if for all s and s′,
[L1D(·, s′)](s) = [L2D(s, ·)](s′).
If L1 = L2, we have self-duality.
Let g = (g, [·, ·]) be a complex Lie algebra with a ∗-structure, i.e., there exists an involution
∗ : X −→ X∗ such that for any X,Y ∈ g, a, b ∈ C,
(aX + bY )∗ = aX∗ + bY ∗, [X,Y ]∗ = [Y ∗, X∗].
Let U(g) be the universal enveloping algebra of g.
Given a Hilbert space (H, ⟨·, ·⟩) and a representation ρ of g on H, we call ρ a ∗-representation
if for any f, g ∈ H and any X ∈ g,
⟨ρ(X)f, g⟩ = ⟨f, ρ(X∗)g⟩.
Suppose we have state spaces Ω1 and Ω2 of configurations on L sites given by Ω1 = E1×· · ·×EL
and Ω2 = F1×· · ·×FL. Let µ = µ1⊗· · ·⊗µL and ν = ν1⊗· · ·⊗ νL be product measures on Ω1
and Ω2.
For 1 ≤ x ≤ L, let ρx and σx be unitarily equivalent ∗-representations of a Lie algebra g
on L2(Ex, µx) and L2(Fx, νx), respectively. Then ρ = ρ1 ⊗ · · · ⊗ ρL and σ = σ1 ⊗ · · · ⊗ σL are
∗-representations of g. We assume that the corresponding unitary intertwiner Λx : L
2(Ex, µx) −→
L2(Fx, νx) has the following form:
(Λxf)(z2) =
∫
Ex
f(z1)Kx(z1, z2) dµx(z1), for νx-almost all z2 ∈ Fx,
for some kernel Kx ∈ L2(Ex × Fx, µx ⊗ νx) satisfying the relation
[ρx(X
∗)Kx(·, z2)](z1) = [σx(X)Kx(z1, ·)](z2), (z1, z2) ∈ Ex × Fx, X ∈ g.
With all the above structures, the following theorem provides a way to construct duality
functions.
Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) 3
Theorem 2.2 ([12]). Suppose L1 and L2 are self-adjoint operators on L2(Ω1, µ) and L2(Ω2, ν),
respectively, given by
L1 = ρ(Y ), L2 = σ(Y ),
for some self-adjoint Y ∈ U(g)⊗L. Then L1 and L2 are in duality, with duality function
D(z1, z2) =
L∏
x=1
Kx(z1x, z2x), z1 = (z11, . . . , z1L) ∈ Ω1, z2 = (z21, . . . , z2L) ∈ Ω2.
3 Multi-species SEP(2j) and Lie algebra sln+1
In this section, we study the multi-species version of the SEP(2j) with 2j ∈ N on a finite
undirected graph G = (V,E), where V = {1, . . . , L} with L ∈ N and L > 2 is the set of sites
(vertices) and E is the set of edges. In what follows, we write site x ∈ G instead of mentioning V
for ease of notation.
The state space S(n, 2j,G) of particle configurations consists of variables ξ =
(
ξxi : 0 ≤ i ≤ n,
x ∈ G
)
, where ξxi denotes the number of particles of species i at site x, and
ξx =
(
ξx0 , . . . , ξ
x
n
)
∈ Ω2j :=
{
ξ = (ξ0, . . . , ξn)
∣∣∣∣∣
n∑
i=0
ξi = 2j, ξi ≥ 0
}
for any site x ∈ G. One can think of ξx0 as the number of holes at site x.
Definition 3.1. The generator of the multi-species SEP(2j) on a finite undirected graph G =
(V,E) is given by
Lf(ξ) =
∑
edge{x,y}∈E
Lx,yf(ξ),
Lx,yf(ξ) =
∑
0≤k<l≤n
ξxl ξ
y
k
[
f
(
ξx,yl,k
)
− f(ξ)
]
+ ξyl ξ
x
k
[
f
(
ξy,xl,k
)
− f(ξ)
]
,
(3.1)
where ξx,yl,k denotes the particle configuration obtained by switching a particle of the lth species
at site x with a particle of the kth species at site y if ξx,yl,k ∈ S(n, 2j,G).
Note that when n = 1, this process reduces to the single-species SEP(2j) defined in [10].
Suppose p = (p0, . . . , pn) is a probability distribution, the multinomial distribution on Ω2j is
defined as
wp(ξ) =
(
2j
ξ
) n∏
i=0
pξii ,
where
(
2j
ξ
)
denotes the multinomial coefficient (2j)!∏n
i=0 ξi!
. Following a simple detailed balance
computation, we can show that the product measure with marginals wp(ξ) for any fixed p being
a distribution is a reversible measure of the multi-species SEP(2j), i.e., ⊗Gwp is a reversible
measure of the multi-species SEP(2j) when p is the same for all sites.
3.1 Multivariate Krawtchouk polynomials and Lie algebra sln+1
First, we introduce the n-variable Krawtchouk polynomials defined by Griffiths [11]. We shall
adopt the notation of Iliev [13] in the following.
4 Z. Zhou
Definition 3.2. Let Kn be the set of 4-tuples
(
ν, P, P̂ , U
)
such that P , P̂ , U are (n+1)×(n+1)
complex matrices satisfying the following conditions:
(1) P = diag(p0, . . . , pn), P̂ = diag(p̂0, . . . , p̂n) and p0 = p̂0 =
1
ν ̸= 0,
(2) U = (ukl)0≤k,l≤n with Uk0 = U0k = 1 for all 0 ≤ k ≤ n,
(3) νPUP̂UT = In+1.
It follows from the above definition that p = (p0, . . . , pn) and p̂ = (p̂0, . . . , p̂n) satisfy that∑n
k=0 pk =
∑n
k=0 p̂k = 1 and pk, p̂k ̸= 0 for any k.
For all points κ ∈ Kn, Griffith constructed multivariate Krawtchouk polynomials using a gen-
erating function as follows.
Definition 3.3 ([11]). For ξ, η ∈ Ω2j and κ ∈ Kn, the multivariate Krawtchouk polynomial
K(ξ, η, κ, j) is defined by
∑
ξ∈Ω2j
(
2j
ξ
)
K(ξ, η, κ, j)zξ11 · · · zξnn =
n∏
k=0
(
1 +
n∑
l=1
uklzl
)ηk
.
Although it’s not obvious to tell from the generating function, K(ξ, η, κ, j) depends on P
and P̂ because the matrix U ∈ κ satisfies the condition (3) in Definition 3.2.
In what follows, we fix a 4-tuple κ ∈ Kn as in Definition 3.2, we also write K(ξ, η, κ, j)
as K(ξ, η) for simplicity.
In [13], Iliev interpreted multivariate Krawtchouk polynomials with representations of the
Lie algebra sln+1. We recall some of the essential results. Let’s start by introducing some
basic notations. Let z0, . . . , zn be mutually commuting variables, we set z = (z0, . . . , zn). For
each ξ = (ξ0, . . . , ξn) ∈ Ω2j , we denote zξ = zξ00 zξ11 · · · zξnn and ξ! = ξ0! · · · ξn!. Also define
V2j = span
{
zξ|ξ ∈ Ω2j
}
⊂ C[z], which is the space consisting of all homogeneous complex
polynomials of total degree 2j.
Let In+1 denote the (n + 1) × (n + 1) identity matrix. For 0 ≤ k, l ≤ n, let ek,l denote the
(n+1)×(n+1) matrix with (k, l)th entry 1 and other entries 0. The special linear Lie algebra of
order n+1 denoted by sln+1 consists of (n+1)×(n+1) matrices with trace zero and has the Lie
bracket [X,Y ] = XY −Y X. It has basis {ekl}0≤k ̸=l≤n and {hl}0<l≤n, where hl = ell− 1
n+1In+1.
The ∗-structure of sln+1 is given by
e∗kl = elk, k ̸= l, h∗l = hl. (3.2)
Let gl(V2j) denote the space of endomorphisms of V2j , we consider the representation ρ : sln+1
−→ gl(V2j) defined by
ρekl = zk∂zl, k ̸= l, ρhl = zl∂zl −
2j
n+ 1
. (3.3)
Next, define an antiautomorphism a on sln+1 by a(X) = P̂XTP̂−1. It follows easily that
a(ekl) =
p̂l
p̂k
elk, k ̸= l, a(hl) = hl. (3.4)
We define a symmetric bilinear form ⟨ , ⟩κ on V2j by
〈
zξ, zη
〉
κ
= δξ,η
ξ!
p̂ξ
θ2j .
Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) 5
Then it is easy to check that for any X ∈ sln+1 and v1, v2 ∈ V2j
⟨ρXv1, v2⟩κ = ⟨v1, ρa(X)v2⟩κ. (3.5)
Let R be the matrix
R = θ̂P̂UT, (3.6)
where θ̂ ∈ C such that det(R) = 1. Next, we define ẑ = (ẑ0, . . . , ẑn) by ẑ = zR.
Lemma 3.4. Define operator AdR on sln+1 by AdR(X) = R−1XR, where R = (rkl)0≤k,l≤n is
defined in equation (3.6). Then AdR is a Lie algebra automorphism of sln+1.
Proof. It can be checked directly. ■
Last, we list some properties of the multivariate Krawtchouk polynomial whose proof could
be found in [13].
Proposition 3.5 ([13, Corollary 5.2]). For ξ, η ∈ Ω2j, the multivariate polynomial K has the
following bilinear form,
K(ξ, η) =
p2j0
(2j)!
〈
zξ, ẑη
〉
κ
.
Proposition 3.6 ([13, Corollary 5.3]). For ξ, η, ζ ∈ Ω2j we have the following relations:
∑
ξ∈Ω2j
K(ξ, η)K(ξ, ζ)wp̂(ξ) =
p2j0 δη,ζ
wp(η)
,
∑
ξ∈Ω2j
K(η, ξ)K(ζ, ξ)wp(ξ) =
p2j0 δη,ζ
wp̂(η)
.
Remark 3.7. If U ∈ κ is a real matrix, then it follows from the generating function that
the multivariate Krawtchouk polynomial is real valued. In this case, Proposition 3.6 is the
orthogonality relation for the multivariate Krawtchouk polynomial in l2(wp) and l2(wp̂).
3.2 Self-duality of the multi-species SEP(2j)
In this subsection, we show that the multi-species SEP(2j) is self dual with respect to duality
functions given by homogeneous products of multivariate Krawtchouk polynomials. Suppose p
and p̂ in the 4-tuple κ ∈ Kn as in Definition 3.2 are both probability measures.
Let l2(wp) be a Hilbert space with inner product (f, g)p =
∑
ξ∈Ω2j
f(ξ)g(ξ)wp(ξ). Now we
define a ∗-representations ρp of sln+1 on l2(wp) by
ρp(ekl)f(ξ) =
√
pk
pl
ξlf
(
ξ−1,+1
l,k
)
for 0 ≤ k ̸= l ≤ n,
ρp(hl)f(ξ) =
(
ξl −
2j
n+ 1
)
f(ξ) for 0 < l ≤ n,
where ξ+1,−1
l,k represents the variable with ξl increased by 1 and ξk decreased by 1. Recalling
the ∗-structure defined in equation (3.2), it is straightforward to check that (ρp(X)f, g)p =
(f, ρp(X
∗)g)p for all X ∈ sln+1.
Next, we introduce another non-trivial ∗-representation σp of sln+1 on l2(wp) that is unitarily
equivalent to ρp̂.
6 Z. Zhou
Definition 3.8. For each ρp̂, define a corresponding representation by σp = ρp̂◦AdR, where AdR
is the automorphism defined in Lemma 3.4.
Proposition 3.9. If the matrix U in the 4-tuple κ ∈ Kn is real, then the representation σp
defined in Definition 3.8 is a ∗-representation of sln+1 on l2(wp).
Proof. By definitions of the matrices U and R, when U is a real matrix, then R and R−1 are
all real matrices. For ease of notation, we write Q = R−1 = (qi,m)0≤i,m≤n. Then, computing σp
explicitly, we have
σp(eim)f(η) =
√
p̂i
p̂m
n∑
k,l=0
qkirmlηlf
(
η+1,−1
k,l
)
. (3.7)
Next we verify that for any X ∈ sln+1, (σp(X)f(η), g(η))p = (f(η), σp(X
∗)g(η))p. First, we
plug equation (3.7) in the inner products, when i ̸= m,
(
σp(eim)f(η), g(η)
)
p
=
√
p̂i
p̂m
n∑
k,l=0
qkirml
(
ηlf
(
η+1,−1
k,l
)
, g(η)
)
p
,
(
f(η), σp(emi)g(η)
)
p
=
√
p̂m
p̂i
n∑
k,l=0
qkmril
(
f(η), ηlg
(
η+1,−1
k,l
))
p
. (3.8)
By switching k and l in equation (3.8), we have that
(
f(η), σp(emi)g(η)
)
p
=
√
p̂m
p̂i
n∑
k,l=0
qlmrik
(
ηkf(η), g
(
η+1,−1
l,k
))
p
.
Now define η̃ = η+1,−1
l,k , we get
(
f(η), σp(emi)g(η)
)
p
=
√
p̂m
p̂i
n∑
k,l=0
qlmrik
pk
pl
(
η̃lf
(
η̃−1,+1
l,k
)
, g(η̃)
)
p
.
Computing the entries of matrices R and Q explicitly, we have rki = θ̂p̂kuik and qki =
1
p0θ̂
pkuki. Plugging in r and q, we have that p̂i
p̂m
qkirml =
pk
pl
qlmrik, which gives that(
σp(eim)f(η), g(η)
)
p
=
(
f(η), σp(emi)g(η)
)
p
.
The proof for hl is similar. ■
Remark 3.10. Proposition 3.9 is nontrivial since the automorphism AdR does not preserve the
∗-structure of sln+1, i.e., there exists X ∈ sln+1 such that AdR(X
∗) ̸= AdR(X)∗. For example,
AdR(e
∗
12) = AdR(e21) =
∑
k,l
qk2r1lekl,
while
AdR(e12)
∗ =
(∑
k,l
qk1r2lekl
)∗
=
∑
k,l
qk1r2lelk =
∑
k,l
ql1r2kekl,
which are not equal for general κ.
Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) 7
Proposition 3.11. When the matrix U in the 4-tuple κ is real, ρp̂ and σp satisfy the following
property: [ρp̂(X
∗)K(·, η)](ξ) = [σp(X)K(ξ, ·)](η) for any X ∈ sln+1 and (ξ, η) ∈ Ω2j × Ω2j.
Proof. Recall from Proposition 3.5, the multivariate Krawtchouk polynomials can be written
in bilinear form. Also recall that the antiautomorphism a defined in (3.4) has property (3.5).
Notice that for function zξ, we can write ρp in terms of the representation ρ defined in (3.3),
ρp̂(eim)zξ =
√
p̂i
p̂m
ρ(eim)zξ.
Thus, for i ̸= m,
[ρp̂(eim)K(·, η)](ξ) = p2j0
(2j)!
√
p̂i
p̂m
〈
ρ(eim)zξ, ẑη
〉
κ
=
p2j0
(2j)!
√
p̂i
p̂m
〈
zξ, ρa(eim)ẑη
〉
κ
=
p2j0
(2j)!
√
p̂m
p̂i
〈
zξ, ρ(emi)ẑ
η
〉
κ
= [σp(emi)K(ξ, ·)](η),
where the last equality follows form the fact that eim = AdR(êim) and ρ(êim)ẑη = ẑi∂ẑm ẑ
η [13],
thus √
p̂m
p̂i
ρ(emi)ẑ
η =
√
p̂m
p̂i
ρ ◦AdR(êmi)ẑ
η = σp(emi)ẑ
η.
The proof for hl follows from the same argument. ■
Proposition 3.12. If the matrix U in the 4-tuple κ is real, define the operator Λ: l2(wp̂) −→
l2(wp) by
(Λf)(η) = p−j
0
∑
ξ∈Ω2j
wp̂(ξ)f(ξ)K(ξ, η).
Then Λ is an unitary operator and intertwines ρp̂ with σp. The kernel K(ξ, η) satisfies
[ρp̂(X
∗)K(·, η)](ξ) = [σp(X)K(ξ, ·)](η). (3.9)
Proof. It follows directly from equation (3.9) that Λ[ρp̂(X)f ] = σp(X)Λ(f) for all X ∈ sln+1,
thus Λ intertwines ρp̂ with σp. Recall that if a process has reversible measure, then it’s self dual
with respect to the cheap duality function, which comes from the reversible measure. For the
multi-species SEP(2j), the cheap duality function is given by δζ(ξ) =
δζ,ξ
wp̂(ξ)
, which has squared
norm 1
wp̂(ζ)
in l2(wp̂).
On the other hand, Λ(δζ)(η) = p−j
0 K(ζ, η) has squared norm 1
wp̂(ζ)
in l2(wp). Thus Λ maps
an orthogonal basis to another orthogonal basis preserving the norm, hence Λ is unitary. ■
Last we show the generator defined in (3.1) is the image of some self-adjoint element in
U(sln+1)
⊗L under the ∗-representations ρ⊗L
p̂ and σ⊗L
p . We generalize the construction of the
Markov generator in terms of the co-product of a Casimir element (see, e.g., [10, 12]) to the
multi-species cases.
We start by constructing a Casimir element of U(sln+1). Under the non-degenerate bilinear
form B(X,Y ) = tr(XY ), the dual basis of sln+1 is given by
e⋆lk = ekl, k ̸= l, h⋆l = ell − e00 = hl +
n∑
k=1
hk.
8 Z. Zhou
The Casimir element Ω of U(sln+1) is given by
Ω =
∑
0≤k<l≤n
(eklelk + elkekl) +
∑
0<l≤n
hlh
⋆
l .
It is easy to verify that Ω is self-adjoint, i.e., with the ∗-structure given in (3.2), Ω∗ = Ω. Next,
define the coproduct for the basis {ekl}0≤k ̸=l≤n and {hl}0<l≤n as ∆(X) = 1 ⊗X +X ⊗ 1, and
define an element
Y = ∆(Ω)− Ω⊗ 1− 1⊗ Ω.
Lemma 3.13. Lx,y is the image of a self-adjoint element in U(sln+1)
⊗2 under the representation
ρp̂ ⊗ ρp̂ and σp ⊗ σp. Specifically, there exists a constant c ∈ R such that
Lx,y =
1
2
ρp̂ ⊗ ρp̂(Yx,y)− c (3.10)
=
1
2
σp ⊗ σp(Yx,y)− c. (3.11)
Proof. To prove (3.10), we make use of the following identity:∑
0≤k<l≤n
ξxkξ
y
l + ξxkξ
y
l =
( ∑
0≤l≤n
ξxl
)( ∑
0≤l≤n
ξyl
)
−
∑
0≤l≤n
ξxl ξ
y
l = (2j)2 −
∑
0≤l≤n
ξxl ξ
y
l .
Expanding Ω in Y , we have
Y = 2
∑
0≤k<l≤n
(elk ⊗ ekl + ekl ⊗ elk) +
∑
1≤l≤n
(hl ⊗ h⋆l + h⋆l ⊗ hl).
Now we can compute the right-hand side of (3.10) using the above identities and the represen-
tation ρp̂ to see that it agrees with Lx,y. Note that Lx,y does not depend on p̂ since all terms
with p̂ get cancelled.
To prove (3.11), it suffices to show AdR ⊗AdR(Y ) = Y . First, we can check that AdR(Ω) = Ω
by direct calculation. Using the fact that AdR ⊗AdR ◦∆ = ∆ ◦ AdR, we have AdR ⊗AdR(Y )
= Y , thus
σp ⊗ σp(Yx,y) = (ρp̂ ⊗ ρp̂) ◦ (AdR ⊗AdR)(Yx,y) = ρp̂ ⊗ ρp̂(Yx,y). ■
Applying Theorem 2.2 yields the self duality for the multi-species SEP(2j).
Theorem 3.14. The multi-species SEP(2j) defined in Definition 3.1 is self dual with respect to
duality functions∏
x∈G
K
(
ξx, ηx, κ, 2j
)
,
for any κ ∈ Kn such that U in κ is real and p, p̂ in κ are probability measures.
4 Multi-species IRW and Heisenberg Lie algebra hn
In this section, we find a family of self-duality functions of the multi-species independent random
walk (multi-species IRW) using the Heisenberg Lie algebra hn.
The n-species independent random walk is a generalization of the usual IRW to n species on
a finite undirected graph G = (V,E), where V = {1, . . . , L} with L ∈ N and L > 2 is the set
Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) 9
of sites (vertices) and E is the set of edges. It is a Markov process where n species of particles
move independently between L sites. The jump rate for a particle of species i from a site is
proportional to the number of species i particles at that site.
The state space S(n,G) of particle configurations consists of variables ξ =
(
ξxi : 1 ≤ i ≤ n,
x ∈ G
)
, where ξxi ∈ N0 (non-negative integers) denotes the number of species i particles at
site x.
Definition 4.1. The generator of n-species IRW on G = (V,E) is given by
Lf(ξ) =
∑
edge{x,y}∈E
Lx,y,
Lx,y =
n∑
i=1
[
ξxi
(
f
(
ξx,yi
)
− f(ξ)
)
+ ξyi
(
f
(
ξy,xi
)
− f(ξ)
)]
,
where ξx,yi denotes the particle configuration obtained by moving a particle of species i from
site x to site y if ξx,yi ∈ S(n,G).
Define measure µλ(ξ) =
∏n
i=1
λξi
ξi!
e−λ with λ > 0. Following a simple detailed balance com-
putation, we can show that the product measure ⊗Gµλ is a reversible measure of the n-species
IRW when λ is the same for all sites.
Next, we mention here that the space l2(µλ) is equipped with inner product
(f, g)λ =
∑
ξ∈Nn
0
µλ(ξ)f(ξ)g(ξ).
4.1 The Charlier polynomials and Heisenberg Lie algebra hn
Definition 4.2. The Heisenberg Lie algebra hn is the 2n+ 1 dimensional complex Lie algebra
with generators {P1, . . . , Pn, Q1, . . . , Qn, Z} and commutation relations: for 1 ≤ i, l ≤ n,
[Pi, Pl] = [Qi, Ql] = [Pi, Z] = [Qi, Z] = 0, [Pi, Ql] = δi,lZ.
The Heisenberg Lie algebra hn is nilpotent but not semisimple. It has a ∗-structure given by
P ∗
i = Qi, Q∗
i = Pi, Z∗ = Z.
The Charlier polynomials are given by
Cm(z, λ) = 2F0
(
−m, −z
−
∣∣∣∣− 1
λ
)
.
Here we list some properties of Charlier polynomials that will be used later on. First, Charlier
polynomials are Orthogonal,∑
z∈N0
Cm(z, λ)Cm̃(z, λ)
λz
z!
e−λ = δm,m̃λ−m̃m̃!.
They have the following raising and lowering property,
mCm−1(z, λ) = λCm(z, λ)− λCm(z + 1, λ),
λCm+1(z, λ) = λCm(z, λ)− zCm(z − 1, λ).
To construct unitary operator later, we define function C(ξ, η, λ) for ξ, η ∈ Nn
0 by
C(ξ, η, λ) =
n∏
i=1
eλCξi(ηi, λ). (4.1)
10 Z. Zhou
4.2 Self duality of the multi-species IRW
Now we define the ∗-representation ρλ of hn on l2(µλ) by
[ρλ(Qi)f ](ξ) = ξif
(
ξ−1
i
)
,
[ρλ(Pi)f ](ξ) = λf
(
ξ+1
i
)
,
[ρλ(Z)f ](ξ) = λf(ξ),
where ξ+1
i
(
ξ−1
i
)
means that ξi is increased (decreased) by 1.
Next, we define the map θ by
θ(Pi) = Z − Pi, θ(Qi) = Z −Qi, θ(Z) = Z,
then θ extends to a Lie algebra isomorphism of hn, preserving the ∗-structure.
Proposition 4.3. For any X ∈ hn, we have
ρλ(X
∗)C(·, η, λ)(ξ) = ρλ(θ(X))C(ξ, ·, λ)(η).
Proof. It’s easy to verify by definition and the raising and lowering property,
ρλ(Qi)C(·, η, λ)(ξ) = ξiC
(
ξ−1
i , η, λ
)
= λC(ξ, η, λ)− λC
(
ξ, η+1
i , λ
)
= ρλ(θ(Pi))C(ξ, ·, λ)(η),
ρλ(Pi)C(·, η, λ)(ξ) = λC
(
ξ+1
i , η, λ
)
= λC(ξ, η, λ)− ηiC
(
ξ, η−1
i , λ
)
= ρλ(θ(Qi))C(ξ, ·, λ)(η). ■
Proposition 4.4. Define the operator Λ: l2(µλ) −→ l2(µλ) by
(Λf)(η) =
∑
ξ∈Nn
0
µλ(ξ)f(ξ)C(ξ, η, λ),
then Λ is an unitary operator and intertwines ρλ with ρλ ◦ θ.
Proof. It follows directly from Proposition 4.3 that Λ[ρλ(X)f ] = ρλ ◦ θ(X)Λ(f) for all X ∈ hn,
thus Λ intertwines ρλ with ρλ ◦ θ. The cheap duality functions for the n-species IRW given by
δζ(ξ) =
δζ,ξ
µλ(ξ)
form an orthogonal basis for l2(µλ) with squared norm 1
µλ(ζ)
, while Λ(δζ)(η) =
C(ζ, η, λ) also has squared norm 1
µλ(ζ)
in l2(µλ). By the fact that all C(ζ, η, λ) form an orthogonal
basis for l2(µλ), Λ is unitary. ■
Finally, to show self duality, we define Y ∈ U(hn)⊗2 by
Y =
n∑
i=1
(1⊗Qi −Qi ⊗ 1)(Pi ⊗ 1− 1⊗ Pi).
Lemma 4.5. The generator for the multi-species IRW can be written as the following:
Lx,y = λ−1ρλ ⊗ ρλ(Yx,y) (4.2)
= λ−1(ρλ ◦ θ)⊗ (ρλ ◦ θ)(Yx,y). (4.3)
Proof. (4.2) is obtained by plugging in definitions, and to prove (4.3), we show that (ρλ ◦ θ)⊗
(ρλ ◦ θ)(Y ) = ρλ ⊗ ρλ(Y ), which follows from [12, Lemma 3.5]. ■
Again, applying Theorem 2.2, we obtain the self duality for the multi-species IRW.
Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) 11
Theorem 4.6. The multi-species IRW defined in Definition 4.1 is self dual with respect to
duality functions∏
x∈G
C
(
ξx, ηx, λ
)
, λ > 0. (4.4)
Remark 4.7. These duality functions could be obtained by the independence of the evolution of
each species of particles and the fact that the duality functions given by (4.1) and (4.4) suitably
factorize over species.
Acknowledgments
The author is very grateful to Jeffrey Kuan and anonymous referees for helpful discussions and
insightful comments.
References
[1] Ayala M., Carinci G., Redig F., Quantitative Boltzmann–Gibbs principles via orthogonal polynomial duality,
J. Stat. Phys. 171 (2018), 980–999, arXiv:1712.08492.
[2] Ayala M., Carinci G., Redig F., Higher order fluctuation fields and orthogonal duality polynomials, Elec-
tron. J. Probab. 26 (2021), 27, 35 pages, arXiv:2004.08412.
[3] Borodin A., Corwin I., Sasamoto T., From duality to determinants for q-TASEP and ASEP, Ann. Probab.
42 (2014), 2314–2382, arXiv:1207.5035.
[4] Caputo P., On the spectral gap of the Kac walk and other binary collision processes, ALEA Lat. Am. J.
Probab. Math. Stat. 4 (2008), 205–222, arXiv:0807.3415.
[5] Carinci G., Franceschini C., Giardinà C., Groenevelt W., Redig F., Orthogonal dualities of Markov processes
and unitary symmetries, SIGMA 15 (2019), 053, 27 pages, arXiv:1812.08553.
[6] Chen J.P., Sau F., Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems,
Markov Process. Related Fields 27 (2021), 339–380, arXiv:2008.13403.
[7] Corwin I., Shen H., Tsai L.-C., ASEP(q, j) converges to the KPZ equation, Ann. Inst. Henri Poincaré
Probab. Stat. 54 (2018), 995–1012, arXiv:1602.01908.
[8] Dermoune A., Heinrich P., Spectral gap for multicolor nearest-neighbor exclusion processes with site disorder,
J. Stat. Phys. 131 (2008), 117–125.
[9] Franceschini C., Giardinà C., Stochastic duality and orthogonal polynomials, arXiv:1701.09115.
[10] Giardinà C., Kurchan J., Redig F., Vafayi K., Duality and hidden symmetries in interacting particle systems,
J. Stat. Phys. 135 (2009), 25–55, arXiv:0810.1202.
[11] Griffiths R.C., Orthogonal polynomials on the multinomial distribution, Aust. J. Stat. 13 (1971), 27–35.
[12] Groenevelt W., Orthogonal stochastic duality functions from Lie algebra representations, J. Stat. Phys. 174
(2019), 97–119, arXiv:1709.05997.
[13] Iliev P., A Lie-theoretic interpretation of multivariate hypergeometric polynomials, Compos. Math. 148
(2012), 991–1002, arXiv:1101.1683.
[14] Kuan J., A multi-species ASEP(q, j) and q-TAZRP with stochastic duality, Int. Math. Res. Not. 2018
(2018), 5378–5416, arXiv:1605.00691.
[15] Kuan J., Stochastic fusion of interacting particle systems and duality functions, arXiv:1908.02359.
[16] Redig F., Sau F., Factorized duality, stationary product measures and generating functions, J. Stat. Phys.
172 (2018), 980–1008, arXiv:1702.07237.
[17] Zhou Z., Hydrodynamic limit for a d-dimensional open symmetric exclusion process, Electron. Commun.
Probab. 25 (2020), 76, 8 pages, arXiv:2004.14279.
https://doi.org/10.1007/s10955-018-2060-7
https://arxiv.org/abs/1712.08492
https://doi.org/10.1214/21-EJP586
https://doi.org/10.1214/21-EJP586
https://arxiv.org/abs/2004.08412
https://doi.org/10.1214/13-AOP868
https://arxiv.org/abs/1207.5035
https://arxiv.org/abs/0807.3415
https://doi.org/10.3842/SIGMA.2019.053
https://arxiv.org/abs/1812.08553
https://arxiv.org/abs/2008.13403
https://doi.org/10.1214/17-AIHP829
https://doi.org/10.1214/17-AIHP829
https://arxiv.org/abs/1602.01908
https://doi.org/10.1007/s10955-008-9496-0
https://arxiv.org/abs/1701.09115
https://doi.org/10.1007/s10955-009-9716-2
https://arxiv.org/abs/0810.1202
https://doi.org/10.1111/j.1467-842x.1971.tb01239.x
https://doi.org/10.1007/s10955-018-2178-7
https://arxiv.org/abs/1709.05997
https://doi.org/10.1112/S0010437X11007421
https://arxiv.org/abs/1101.1683
https://doi.org/10.1093/imrn/rnx034
https://arxiv.org/abs/1605.00691
https://arxiv.org/abs/1908.02359
https://doi.org/10.1007/s10955-018-2090-1
https://arxiv.org/abs/1702.07237
https://doi.org/10.1214/20-ecp350
https://doi.org/10.1214/20-ecp350
https://arxiv.org/abs/2004.14279
1 Introduction
2 Background
3 Multi-species SEP(2j) and Lie algebra sl_{n+1}
3.1 Multivariate Krawtchouk polynomials and Lie algebra sl_{n+1}
3.2 Self-duality of the multi-species SEP(2j)
4 Multi-species IRW and Heisenberg Lie algebra h_n
4.1 The Charlier polynomials and Heisenberg Lie algebra h_n
4.2 Self duality of the multi-species IRW
References
|
| id | nasplib_isofts_kiev_ua-123456789-211414 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T04:33:38Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zhou, Zhengye 2026-01-02T08:27:24Z 2021 Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW. Zhengye Zhou. SIGMA 17 (2021), 113, 11 pages 1815-0659 2020 Mathematics Subject Classification: 60K35 arXiv:2110.07042 https://nasplib.isofts.kiev.ua/handle/123456789/211414 https://doi.org/10.3842/SIGMA.2021.113 We obtain orthogonal polynomial self-duality functions for the multi-species version of the symmetric exclusion process (SEP(2)) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have > 1 species of particles. In addition, we allow up to 2 particles to occupy each site in the multi-species SEP(2). The duality functions for the multi-species SEP(2) and the multi-species IRW come from unitary intertwiners between different ∗-representations of the special linear Lie algebra ₙ₊₁ and the Heisenberg Lie algebra ₙ, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP(2) and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW. The author is very grateful to Jeffrey Kuan and anonymous referees for helpful discussions and insightful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW Article published earlier |
| spellingShingle | Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW Zhou, Zhengye |
| title | Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW |
| title_full | Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW |
| title_fullStr | Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW |
| title_full_unstemmed | Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW |
| title_short | Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2) and Multi-Species IRW |
| title_sort | orthogonal polynomial stochastic duality functions for multi-species sep(2) and multi-species irw |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211414 |
| work_keys_str_mv | AT zhouzhengye orthogonalpolynomialstochasticdualityfunctionsformultispeciessep2andmultispeciesirw |