SPECIAL FUNCTIONS FOR HYPEROCTAHEDRAL GROUPS USING BOSONIC TRIGONOMETRIC SIX-VERTEX MODELS BEN BRUBAKER WILL GRODZICKI AND ANDREW SCHULTZ

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SPECIAL FUNCTIONS FOR HYPEROCTAHEDRAL GROUPS USING

BOSONIC, TRIGONOMETRIC SIX-VERTEX MODELS

BEN BRUBAKER, WILL GRODZICKI, AND ANDREW SCHULTZ

Abstract. Recent works have sought to realize certain families of orthogonal, symmetric polynomials as

partition functions of well-chosen classes of solvable lattice models. Many of these use Boltzmann weights

arising from the trigonometric six-vertex model R-matrix (or generalizations or specializations of these

weights). In this paper, we seek new variants of bosonic models on lattices designed for type B/C root

systems, whose partition functions match the zonal spherical function in type C. Under general assumptions,

we ﬁnd that this is possible for all highest weights in rank 2 and 3, but not for higher rank.

1. Introduction

A number of recent papers have studied symmetric function theory and related special functions using

solvable lattice models, including [4, 5, 8, 10, 11, 13, 24]. The word “solvable” here indicates that the models

satisfy certain local relations, principally the “Yang-Baxter equations” or “RTT relations” (to be described

in more detail in the subsequent sections). Such relations are known to arise naturally from modules for

quantum groups and it is an interesting problem to associate families of symmetric functions to quantum

group modules via the partition functions of solvable lattice models in this way. In the present paper, we

explore this connection when the resulting R-matrix is, up to scaling, the trigonometric R-matrix for the

six-vertex model coming from the standard module for the quantum group Uq(ˆ

sl2) and the lattices are

constructed to reﬂect symmetries of the Weyl group in types B/C – the so-called hyperoctahedral groups.

The two-dimensional lattice models we consider will consist primarily of ﬁnite rectangular arrays of vertices

on a square lattice, with four edges (two vertical and two horizontal) adjacent to each vertex. Models for

classical Cartan types outside of type Amay have additional boundary vertices which roughly reﬂect the

embeddings of classical groups into the general linear group and are inspired by Kuperberg’s models for

symmetry classes of alternating sign matrices [16]. These will be deﬁned precisely in Section 3.

Admissible states of these models for given boundary data will consist of paths of particles beginning

from the bottom boundary, moving upward and leftward, and exiting out the left boundary. There’s an

interesting dichotomy here in whether we allow superposition of particles along vertical edges - such models

are called “bosonic” models when superposition is allowed, and “fermionic” when not allowed. Superposition

of particles on horizontal edges will always be forbidden in this context, in keeping with our requirement

that the R-matrix in the RTT-relation involving a pair of adjacent rows is a six-vertex model.

We may keep track of initial particle locations along the bottom boundary using an integer partition λ

whose parts λiare associated to column indices where particles appear. In particular the parts of λare

weakly decreasing in the bosonic case and strictly decreasing in the fermionic case. All other boundaries will

be regular, so λsuﬃces to determine the boundary conditions. An example of an admissible state in a type

Abosonic model is pictured in Figure 1 below. Note in particular the superposition of two particles on a

lone edge in the column labeled ‘2’ at the bottom, as depicted by the particle number in the box. Let Sλ

denote the set of all admissible states corresponding to the bottom boundary in the model. Full details on

these models will be provided in Sections 2 and 3.

Broadly speaking, the goal is to represent families of symmetric functions indexed by partitions λas

generating functions on the admissible states Sλ. These generating functions come from choices of local

weights, the “Boltzmann weights” at each vertex depending on occupancy in adjacent edges. Taking the

product of weights at all vertices in the admissible state, and summing these products over all admissible

states in Sλ, results in a generating function over states called the “partition function,” denoted here as

Z(Sλ). As indicated in the example above, each row of the model is associated to an indeterminate xi(the

so-called “spectral parameters”) and the Boltzmann weights of vertices in the row will be allowed to depend

arXiv:2210.13174v2 [math.CO] 19 Sep 2023

0 1 2 3

Figure 1. A sample state for the type Amodel when λ= (3,2,0)

on these xi. In this way, we will see that Z(Sλ) will be a polynomial in the parameters xiaccording to the

choice of Boltzmann weights.

The symmetric functions we seek to represent as partition functions of lattice models are a particular

family of orthogonal polynomials in several variables x:= (x1, . . . , xr) indexed by λ∈Λrassociated to

aﬃne root systems. More precisely, they are one parameter specializations of the two parameter Macdonald

polynomials P(ϵ)

λ(x) deﬁned in Section 5.7 of [18]. If the aﬃne Weyl group associated to the root system

is W=W0⋉Λrwith W0the ﬁnite Weyl group and Λrthe weight lattice of rank r, then the family of

polynomials in rvariables xand a linear character ϵof W0. In our specialization, the P(ϵ)

λare constructed

from certain Hecke algebra symmetrizers (cf. (5.5.6) in [18]) built from ϵas sums over W0, and acting on

monomials xλ=xλ1

1· · · xλr

r. Our primary example will be the case where ϵis the trivial character, where

P(ϵ)

λis the usual two parameter Macdonald polynomial and in our specialization to one parameter qis

Macdonald’s zonal spherical function.1In type A, this is also known as the Hall-Littlewood polynomial. For

any root system and associated ﬁnite Weyl group W0, it takes the form:

(1.1) Pλ(x1, . . . , xr;q) = 1

Qλ(q)X

σ∈W0

σ xλY

α∈Φ+

1−qx−α

1−x−α!,

for some polynomial Qλ. For other choices of linear character ϵ, we achieve similar averaging formulas over

W0; see for example Section 8 of [9] for a discussion of this. Attempting to use lattice models derived

from quantum group modules to represent Macdonald polynomials made from Hecke algebras is perhaps

natural in light of Jimbo’s generalization of Schur-Weyl duality [15]. Making this connection precise in the

language of partition functions, viewed as matrix coeﬃcients of quantum group modules, is an interesting

open question. At present, we don’t even know whether solvable lattice models exist for all such polynomials

in one deformation parameter (nevermind two). This paper is an attempt to understand this question better

for classical groups, and we begin by reviewing what is known.

In type A, the connections between such orthogonal polynomials and solvable lattice models are well devel-

oped. The Weyl groups W0of simply laced types have just two linear characters - the sign and trivial charac-

ters - which produce spherical Whittaker functions and Hall-Littlewood polynomials, respectively. Families

of solvable lattice models whose partition functions are spherical Whittaker functions and the aforemne-

tioned Hall-Littlewood polynomials appeared previously in the literature in [8] and [24], respectively. The

Whittaker functions use a fermionic lattice model while those for Hall-Littlewood polynomials are bosonic.

In fact, this connection between symmetric functions and lattice models extends to the nonsymmetric ana-

logues. Indeed if we distinguish each path in an admissible state of the model by recording its color, then the

resulting lattice models with a ﬁxed set of colors along the boundary realize the non-symmetric analogues

above. These are the non-symmetric Hall-Littlewood polynomials and Iwahori-Whittaker functions, which

were studied in [4] and [6], respectively. These latter Whittaker functions arise in representation theory of

p-adic algebraic groups; further specializing the one parameter in the Boltzmann weights, they degenerate

1Macdonald polynomials are commonly written in terms of parameters qand t, though conventions diﬀer about their roles.

Our conventions match those of p-adic representation theory. In particular, the usual role of tin the literature is played by q

for us, owing to its connection with the cardinality of the residue ﬁeld of the p-adic ﬁeld which is typically notated with the

same letter. No confusion will arise since our polynomials have a single deformation parameter and its role will always appear

through explicit formulas like (1.1).

to so-called “Demazure atoms” as in [19], originally called “standard bases” by Lascoux and Sch¨utzenberger

in [17], and their connection to lattice models is explored in [7].

In types Band C, less is known. There are results for spherical Whittaker functions – that is, P(ϵ)

λwith

ϵthe sign character – in [14] using fermionic lattice models with bends along one side and bivalent vertices

at each bend (Figure 5 gives one such example). In [24], bosonic models with the same underlying lattice

shape are shown to give certain two-parameter generalizations of polynomials P(ϵ)

λwhere ϵis the character

of W0sending long roots to 1 and short roots to −1. The two parameters, called γand δ, arise from a pair of

“bonus” columns introduced into the lattice models next to the bends. The Boltzmann weights for vertices

in each of these two extra columns depend on one of these parameters, but setting γ=δ= 0 is equivalent

to omitting these two bonus columns entirely from the model. In this case, according to (64) in [24], one

obtains P(ϵ)

λfor this character ϵ(see Theorem 4.2). If instead one chooses γand δwith γ+δ= 0 and

γδ =q, this results in the zonal spherical function in type C. So it is natural to ask which special functions

are achievable from a solvable lattice model with bends in which there are no “bonus” columns, and for

which the rectangular (tetravalent) vertices satisfy a Yang-Baxter equation with the trigonometric six-vertex

model. This is the main question we try to answer in the present paper. In particular, we address whether

one can obtain the zonal spherical function in these non-simply-laced types via such solvable lattice models.

As we alluded to above, the lattice models in types Band Cconsist almost entirely of tetravalent vertices,

together with a set of bivalent vertices appearing as bends connecting pairs of adjacent rows along one side

of the model. The example in Figure 5 shows one such admissible state in rank two. Thus our question

becomes one of exploring choices of these Boltzmann weights at the bend vertices (or “bend weights,” for

short) which simultaneously preserve the solvability of the model and achieve our desired special functions

as partition functions of the model. In doing so, we are forced to consider bend weights which are not

uniformly deﬁned in terms of spectral parameters. For example in rank two models, the function deﬁning

the Boltzmann weight for the top bend may diﬀer from the one for the bottom bend. Details are given in

Section 3.

In later sections of the paper, we provide conditions on the bend weights to guarantee the solvability of

the model. This leads to the following result.

Main Theorem. Let Bλbe a solvable lattice model of type B/C with boundary conditions corresponding

to a dominant weight λ, and having Boltzmann weights from the bosonic, trigonometric six-vertex model at

all tetravalent vertices. Then the existence of bend weights such that Z(Bλ) = Pλ(x;q), the zonal spherical

function in type C, for all dominant weights λdepends on the rank as follows:

•For rank r≤3, there exists a choice of Boltzmann weights for the bend vertices that realizes the

zonal spherical function at dominant weights λ. These bend weights depend on the row, and are not

uniformly dependent on the (spectral) parameters xi.

•In rank r≥4, no such choice of Boltzmann weights for bend vertices exists.

We brieﬂy outline the contents of each section and their role in leading up to this main result. In Section 2,

we ﬁx notation, recall the deﬁnition of triangular weights and review the concept of solvability (the existence

of a Yang-Baxter equation for tetravalent vertices) for bosonic models in type A. In Section 3, we extend

this notion of solvability to type B/C models. See Deﬁnition 3.2 for a precise deﬁnition of solvability for

lattice models of types B/C. The deﬁnition ensures that the row-to-row transfer matrices (taken in pairs

of rows connected by a bend vertex) commute, as shown in Lemmas 3.5 and 3.6. Solvability in these

types requires certain additional relations, as has been long understood in lattice models of this geometry

[10, 14, 16, 22, 24]. Following Kuperberg, these relations tend to be named after animals – ﬁsh, caduceus,

etc. We prove necessary conditions on bend weights to satisfy the required relations in a series of lemmas in

Section 3. In particular, we ﬁnd new solutions to the caduceus relation (e.g., Lemma 3.2) using non-uniform

choices of bend weights. Section 4 provides methods for explicit evaluations of solvable lattice models in

type B/C, in the style of [24]. The later sections of the paper then apply these methods to lattice models

of various ranks, as the analysis is rather diﬀerent in each case, and results in the main theorem above. In

the process, we evaluate all solvable lattice models in rank at most 3 with trigonometric six-vertex model

weights at tetravalent vertices.

Our main theorem handles the most general set of bend weights that is consistent with the paradigm of

partition functions as matrix coeﬃcients for quantum group modules – in particular, we require the caduceus

equation for bend weights to be consistent with Cherednik and Sklyanin’s reﬂection equation and we require

monomial dependence in the spectral parameters xi, consistent with formal group laws for parametrized

Yang-Baxter equations from families of aﬃne quantum group representations. Thus any solution for bend

weights outside of our assumptions would represent a break from this point of view. It is possible that our

analysis could be used as a starting point for developing combinatorial solutions outside of this paradigm,

and one might hope the resulting algebraic structures would be of fundamental interest. This work was

partially supported by NSF grant DMS-2101392 (Brubaker).

2. Bosonic Models for Cartan Type A

We begin by describing the general features of six-vertex lattice models in type A, all of which will be

needed in subsequent sections on other classical Cartan types. Each lattice model is indexed by an integer

partition λ= (λ1⩾λ2⩾· · · ⩾λr) (with λr≥0) which determines its size, shape, and boundary conditions.

Given such a λ, each lattice model will be built from two-dimensional grids of tetravalent vertices with nrows,

numbered 1 to nfrom bottom to top, and λ1+1 columns, numbered 0 to λ1from left to right. An admissible

state of such a model is a conﬁguration of rpaths along the lattice edges, with the i-th path starting from

the λi-th column along the bottom boundary edges and ending on an edge at the far left side of the diagram.

At each vertex, paths are allowed to either move up or left. Paths are allowed to overlap without restriction

along vertical edges, but any given horizontal edge can allow at most one path. Alternatively, we could view

columns between each row as recording the position of a family of particles (or a ﬁnite window of interest

among a semi-inﬁnite collection of particles) and the admissible states record a discrete time evolution of

these particles with each new row. The fact that columns may contain multiple paths then translates as the

superposition of these particles, so we refer to the associated models as “bosonic” (as opposed to “fermionic”

where no superposition is permitted). We will denote the set of all such (admissible) states as Aλ.

Ultimately we will attach a polynomial to each particular model. To do this, each vertex vis assigned

a Boltzmann weight B(v); that weight will be a function depending on the edge conﬁgurations adjacent to

that vertex vand on a transcendental parameter xicorresponding to the row iin which the vertex occurs.

For an admissible state Sof the system, its weight wt(S) is then deﬁned as the product of weights over all

its vertices: wt(S) = Qv∈SB(v). The partition function Z(Aλ) for the model is then recovered by summing

these weights over all states of the model:

Z(Aλ) = X

S∈Aλ

wt(S).

This is a slight abuse of notation, as we’ve continued to use Aλrather than introduce a new notation for the

set of admissible states together with an associated collection of Boltzmann weights. We repeat this often in

what follows when the underlying Boltzmann weights are clear from context.

Thus to specify the partition function Z(Aλ) it remains to deﬁne the Boltzmann weights. The Boltzmann

weights at each vertex are given in the table in Figure 2. They agree with the weights in Equation (35) of

[24] and arise naturally from evolution operators on bosonic Fock space made by creation and annihilation of

particles (see Section 2 of [24] for details). For us, the key point is that these weights satisfy certain relations

known as “Yang-Baxter equations” or “RT T relations,” which we address shortly below.

m−1

m+ 1

1xj1−qmxj

Figure 2. Rectangular lattice Boltzmann weights - the boxed integer mdenotes the mul-

tiplicity of paths along the associated vertical edge.

This choice of Boltzmann weights gives the following explicit evaluations of the partition function:

Theorem 2.1 (Wheeler-Zinn-Justin [24], Thm. 2).For any partition λand weights as in Figure 2, then

Z(Aλ)=(x1· · · xr)Pλ(x;q)

where Pλdenotes the Hall-Littlewood polynomial as in (1.1).

To prove this, one must demonstrate that the partition function Z(Aλ) satisﬁes certain symmetries. In

this case, we want to ensure that Z(Aλ) is left unchanged after an exchange of variables xi↔xi+1. Recalling

that the weights depend on xiin row i, this is equivalent to saying the partition function is symmetric upon

switching the roles of the rows iand i+ 1, a property known as “commuting transfer matrices” for the

row-to-row transfer matrix.

Following Baxter [1], one may do this by taking advantage of certain “local symmetries” enjoyed by these

Boltzmann weights. Consider an auxiliary family of Boltzmann weights for the six so-called “R-vertices,”

or R-matrix weights depicted in Figure 3 below. Their graphical depiction reﬂects the fact that each such

vertex is the intersection of two horizontal edges, and hence each adjacent edge may admit at most one path.

Their Boltzmann weights are a scaled version of the entries in the trigonometric R-matrix of the six-vertex

model, now depending on the pair of parameters xjand xkon the respective rows. In addition to providing

the explicit description of each weight in the bottom row of the table, the middle row gives an abstract name

for each vertex, to be used in later computations.

a2(k, j)a1(k, j)b2(k, j)b1(k, j)c2(k, j)c1(k, j)

1 1 q(xk−xj)

xk−qxj

xk−xj

xk−qxj

(1−q)xj

xk−qxj

(1−q)xk

xk−qxj

Figure 3. R-matrix weights from the trigonometric six-vertex model

Proposition 2.2 (Yang-Baxter equation, RTT relation).For any choice of edge labels α, β, γ, δ, ϵ, ϕ and

Boltzmann weights as in Figures 2 and 3, one has the equality of partition functions

























This may be checked explicitly, as the weights in Figure 2 are uniformly expressed in terms of the number

of particles m. Such an identity was asserted in Equation 3.3 of [3] and in Equation (14) of [24].

In addition to the previous result, it will be useful to know the following result; compare Equation (11)

in [24]. Its proof is a direct computation and left to the reader.

Proposition 2.3 (Unitarity relation).For any choice of edge labels αand βand using Boltzmann weights

as in Figure 3, one has

Z









= 1.

We say that a type Amodel is “solvable” if its weights possess a solution Rto the Yang-Baxter equation

above. It implies that the row-to-row transfer matrices commute, via the now familiar “train argument,” and

hence the partition function Z(Aλ) is symmetric under the exchange of variables xi↔xi+1. A graphical

depiction of this argument is shown in Figure 4, which highlights both the repeated use of the Yang-Baxter

equation and that the symmetry of Zhinges on the fact that the Rmatrix weights a1(k, k + 1) = a2(k, k +1)

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SPECIALFUNCTIONSFORHYPEROCTAHEDRALGROUPSUSINGBOSONIC,TRIGONOMETRICSIX-VERTEXMODELSBENBRUBAKER,WILLGRODZICKI,ANDANDREWSCHULTZAbstract.Recentworkshavesoughttorealizecertainfamiliesoforthogonal,symmetricpolynomialsaspartitionfunctionsofwell-chosenclassesofsolvablelatticemodels.ManyoftheseuseBoltzmannwe...

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SPECIAL FUNCTIONS FOR HYPEROCTAHEDRAL GROUPS USING BOSONIC TRIGONOMETRIC SIX-VERTEX MODELS BEN BRUBAKER WILL GRODZICKI AND ANDREW SCHULTZ

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