Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice

Abstract

We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in finite lattice. We introduce the L-operator related with the U q (sl 2 ) R-matrix, and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence between the wavefunctions and symmetric polynomials, we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the B-operators. *

Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model

“…And in our very recent work [30], we extended the Izergin-Korepin analysis to study the projected wavefunctions of the ( 2 ) six-vertex model. The resulting symmetric polynomials representing the projected wavefunctions contains the Grothendieck polynomials as a special case when the six-vertex model reduces to the five-vertex model [31][32][33]. We apply this technique to study the free-fermion model in an external field.…”

Symmetric elliptic functions, IRF models, and dynamic exclusion processes

“…For the step initial condition, we evaluate averages of observables for them as well, and use those to investigate one-point asymptotics of the dynamic SSEP.The construction and proofs are based on remarkable properties (branching and Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric elliptic functions that arise as matrix elements in an infinite volume limit of the algebraic Bethe ansatz for E τ,η (sl 2 ).SYMMETRIC ELLIPTIC FUNCTIONS, IRF MODELS, AND DYNAMIC EXCLUSION PROCESSES 2 foundational works. This article is concerned with two of its very recent and closely related applications to:(a) theory of symmetric functions, with new families of symmetric functions being introduced and new summation identities for older ones being proved, see [7,35,31,32,15,16,26];(b) deriving new exact formulas for averages of observables in two-dimensional integrable lattice models and (1+1)-dimensional random growth models, and using those to study large scale and time asymptotics [18,15,16,4,1,8,14,9,22].All the above cited papers deal with the R-matrix for the (higher spin) six vertex model and its degenerations (in other words, with representations of the affine quantum group U q (ŝl 2 ) and their limits). Certain progress has been achieved for the U q (ŝl n ) case as well, with new Markov chains introduced via the corresponding R-matrices [30,17] and a duality functional for them provided in [29].The goal of this work is to climb higher in the hierarchy and to extend some of the recent progress from vertex models to the so-called Interaction-Round-a-Face (IRF) models, also known as face and solid-on-solid (SOS) models.…”

“…(a) theory of symmetric functions, with new families of symmetric functions being introduced and new summation identities for older ones being proved, see [7,35,31,32,15,16,26];…”

Symmetric functions and wavefunctions of XXZ-type six-vertex models and elliptic Felderhof models by Izergin–Korepin analysis

“…Our result can be regarded as an extension of these combinatorial objects by using the six-vertex U q (sl 2 ) R-matrix as the bulk weights, and more general triangular K-matrix as the boundary weights. The ordinary wavefunctions [24,25] is represented by the Grothendieck polynomials of type A Grassmannian variety and their quantum group deformations. See the Appendix for a proof of the ordinarywavefunctions based on the Izergin-Korepin analysis.…”

“…There are now a lot of papers on this subject. See [19,20,21,22,23,24,25,26,27,28,29,30,31,32] for examples which investigate symmetric functions by using the XXZ model and the q-boson model, and [33,34,35,36,37,38] by using the free-fermion model in an external field .…”