Partition functions of integrable lattice models and combinatorics of symmetric polynomials

Abstract

We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In particular, we examine the correspondence between the wavefunctions of the XXZ type, Felderhof type and the boson type integrable models and symmetric polynomials such as the Schur, Grothendieck and symplectic Schur functions. We also give a brief report of our work on generalizing the correspondence between the Felderhof models and factorial Schur and symplectic Schur functions.

Dual wavefunction of the Felderhof model

“…We also generalized the relation between the dual wavefunction to the Felderhof model with inhomogeneous parameters in the quantum space and the factorial Schur polynomials, which is motivated by the fact that the wavefunction of the Felderhof model with inhomogeneties are given by the factorial Schur polynomials [16]. The expression can be extended furthermore to the Felderhof model with two types of inhomogeneous parameters, and there are correspondences between the original and the dual wavefunctions and a generalization of the factorial Schur polynomials [22,23].…”

“…The integrable five-vertex model which is the t = 0 limit of the L-operator (2.8), which gives the Schur polynomials, can be regarded as special limits of both the Felderhof model and the XXZ model. See [21,22,29,30,31,32,33] for examples on the recent investigations on the combinatorics of the symmetric polynomials from the viewpoint of partition functions, in which the combinatorial identities of various symmetric polynomials such as the Schur, Grothendieck, Hall-Littlewood and their noncommutative versions are derived.…”

“…We give another proof of Theorem 4.1 by using a modern statistical mechanical method and an analysis on a fundamental object in quantum integrable models, i.e., we use the matrix product method and the domain wall boundary partition function, as was done in the case of the Grothendieck polynomials in [21] (see also [22] in which we demonstrate a proof of Theorem 3.2 by using the same arguments given in this section). We prove Theorem 4.1 as follows.…”

Dual wavefunction of the symplectic ice

“…The result is a symplectic version of the result in [48], where the correspondence between the wavefunction without reflecting boundary and the factorial Schur polynomials is established. We extended furthermore to the free-fermion model with two types of inhomogeneous parameters, and there are correspondences between the original and the dual wavefunctions and a generalization of the factorial Schur polynomials and factorial symplectic Schur functions [59,60]. Details will appear elsewhere.…”

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

“…This fact allowed us to extract various properties of the Grothendieck polynomials. This is just an example, and there are increasing interests on the studies of symmetric polynomials from the point of view of integrable lattice models today (see [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] for examples on these subjects) . One of the interesting topics is the study on symmetric polynomials by investigating integrable boson models in half-infinite lattice initiated in [24], which resembles the q-vertex operator approach.…”