Vertex models, TASEP and Grothendieck polynomials

Abstract

We examine the wavefunctions and their scalar products of a one-parameter family of integrable five vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on/off-shell wavefunctions of the five vertex models are represented as a certain determinant form. Up to some normalization factors, we find the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of Grothendieck polynomials.

Dual wavefunction of the Felderhof model

“…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.…”

“…We can easily show by its graphical description that Φ({z} N )|x 1 = 1, · · · , x N = N can be factorized as 21) where Z N ({z} N ) is the domain wall boundary partition function on an N × N lattice. The domain wall boundary partition function on an M × M lattice is defined as …”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…107. The elements {T (ij) , 1 ≤ i < j < n − 1} satisfy the defining relations of the non-local Kohno-Drinfeld algebra N L4T n−1 , see Definition 4.…”

Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

“…. , n), the correspondence at q = 0 (2.10) becomes the following correspondence between the wavefunctions of the five-vertex model and the β = −1 factorial Grothendieck polynomials [9,12] W m+n−k,n (u 1 , . .…”