Quantum integrability and generalised quantum Schubert calculus

2017

Lett Math Phys

We study the Felderhof free-fermion six-vertex model, whose wavefunction recently turned out to possess rich combinatorial structure of the Schur polynomials. We investigate the dual version of the wavefunction in this paper, which seems to be a harder object to analyze. We evaluate the dual wavefunction in two ways. First, we give the exact correspondence between the dual wavefunction and the Schur polynomials, for which two proofs are given. Next, we make a microscopic analysis and express the dual wavefunction in terms of strict Gelfand-Tsetlin pattern. As a consequence of these two ways of evaluation of the dual wavefunction, we obtain a dual version of the Tokuyama combinatorial formula for the Schur polynomials. We also give a generalization of the correspondence between the dual wavefunction of the Felderhof model and the factorial Schur polynomials.Mathematics Subject Classification. 05E05, 05E10, 16T25, 16T30, 17B37.

Section: Resultsmentioning

confidence: 99%

Dual wavefunction of the Felderhof model

2017

Lett Math Phys

“…. , 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 , . .…”

Section: Introductionmentioning

confidence: 99%

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

2020

Nuclear Physics B

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive a new identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a q-deformation of the Guo-Sun identity. *

“…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 .…”

Section: Introductionmentioning

confidence: 99%

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

Journal of Mathematical Physics

2018

We analyze wavefunctions of the six-vertex model by extending the Izergin-Korepin analysis on the domain wall boundary partition functions. We particularly focus on the case with triangular boundary. By using the U q (sl 2 ) R-matrix and a special class of the triangular K-matrix, we first introduce an analogue of the wavefunctions of the integrable six-vertex model with triangular boundary. We first give a characterization of the wavefunctions by extending our recent work of the Izergin-Korepin analysis of the domain wall boundary partition function with triangular boundary, and then determine the explicit form of the symmetric functions representing the wavefunctions by showing that it satisfies all the required properties. We also illustrate the Izergin-Korepin analysis for the case of ordinary wavefunctions as it is the basic case. *