scholarly journals Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

2019 ◽  
Vol 2019 (757) ◽  
pp. 159-195 ◽  
Author(s):  
Michael Wheeler ◽  
Paul Zinn-Justin
Keyword(s):  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Structure Constants ◽  
Product Rules ◽  
Grothendieck Polynomials

AbstractWe study the Littlewood–Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang–Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood–Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson–Tao and Vakil.

scholarly journals Integrable probability: stochastic vertex models and symmetric functions

2018 ◽  
Author(s):  
Alexei Borodin ◽  
Leonid Petrov
Keyword(s):  
Symmetric Functions ◽  
Height Function ◽  
Rational Functions ◽  
Universality Class ◽  
Integral Representations ◽  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Schur Polynomials ◽  
Higher Spin

This chapter presents the study of a homogeneous stochastic higher spin six-vertex model in a quadrant. For this model concise integral representations for multipoint q-moments of the height function and for the q-correlation functions are derived. At least in the case of the step initial condition, these formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six-vertex model, ASEP, various q-TASEPs, and associated zero-range processes. The arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the higher spin six-vertex model for suitable domains; they generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six-vertex model.

scholarly journals On the Yang–Baxter equation for the six-vertex model

2014 ◽  
Vol 882 ◽  
pp. 70-96 ◽  
Author(s):  
Vladimir V. Mangazeev
Keyword(s):  
Baxter Equation ◽  
Vertex Model

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

A FAMILY OF FACE MODELS RELATED TO ZN-SYMMETRIC VERTEX MODEL OF BELAVIN

2004 ◽  
Vol 19 (supp02) ◽  
pp. 478-509
Author(s):  
Y. YAMADA
Keyword(s):  
Baxter Equation ◽  
Vertex Model ◽  
Face Type ◽  
The Face ◽  
Elliptic Solutions ◽  
Intertwining Relation

We present face-type elliptic solutions to the Yang-Baxter equation. They have 2N-2 real parameters. When specializing them to definite values, we recover the various models so far known. The intertwining relation between the face models above and the ZN-symmetric vertex model of Belavin is also given.

FUSION MATRICES AND C<1 (QUASI) LOCAL CONFORMAL THEORIES

1990 ◽  
Vol 05 (14) ◽  
pp. 2721-2735 ◽  
Author(s):  
P. FURLAN ◽  
A. CH. GANCHEV ◽  
V.B. PETKOVA
Keyword(s):  
Explicit Expression ◽  
Local Theory ◽  
Partition Functions ◽  
Two Dimensional ◽  
Modular Invariant ◽  
General Properties ◽  
Structure Constants ◽  
6J Symbols

General properties of the fusion matrices and their explicit expression given by the Uq( sl (2)) quantum 6j-symbols are exploited to analyze some two dimensional c<1 conformal theories. The primary fields structure constants of the local theory, corresponding to the (A10, E6) modular invariant, and of the Z2 quasilocal analogs of the (A, D) and (A10, E6) series, are computed. The list of the Γ0(2) submodular invariant partition functions on the torus is extended.

scholarly journals Characterizing partition functions of the vertex model

2012 ◽  
Vol 350 (1) ◽  
pp. 197-206 ◽  
Author(s):  
Jan Draisma ◽  
Dion C. Gijswijt ◽  
László Lovász ◽  
Guus Regts ◽  
Alexander Schrijver
Keyword(s):  
Partition Functions ◽  
Vertex Model

Solutions of Yang–Baxter equation in the vertex model and the face model for octet representation

1991 ◽  
Vol 32 (8) ◽  
pp. 2210-2218 ◽  
Author(s):  
Bo‐Yu Hou ◽  
Bo‐Yuan Hou ◽  
Zhong‐Qi Ma ◽  
Yu‐Dong Yin
Keyword(s):  
Baxter Equation ◽  
Vertex Model ◽  
Face Model ◽  
The Face

scholarly journals Cylindric Hecke Characters and Gromov–Witten Invariants via the Asymmetric Six-Vertex Model

2020 ◽  
Author(s):  
Christian Korff
Keyword(s):  
Hecke Algebras ◽  
Square Lattice ◽  
Vertex Operators ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Genus Zero ◽  
Planar Lattice ◽  
Infinite Dimensional ◽  
Structure Constants ◽  
Restriction Functor

AbstractWe construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov–Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual characters for whose computation we give several explicit combinatorial formulae. One of these expressions is a generalisation of Ram’s formula for irreducible Hecke characters and uses cylindric broken rim hook tableaux. We show that the latter are in bijection with so-called ‘ice configurations’ on a cylindrical square lattice, which define the asymmetric six-vertex model in statistical mechanics. A key ingredient of our construction is an extension of the boson-fermion correspondence to Hecke algebras and employing the latter we find new expressions for Jing’s vertex operators of Hall–Littlewood functions in terms of the six-vertex transfer matrices on the infinite planar lattice.

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