[Paper Review] Partition functions of integrable lattice models and combinatorics of symmetric polynomials

Kohei Motegi, Kazumitsu Sakai+1 more|arXiv (Cornell University)|Dec 25, 2015

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper establishes a deep correspondence between partition functions of integrable lattice models—specifically XXZ-type and Felderhof models—and symmetric polynomials such as Schur, Grothendieck, and symplectic Schur functions. By deriving wavefunctions from L-operators and boundary conditions, the authors prove generalized dual Cauchy identities that link these partition functions to factorial and symplectic Schur functions, revealing new combinatorial structures via quantum integrability.

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.

Motivation & Objective

• To establish a systematic correspondence between integrable lattice model partition functions and symmetric polynomials.
• To explore how wavefunctions in XXZ-type and Felderhof models generate Schur, Grothendieck, and symplectic Schur functions.
• To generalize the correspondence to include colored representations and extended spectral parameters.
• To derive new dual Cauchy identities for factorial and symplectic Schur functions using domain wall and Tsuchiya boundary conditions.
• To demonstrate that quantum integrability provides a powerful framework for discovering and proving combinatorial identities in symmetric function theory.

Proposed method

• Derive wavefunctions from L-operators satisfying the RLL relation, using spectral parameters and additional deformation parameters.
• Construct partition functions with domain wall and Tsuchiya boundary conditions to generate symmetric polynomials.
• Apply the Yang-Baxter relation and R-matrix formalism to ensure integrability and consistency of the model.
• Use determinant representations and generating functions to express generalized Schur and Grothendieck polynomials.
• Derive dual Cauchy identities by evaluating scalar products and partition functions under specific boundary conditions.
• Generalize the correspondence to include factorial and symplectic Schur functions via extended parameter sets and modified L-operators.

Experimental results

Research questions

• RQ1 How do wavefunctions of XXZ-type and Felderhof models correspond to Schur and Grothendieck polynomials?
• RQ2 What is the role of spectral parameters and additional deformation parameters in generating symmetric polynomials from lattice partition functions?
• RQ3 How do domain wall and Tsuchiya boundary conditions lead to dual Cauchy identities for factorial and symplectic Schur functions?
• RQ4 What is the algebraic structure underlying the generalized correspondence between integrable models and symmetric polynomials?
• RQ5 Can quantum integrability provide new combinatorial formulae for symmetric functions that are otherwise difficult to derive?

Key findings

• The wavefunction of the XXZ-type model is identified as a q-deformation of the β-Grothendieck polynomial, with the q-parameter tied to the quantum group Uq(sl2).
• For the Felderhof model, the partition function with domain wall boundaries yields a dual Cauchy identity involving factorial Schur functions with parameters α and γ.
• The Tsuchiya boundary condition leads to the appearance of symplectic Schur functions in the partition function, generalizing earlier results.
• A new dual Cauchy identity is derived for generalized factorial symplectic Schur functions, involving products over (1 + αj(γk − γj)) and (1 − γjγk) terms.
• The generalized symplectic Schur function is defined via a determinant of a matrix involving gμ(z|{α}, {γ}) functions, providing a new representation for these symmetric polynomials.
• The correspondence reveals that integrable models naturally encode deep combinatorial identities in algebraic combinatorics, especially in Schubert calculus.