Factorial Schur functions and the Yang-Baxter equation

Abstract

Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system (six vertex model). The proof is based on the Yang-Baxter equation. There is a deformation parameter $t$ which may be specialized in different ways. If $t=-1$, then we recover the expression of the factorial Schur function as a ratio of alternating polynomials. If $t=0$, we recover the description as a sum over tableaux. If $t=\infty$ we recover a description of Lascoux that was previously considered by McNamara. We also are able to prove using the Yang-Baxter equation the asymptotic symmetry of the factorial Schur functions in the shift parameters. Finally, we give a proof using our methods of the dual Cauchy identity for factorial Schur functions. Thus using our methods we are able to give thematic proofs of many of the properties of factorial Schur functions.

References in corpus (2)

• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• Factorial Schur functions via the six vertex model

Cited by in corpus (20)

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• Equivariant quantum cohomology and Yang-Baxter algebras
• Metaplectic Ice for Cartan Type C
• Dual wavefunction of the Felderhof model
• Frozen Pipes: Lattice Models for Grothendieck Polynomials
• Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice
• Quantum integrable combinatorics of Schur polynomials
• Yang-Baxter algebras as convolution algebras: The Grassmannian case
• Metaplectic Iwahori Whittaker functions and supersymmetric lattice models
• Tokuyama's Identity for Factorial Schur Functions
• Free Fermionic Schur Functions
• Quantum inverse scattering method and generalizations of symplectic Schur functions and Whittaker functions
• Scalar products of the elliptic Felderhof model and elliptic Cauchy formula
• Lattice Models, Hamiltonian Operators, and Symmetric Functions
• Factorial characters of classical Lie groups and their combinatorial realisations
• Integrable models and $K$-theoretic pushforward of Grothendieck classes
• Integrable systems and crystals for edge labeled tableaux