Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices

Abstract

We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry properties. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity, allowing them to be interpreted as generating series for plane partitions. The right hand side of each identity is a partition function of the six-vertex model, on a relevant domain. These can be interpreted as generating series for alternating sign matrices, using the well known bijection with six-vertex model configurations.

29 pages, 20 figures, new subsections on partition functions of rectangular-shaped lattices added in v2

References in corpus (1)

• Multivariate Quadratic Transformations and the Interpolation Kernel

Cited by in corpus (17)

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• Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Symmetric functions and wavefunctions of the six-vertex model by Izergin-Korepin analysis
• Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators
• Dual wavefunction of the Felderhof model
• Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice
• Quantum integrable combinatorics of Schur polynomials
• Integral formulas and antisymmetrization relations for the six-vertex model
• Dual wavefunction of the symplectic ice
• Izergin-Korepin analysis on the projected wavefunctions of the generalized free-fermion model
• Symmetry classes of alternating sign matrices in the nineteen-vertex model
• The domain wall partition function for the Izergin-Korepin 19-vertex model at a root of unity
• Quantum inverse scattering method and generalizations of symplectic Schur functions and Whittaker functions
• A class of partition functions associated with $E_{\tau ,\eta }(g{l}_3)$ by Izergin-Korepin analysis
• Scalar products of the elliptic Felderhof model and elliptic Cauchy formula
• Integrable models and $K$-theoretic pushforward of Grothendieck classes