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Integrable systems and crystals for edge labeled tableaux

Ajeeth Gunna, Travis Scrimshaw

Abstract

We introduce the edge Schur functions $E^λ$ that are defined as a generating series over edge labeled tableaux. We formulate $E^λ$ as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of $E^λ$ and show it intertwines with an uncrowding algorithm.

Integrable systems and crystals for edge labeled tableaux

Abstract

We introduce the edge Schur functions that are defined as a generating series over edge labeled tableaux. We formulate as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of and show it intertwines with an uncrowding algorithm.
Paper Structure (12 sections, 8 theorems, 79 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tableau generating functions
  3. Factorial Schur functions
  4. Edge Schur functions
  5. Lattice models
  6. Conventions
  7. Lattice model for edge Schur functions
  8. Lattice model for factorial Schur polynomials
  9. Cauchy identity
  10. Basis properties and variations
  11. Crystallization
  12. Crystal preliminaries

Key Result

Proposition 3.3

This vertex model is integrable, which means there exists an $R$-matrix that satisfies the $RLL$ form of the Yang--Baxter equation in $\mathop{\mathrm{End}}\nolimits(H_i \otimes H_j \otimes V_k)$: which are represented pictorially, respectively, as

Figures (1)

  • Figure 1:

Theorems & Definitions (27)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4: edge Schur functions
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • Remark 3.5
  • ...and 17 more