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Closed Colored Models and Demazure Crystals

Yingzi Yang

TL;DR

The paper constructs closed colored five-vertex lattice models whose partition functions realize Demazure characters, and embeds model states into Kashiwara-Nakashima crystals. It proves a state-level bijection between closed states with a fixed Gelfand–Tsetlin pattern and the corresponding Demazure crystal B_λ(y), extending known open-case results. The proof relies on two interrelated state-adjustment procedures, Bruhat-order arguments, and crystal-operator analysis of GT-patterns and recolorings. Together, these results provide a crystal-theoretic interpretation of closed-state spaces and deepen the link between solvable lattice models and representation theory.

Abstract

We will construct solvable lattice models whose partition functions are Demazure characters. We will construct a crystal structure on the states of the model and prove that the states of the closed model form a Demazure crystal.

Closed Colored Models and Demazure Crystals

TL;DR

The paper constructs closed colored five-vertex lattice models whose partition functions realize Demazure characters, and embeds model states into Kashiwara-Nakashima crystals. It proves a state-level bijection between closed states with a fixed Gelfand–Tsetlin pattern and the corresponding Demazure crystal B_λ(y), extending known open-case results. The proof relies on two interrelated state-adjustment procedures, Bruhat-order arguments, and crystal-operator analysis of GT-patterns and recolorings. Together, these results provide a crystal-theoretic interpretation of closed-state spaces and deepen the link between solvable lattice models and representation theory.

Abstract

We will construct solvable lattice models whose partition functions are Demazure characters. We will construct a crystal structure on the states of the model and prove that the states of the closed model form a Demazure crystal.

Paper Structure

This paper contains 6 sections, 27 theorems, 91 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Strategy of proofs.
  4. Acknowledgements
  5. Colored lattice models
  6. Gelfand-Tsetlin patterns and semistandard Young tableau

Key Result

Theorem 1.1

For $y\in W$, let $\mathfrak{S}^\circ_{\lambda}(y)$ (by abuse of notation) be the set of states for the model $\mathfrak{S}^\circ_{\lambda}(y)$. Let $v\mapsto v'$ be the Lascoux-Schützenberger involution on $\mathcal{B}_\lambda$. If $y\in W$ is the longest element of the coset in $W/W_\lambda$, then

Figures (15)

  • Figure 1: Boundary conditions for $\mathfrak{S}^\bullet_{(3,2,0)}(1\,2\,3)$.
  • Figure 2: A closed state for $\mathfrak{S}^\bullet_{(3,2,0)}(1\,2\,3)$ with $\text{red} > \text{blue} > \text{green}.$
  • Figure 3: An open state for $\mathfrak{S}^\circ_{(3,2,0)}(1\,2\,3)$ with $\text{red} > \text{blue} > \text{green}.$
  • Figure 4: Illustration of a closed adjustment, with red $>$ blue.
  • Figure 5: A closed state for $\mathfrak{S}_{(3,2,0),(1\,2\,3)}.$
  • ...and 10 more figures

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 59 more