[Paper Review] Colored five-vertex models and Demazure atoms

Ben Brubaker, Valentin Buciumas+2 more|arXiv (Cornell University)|Feb 5, 2019

Algebraic structures and combinatorial models45 references4 citations

TL;DR

This paper establishes a bijection between colored five-vertex ice models and Demazure atoms in the crystal basis of a highest-weight representation of a Lie algebra. Using colored spins and vertex weights, it shows that the partition function of the colored model decomposes into components indexed by Weyl group elements, with each component corresponding to a Demazure atom via a Sch"utzenberger involution. The key result is a combinatorial realization of Demazure atoms through statistical mechanics models.

ABSTRACT

Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a biproduct, we construct Demazure atoms on Kashiwara's $\mathcal{B}_\infty$ crystal and give new algorithms for computing Lascoux-Schützenberger keys.

Motivation & Objective

• To establish a combinatorial correspondence between colored five-vertex models and Demazure atoms in crystal bases.
• To refine the understanding of Demazure atoms by indexing them via Weyl group elements and coset representatives.
• To provide a statistical mechanical realization of the Demazure atom decomposition using vertex models with colored edges.
• To prove that the image of colored ice states under a crystal map corresponds precisely to Demazure atoms via the Sch"utzenberger involution.

Proposed method

• Uses a colored five-vertex model where edges carry spins and colors, with weights defined by Figure LABEL:coloredweights.
• Applies a deterministic coloring rule to $-$ spins based on adjacent colored spins and vertex weights, ensuring color conservation at vertices.
• Shows that each colored state belongs to a unique component $\mathfrak{S}_{\mathbf{z},\lambda,w}$ indexed by $w \in W$, the Weyl group.
• Establishes a map $\mathfrak{s} \mapsto \mathfrak{T}(\mathfrak{s})'$ from colored ice states to the crystal $\mathcal{B}_\lambda$.
• Uses the Sch"utzenberger involution $v \mapsto v'$ and the map $\omega$ to relate the Weyl group element $w$ to the image of the state.
• Proves that $\mathfrak{s} \in \mathfrak{S}_{\mathbf{z},\lambda,w}$ if and only if $w_0 \omega(\mathfrak{T}(\mathfrak{s})') = w$, linking the model to Demazure atoms.

Experimental results

Research questions

• RQ1 How can Demazure atoms in a crystal basis be realized combinatorially through statistical mechanics models?
• RQ2 What is the precise correspondence between colored ice states and Weyl group elements in the context of Demazure atoms?
• RQ3 How does the Sch"utzenberger involution relate the state space of colored ice models to the Demazure atom decomposition?
• RQ4 Can the partition function of a colored five-vertex model be decomposed into components indexed by Weyl group elements, each corresponding to a Demazure atom?
• RQ5 What role does the Yang-Baxter equation play in ensuring the consistency of the colored vertex model and its relation to integrability?

Key findings

• The partition function of the colored five-vertex model decomposes as $Z(\mathfrak{S}_{\mathbf{z},\lambda}) = \sum_{w \in W} Z(\mathfrak{S}_{\mathbf{z},\lambda,w})$, showing a direct decomposition into Weyl group-indexed components.
• Each colored ice state belongs to a unique $\mathfrak{S}_{\mathbf{z},\lambda,w}$, with the coloring determined uniquely by vertex rules and color conservation.
• The map $\mathfrak{s} \mapsto \mathfrak{T}(\mathfrak{s})'$ sends $\mathfrak{S}_{\mathbf{z},\lambda,w}$ bijectively to the Demazure atom $\mathcal{B}_\lambda^\circ(w)$, establishing a combinatorial isomorphism.
• The condition $w_0 \omega(\mathfrak{T}(\mathfrak{s})') = w$ characterizes membership of a state $\mathfrak{s}$ in $\mathfrak{S}_{\mathbf{z},\lambda,w}$, providing an algorithmic criterion.
• The Yang-Baxter equation holds for the colored model, ensuring integrability and consistency of the partition function across different vertex configurations.
• The proof relies on a finite case analysis (4096 cases) verified via Sage, confirming the Yang-Baxter equation for all possible color configurations on boundary edges.