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Unveiling Crystal Embeddings: New Perspectives on String Polytopes and Atomic Decompositions

Lara Bossinger, Jacinta Torres

TL;DR

This work constructs a family of embeddings $\phi_j$ between string polytopes $\mathcal{S}_{\mathbf{i}}(\lambda)$ and $\mathcal{S}_{\mathbf{i}}(\lambda+\theta)$ in type $A_{n-1}$, where $\theta$ is the highest root, and analyzes their compatibility with the crystal structure. For the endpoint indices $j=1$ and $j=n-1$, these embeddings are crystal morphisms, while other $\phi_j$ exhibit structured compatibility, all culminating in a conjecture that one can produce $n$ distinct atomic decompositions of crystals with highest weight multiples of $\theta$. The paper develops embeddings for arbitrary reduced expressions, leveraging weight-shifting vectors $z_i$ and projection maps that relate to $\mathfrak{sl}_{n-1}$, and provides explicit realizations of the crystal operators via wiring diagrams and Gleizer–Postnikov rigorous paths. If validated, the conjectures offer a new, geometric mechanism to construct atomic decompositions and positively compute Kostka–Foulkes-type polynomials beyond type $A$, linking crystal theory, affine Hecke algebras, and pre-canonical bases. Overall, the approach extends the landscape of atomic decompositions in crystals and suggests a broad, embedding-based strategy for positive combinatorial formulas in representation theory.

Abstract

We present n-1 different embeddings of string polytopes of type A. We characterize their compatibility with the crystal structure on the string polytopes, and formulate a conjecture describing how to obtain n-1 different atomic decompositions of the crystal with highest weight a multiple of the highest root.

Unveiling Crystal Embeddings: New Perspectives on String Polytopes and Atomic Decompositions

TL;DR

This work constructs a family of embeddings between string polytopes and in type , where is the highest root, and analyzes their compatibility with the crystal structure. For the endpoint indices and , these embeddings are crystal morphisms, while other exhibit structured compatibility, all culminating in a conjecture that one can produce distinct atomic decompositions of crystals with highest weight multiples of . The paper develops embeddings for arbitrary reduced expressions, leveraging weight-shifting vectors and projection maps that relate to , and provides explicit realizations of the crystal operators via wiring diagrams and Gleizer–Postnikov rigorous paths. If validated, the conjectures offer a new, geometric mechanism to construct atomic decompositions and positively compute Kostka–Foulkes-type polynomials beyond type , linking crystal theory, affine Hecke algebras, and pre-canonical bases. Overall, the approach extends the landscape of atomic decompositions in crystals and suggests a broad, embedding-based strategy for positive combinatorial formulas in representation theory.

Abstract

We present n-1 different embeddings of string polytopes of type A. We characterize their compatibility with the crystal structure on the string polytopes, and formulate a conjecture describing how to obtain n-1 different atomic decompositions of the crystal with highest weight a multiple of the highest root.

Paper Structure

This paper contains 9 sections, 8 theorems, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Crystals and string polytopes
  3. Tensor products of crystals
  4. String cones and polytopes
  5. String polytopes via wiring diagrams
  6. Rigorous paths
  7. Crystal structure
  8. Embeddings for any reduced expression
  9. Outlook: Atomic decompositions

Key Result

Theorem 2.3

The cone $\mathcal{S}_{\underline{i}}$ is a semigroup, that is, $x+y \in \mathcal{S}_{\underline{i}}$ for $x,y \in \mathcal{S}_{\underline{i}}$.

Figures (4)

  • Figure 1: The wiring diagram $\operatorname{wd}(\mathbf{i})$ for $n = 5$ and $\mathbf{i} = (1,2,3,4,1,2,3,1,2,1)$ with orientation $(\ell_3,\ell_4)$. In blue, a rigorous path in $\Gamma_3$ determined by its turning points $v_{3,1}$ and $v_{1,4}$.
  • Figure 2: On the left (resp. right) the two red arrows are forbidden in left (resp. right) rigorous paths.
  • Figure 3: Half of the associated crystal graph $\mathcal{B}(\theta)$ for $\mathfrak{sl}(5,\mathbb{C})$ and its adjoint representation with highest root $\theta = \omega_{1} + \omega_{4}$. Arrows $a \overset{i}{\longrightarrow} b$ mean $f_{i}(a) = b$.
  • Figure 4: Notice how the subword of ${\bf i}=(1,3,2,4,1,3,2,4,3,1)$ corresponding to $z_1=f_1f_2f_3f_4.b_{\theta}$ (resp. $z_4=f_4f_3f_2f_1.b_{\theta}$) picks up all crossings along the wire $\ell_1$ (resp. $\ell_5$). This is not true for other $z_i$, $i\not\in\{1,4\}$, as for example, $z_3=f_3f_4f_2f_1.b_\theta$ is parametrized by the crossings marked by ${\color{green}\bullet}$ which are spread over different wires.

Theorems & Definitions (27)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3: Berenstein--Zelevinsky BerensteinZelevinsky
  • Theorem 2.4: Littelmann conescrystalspatterns
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4: Gleizer-Postnikov Corollary 5.8, beaglebvolker Theorem 6.1
  • Definition 3.5
  • Definition 3.6
  • ...and 17 more