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

TL;DR

This paper introduces multiple new embeddings of string polytopes of type A, explores their compatibility with crystal structures, and proposes a conjecture for atomic decompositions of crystals related to highest weights.

Contribution

It presents n-1 novel embeddings of string polytopes, analyzes their compatibility with crystal structures, and formulates a conjecture on atomic decompositions for crystals with specific highest weights.

Findings

Characterized compatibility of embeddings with crystal structures

Formulated a conjecture on atomic decompositions of crystals

Proposed n-1 different embeddings of string polytopes

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.