Relations in doubly laced crystal graphs via discrete Morse theory

TL;DR

This paper investigates the combinatorial structure of crystal graphs from classical Lie algebra representations, revealing new relations among crystal operators using poset topology and discrete Morse theory, especially in doubly laced types.

Contribution

It uncovers previously unknown relations among crystal operators in classical types and links these relations to the M"obius function of crystal posets using discrete Morse theory.

Findings

New relations among crystal operators in type C_n.

Crystals of types B_2 and C_2 are not lattices.

Intervals with nontrivial M"obius function imply new relations.

Abstract

We study the combinatorics of crystal graphs given by highest weight representations of types $A_{n}, B_{n}, C_{n}$, and $D_{n}$, uncovering new relations that exist among crystal operators. Much structure in these graphs has been revealed by local relations given by Stembridge and Sternberg. However, there exist relations among crystal operators that are not implied by Stembridge or Sternberg relations. Viewing crystal graphs as edge colored posets, we use poset topology to study them. Using the lexicographic discrete Morse functions of Babson and Hersh, we relate the M\"obius function of a given interval in a crystal poset of simply laced or doubly laced type to the types of relations that can occur among crystal operators within this interval. For a crystal of a highest weight representation of finite classical Cartan type, we show that whenever there exists an interval whose M\"obius function is not equal to -1, 0, or 1, there must be a relation among crystal operators within this interval not implied by Stembridge or Sternberg relations. As an example of an application, this yields relations among crystal operators in type $C_{n}$ that were not previously known. Additionally, by studying the structure of Sternberg relations in the doubly laced case, we prove that crystals of highest weight representations of types $B_{2}$ and $C_{2}$ are not lattices.