Relations in doubly laced crystal graphs via discrete Morse theory

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 Mobius 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 Mobius 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.