Topological combinatorics of crystal posets
Crystal bases were first introduced by Kashiwara when studying modules of quantum groups. Each crystal base has an associated directed, edge colored graph called a crystal graph. In many cases, these crystal graphs give rise to a natural partial order. In this talk, we study crystal posets associated to highest weight representations. We use lexicographic discrete Morse functions to connect the Möbius function of an interval in a crystal poset with the relations that exist among crystal operators within that interval. We will discuss some further directions for this work.