Properties of plactic monoid centralizers

Let u be a word over the positive integers and consider its
centralizer C(u) in the plactic monoid. Motivated by questions from
crystal theory, Sagan and Wilson conjectured a stability phenomenon:
as we take higher and higher powers of a word u, the collection of
words that commute with it eventually stops changing.

In this talk, I will explain this stabilization phenomenon and present
results proving it for several natural families of words, including
words built from 1s and 2s and permutations. I will also discuss
related counting questions that arise from studying these commuting
words. This is joint work with Bruce Sagan.