Properties of plactic monoid centralizers

TL;DR

This paper investigates the structure of centralizers in the plactic monoid, proving stability properties for certain words and confirming conjectures about polynomial coefficients related to these centralizers.

Contribution

It establishes the stability phenomenon for centralizers of powers of specific words and proves conjectures about the nonnegativity of coefficients in related polynomial expansions.

Findings

Stability of centralizers C(u^k) for various words u.

Proof that coefficients c_{n,m}(u) are nonnegative integers.

Confirmation of conjectures regarding polynomial coefficients in the context of plactic monoid centralizers.

Abstract

Let u be a word over the positive integers P. Motivated by a question involving crystal graphs, Sagan and Wilson initiated the study of the centralizer of u in the plactic monoid which is the set C(u) = {w | uw is Knuth equivalent to wu}. In particular, they conjectured the following stability phenomenon: for any u there is a positive integer K depending only on u such that C(u^k) = C(u^K) for k >= K. We prove that this property holds for various u including words consisting of only ones and twos, as well as permutations. Sagan and Wilson also considered c_{n,m}(u) which is the number of w in C(u) of length n and maximum at most m. They showed that c_{n,m}(1) is a polynomial in m of degree n-1 and conjectured properties of the coefficients when it is expanded in a binomial coefficient basis. We prove some of these conjectures, for example, that the coefficients are always nonnegative integers.