Centralizers in the plactic monoid

TL;DR

This paper investigates the structure of centralizers in the plactic monoid, providing characterizations, necessary conditions, and polynomial formulas for counting elements, with implications for representation theory.

Contribution

It offers new characterizations of centralizers in the plactic monoid and introduces polynomial formulas for counting elements based on word length and maximum value.

Findings

Characterization of C(u) for words with few letters or specific sequences

Polynomial formulas for c_{n,m}(u) depending on word length and maximum value

Dependence of c_{n,m}(u) on relative sizes of u and m for single-letter words

Abstract

Let u be a word over the positive integers. Motivated in part by a question from representation theory, we study the centralizer set of u which is C(u) = {w | uw is Knuth-equivalent to wu}. In particular, we give various necessary conditions for w to be in C(u). We also characterize C(u) when u has few letters, when it has a single repeated entry, or when it is a certain type of decreasing sequence. We consider c_{n,m}(u), the number of w in C(u) of length n with max w at most m. We prove that for |u| = 1 the value of this function depends only on the relative sizes of u and m and not on their actual values. And for various u we use Stanley's theory of poset partitions to show that, for fixed n, c_{n,m}(u) is a polynomial in m with certain degree and leading coefficient. We end with various conjectures and directions for further research.