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Centralizers in the plactic monoid

Bruce E. Sagan, Alexander N. Wilson

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.

Centralizers in the plactic monoid

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.

Paper Structure

This paper contains 6 sections, 19 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Necessary Conditions
  3. Commuting with short $u$
  4. Commuting with longer $u$
  5. Enumeration
  6. Open problems and conjectures

Key Result

Lemma 2.1

Let $a\neq b$ be distinct positive integers and let $w\in{\mathbb P}^*$. Then

Figures (1)

  • Figure 1: A semistandard Young tableau (SSYT), $P$

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 30 more