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arXiv · 2410.20460

Centralizers in the plactic monoid

Bruce E. Sagan·Alexander N. Wilson
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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.

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Bruce E. Sagan, Alexander N. Wilson. 2024-10-27. Centralizers in the plactic monoid. https://arxiv.org/abs/2410.20460

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