Properties of plactic monoid centralizers
Bruce E. Sagan, Chenchen Zhao
TL;DR
This work investigates centralizers in the plactic monoid, focusing on stability phenomena: for a word $u$, whether $C(u^k)$ eventually stabilizes as $k$ grows and, in particular, proving strong stability for words over $\{1,2\}$ and for all permutations. The authors develop insertion-tableau and jeu-de-taquin techniques to characterize centralizers, establishing stability results and obtaining detailed descriptions of $\mathcal C_{n,m}$-type enumerations. They advance the understanding of the invariant $c_{n,m}(u)$, proving polynomial dependence on $m$ for $u=1$ and showing nonnegativity of coefficients in binomial expansions, with combinatorial interpretations via $k$-packed words. These results connect Knuth equivalence, RSK, and Greene’s theorem to stability phenomena with implications in representation theory and algebraic combinatorics.
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.