Properties of plactic monoid centralizers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917169375215616 |
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| author | Sagan, Bruce E. Zhao, Chenchen |
| author_facet | Sagan, Bruce E. Zhao, Chenchen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Properties of plactic monoid centralizers Sagan, Bruce E. Zhao, Chenchen Combinatorics 05E99 (Primary) 05A05, 05A15 (Secondary) 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. |
| title | Properties of plactic monoid centralizers |
| topic | Combinatorics 05E99 (Primary) 05A05, 05A15 (Secondary) |
| url | https://arxiv.org/abs/2512.21401 |