Centralizers in the plactic monoid
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910672392028160 |
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| author | Sagan, Bruce E. Wilson, Alexander N. |
| author_facet | Sagan, Bruce E. Wilson, Alexander N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Centralizers in the plactic monoid Sagan, Bruce E. Wilson, Alexander N. Combinatorics 05E99 (Primary) 05A05, 05A15 (Secondary) 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. |
| title | Centralizers in the plactic monoid |
| topic | Combinatorics 05E99 (Primary) 05A05, 05A15 (Secondary) |
| url | https://arxiv.org/abs/2410.20460 |