Robinson-Schensted shapes arising from cycle decompositions
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arXiv
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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916541427089408 |
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| author | Preez, Martha Du Erickson, William Q. Feigert, Jonathan Hunziker, Markus Meddaugh, Jonathan Minyard, Mitchell Sepanski, Mark R. Rosengartner, Kyle |
| author_facet | Preez, Martha Du Erickson, William Q. Feigert, Jonathan Hunziker, Markus Meddaugh, Jonathan Minyard, Mitchell Sepanski, Mark R. Rosengartner, Kyle |
| contents | In the symmetric group $S_n$, each element $σ$ has an associated cycle type $α$, a partition of $n$ that identifies the conjugacy class of $σ$. The Robinson-Schensted (RS) correspondence links each $σ$ to another partition $λ$ of $n$, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to $σ$. Surprisingly, the relationship between these two partitions, namely the cycle type $α$ and the RS shape $λ$, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes $λ$ that can arise from elements of each cycle type $α$ in cases where $α$ consists of two cycles. To do this, we introduce the notion of an $α$-coloring, where one colors the entries in a certain tableau of shape $λ$, in such a way as to construct a permutation $σ$ with cycle type $α$ and RS shape $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Robinson-Schensted shapes arising from cycle decompositions Preez, Martha Du Erickson, William Q. Feigert, Jonathan Hunziker, Markus Meddaugh, Jonathan Minyard, Mitchell Sepanski, Mark R. Rosengartner, Kyle Combinatorics 05E10 (Primary) 20B30 (Secondary) In the symmetric group $S_n$, each element $σ$ has an associated cycle type $α$, a partition of $n$ that identifies the conjugacy class of $σ$. The Robinson-Schensted (RS) correspondence links each $σ$ to another partition $λ$ of $n$, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to $σ$. Surprisingly, the relationship between these two partitions, namely the cycle type $α$ and the RS shape $λ$, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes $λ$ that can arise from elements of each cycle type $α$ in cases where $α$ consists of two cycles. To do this, we introduce the notion of an $α$-coloring, where one colors the entries in a certain tableau of shape $λ$, in such a way as to construct a permutation $σ$ with cycle type $α$ and RS shape $λ$. |
| title | Robinson-Schensted shapes arising from cycle decompositions |
| topic | Combinatorics 05E10 (Primary) 20B30 (Secondary) |
| url | https://arxiv.org/abs/2412.18058 |