Representation theory and cycle statistics for random walks on the symmetric group
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915678545510400 |
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| author | Arcona, Dominic |
| author_facet | Arcona, Dominic |
| contents | We use representation theory of $S_n$ to analyze the mixing of permutation cycle type statistics $a_j(σ) = ${# of $j$-cycles of $σ$} for any fixed $j$ and $σ$ resulting from a random $i$-cycle walk on $S_n$. We also derive analogous results for the random star transposition walk. Our approach uses the method of moments; a key ingredient is a new formula for the coefficients in the irreducible character decomposition of the $S_n$-class function $(a_j)^r(σ)=\{(\text{# of $j$-cycles of $σ$})^r\}$ for any positive integers $r,j$ when $n\geq 2rj$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13969 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representation theory and cycle statistics for random walks on the symmetric group Arcona, Dominic Combinatorics Probability Representation Theory We use representation theory of $S_n$ to analyze the mixing of permutation cycle type statistics $a_j(σ) = ${# of $j$-cycles of $σ$} for any fixed $j$ and $σ$ resulting from a random $i$-cycle walk on $S_n$. We also derive analogous results for the random star transposition walk. Our approach uses the method of moments; a key ingredient is a new formula for the coefficients in the irreducible character decomposition of the $S_n$-class function $(a_j)^r(σ)=\{(\text{# of $j$-cycles of $σ$})^r\}$ for any positive integers $r,j$ when $n\geq 2rj$. |
| title | Representation theory and cycle statistics for random walks on the symmetric group |
| topic | Combinatorics Probability Representation Theory |
| url | https://arxiv.org/abs/2512.13969 |