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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.10462 |
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| _version_ | 1866910342294011904 |
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| author | Deng, Yongtao Tan, Shi Jie Samuel |
| author_facet | Deng, Yongtao Tan, Shi Jie Samuel |
| contents | In this paper, we present a detailed proof for the exhibition of a cutoff for the one-sided transposition (OST) shuffle on the generalized symmetric group $G_{m,n}$. Our work shows that based on techniques for $m \leq 2$ proven by Matheau-Raven, we can prove the cutoff in total variation distance and separation distance for an unbiased OST shuffle on $G_{m,n}$ for any fixed $m \geq 1$ in time $n \log(n)$. We also prove the branching rules for the simple modules of $G_{m,n}$ and lay down some of the mathematical foundation for proving the conjecture for the cutoff in total variation distance for any general biased OST shuffle on $G_{m,n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10462 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Random Walks on the Generalized Symmetric Group: Cutoff for the One-sided Transposition Shuffle Deng, Yongtao Tan, Shi Jie Samuel Probability Group Theory Representation Theory 60J10, 20B05, 20B25 In this paper, we present a detailed proof for the exhibition of a cutoff for the one-sided transposition (OST) shuffle on the generalized symmetric group $G_{m,n}$. Our work shows that based on techniques for $m \leq 2$ proven by Matheau-Raven, we can prove the cutoff in total variation distance and separation distance for an unbiased OST shuffle on $G_{m,n}$ for any fixed $m \geq 1$ in time $n \log(n)$. We also prove the branching rules for the simple modules of $G_{m,n}$ and lay down some of the mathematical foundation for proving the conjecture for the cutoff in total variation distance for any general biased OST shuffle on $G_{m,n}$. |
| title | Random Walks on the Generalized Symmetric Group: Cutoff for the One-sided Transposition Shuffle |
| topic | Probability Group Theory Representation Theory 60J10, 20B05, 20B25 |
| url | https://arxiv.org/abs/2211.10462 |