Markov chains on Weyl groups from the geometry of the flag variety
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911248500654080 |
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| author | Diaconis, Persi Morton-Ferguson, Calder |
| author_facet | Diaconis, Persi Morton-Ferguson, Calder |
| contents | This paper studies a basic Markov chain, the Burnside process, on the space of flags $G/B$ with $G = GL_n(\mathbb{F}_q)$ and $B$ its upper triangular matrices. This gives rise to a shuffling: a Markov chain on the symmetric group realized via the Bruhat decomposition. Actually running and describing this Markov chain requires understanding Springer fibers and the Steinberg variety. The main results give a practical algorithm for all n and q and determine the limiting behavior of the chain when q is large. In describing this behavior, we find interesting connections to the combinatorics of the Robinson-Schensted correspondence and to the geometry of orbital varieties. The construction and description is then carried over to finite Chevalley groups of arbitrary type, describing a new class of Markov chains on Weyl groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_02285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Markov chains on Weyl groups from the geometry of the flag variety Diaconis, Persi Morton-Ferguson, Calder Probability Combinatorics Representation Theory 60J10, 14M15 (Primary) 05E10, 05A17, 20G40 (Secondary) This paper studies a basic Markov chain, the Burnside process, on the space of flags $G/B$ with $G = GL_n(\mathbb{F}_q)$ and $B$ its upper triangular matrices. This gives rise to a shuffling: a Markov chain on the symmetric group realized via the Bruhat decomposition. Actually running and describing this Markov chain requires understanding Springer fibers and the Steinberg variety. The main results give a practical algorithm for all n and q and determine the limiting behavior of the chain when q is large. In describing this behavior, we find interesting connections to the combinatorics of the Robinson-Schensted correspondence and to the geometry of orbital varieties. The construction and description is then carried over to finite Chevalley groups of arbitrary type, describing a new class of Markov chains on Weyl groups. |
| title | Markov chains on Weyl groups from the geometry of the flag variety |
| topic | Probability Combinatorics Representation Theory 60J10, 14M15 (Primary) 05E10, 05A17, 20G40 (Secondary) |
| url | https://arxiv.org/abs/2510.02285 |