Koszulity, supersolvability, and Stirling representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911141786025984 |
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| author | Almousa, Ayah Reiner, Victor Sundaram, Sheila |
| author_facet | Almousa, Ayah Reiner, Victor Sundaram, Sheila |
| contents | Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik-Solomon algebras and graded Varchenko-Gel'fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_10858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Koszulity, supersolvability, and Stirling representations Almousa, Ayah Reiner, Victor Sundaram, Sheila Combinatorics Commutative Algebra Rings and Algebras 16S37, 05B35 Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik-Solomon algebras and graded Varchenko-Gel'fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality. |
| title | Koszulity, supersolvability, and Stirling representations |
| topic | Combinatorics Commutative Algebra Rings and Algebras 16S37, 05B35 |
| url | https://arxiv.org/abs/2404.10858 |