The ring of stable characters over $\text{GL}_\bullet(q)$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916410707410944 |
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| author | Ernst-West, Danielle Puder, Doron Shomroni, Yotam |
| author_facet | Ernst-West, Danielle Puder, Doron Shomroni, Yotam |
| contents | For a fixed prime power $q$, let $\text{GL}_\bullet(q)$ denote the family of groups $\text{GL}_N(q)$ for $N \in \mathbb{Z}_{\geq 0}$. In this paper we study the $\mathbb{C}$-algebra of "stable" class functions of $\text{GL}_\bullet(q)$, and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple $\mathrm{VI}$-modules in the sense of [arXiv:1408.3694,arXiv:1602.00654]. A second one comes from characters of parabolic representations. The final two, one originally defined in [arXiv:1803.04155] and the other in [arXiv:2110.11099], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [arXiv:2106.11587]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16571 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ring of stable characters over $\text{GL}_\bullet(q)$ Ernst-West, Danielle Puder, Doron Shomroni, Yotam Combinatorics Group Theory Representation Theory 05e05 (Primary) 20c33, 05e10, 05e16 (Secondary) For a fixed prime power $q$, let $\text{GL}_\bullet(q)$ denote the family of groups $\text{GL}_N(q)$ for $N \in \mathbb{Z}_{\geq 0}$. In this paper we study the $\mathbb{C}$-algebra of "stable" class functions of $\text{GL}_\bullet(q)$, and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple $\mathrm{VI}$-modules in the sense of [arXiv:1408.3694,arXiv:1602.00654]. A second one comes from characters of parabolic representations. The final two, one originally defined in [arXiv:1803.04155] and the other in [arXiv:2110.11099], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [arXiv:2106.11587]. |
| title | The ring of stable characters over $\text{GL}_\bullet(q)$ |
| topic | Combinatorics Group Theory Representation Theory 05e05 (Primary) 20c33, 05e10, 05e16 (Secondary) |
| url | https://arxiv.org/abs/2409.16571 |