On a Bruhat decomposition related to the Shalika subgroup of $GL(2n)$
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2025
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| author | Harshitha, C. Venketasubramanian, C. G. |
| author_facet | Harshitha, C. Venketasubramanian, C. G. |
| contents | Let $F$ be a non-archimedean local field or a finite field. In this article, we obtain an explicit and complete set of double coset representatives for $S\backslash GL_{2n}(F)/Q$ where $S$ is the Shalika subgroup and $Q$ a maximal parabolic subgroup of the group $GL_{2n}(F)$ of invertible $2n\times 2n$ matrices. We compute the cardinality of $S\backslash GL_{2n}(F)/Q$ and also give an alternate perspective on the double cosets arising intrinsically from certain subgroups which are relevant for applications in representation theory. Finally, if $Q$ is a maximal parabolic subgroup of the type $(r,2n-r),$ we prove that $S\backslash GL_{2n}(F)/Q$ is in one to one correspondence with $ΔS_n\backslash S_{2n}/S_{r}\times S_{2n-r}$ leading to a Bruhat decomposition. The results and proofs discussed in this article are valid over any arbitrary field $F$ even though our motivation is from representation theory of $p$-adic and finite linear groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Bruhat decomposition related to the Shalika subgroup of $GL(2n)$ Harshitha, C. Venketasubramanian, C. G. Representation Theory Group Theory 20G05, 22E50 Let $F$ be a non-archimedean local field or a finite field. In this article, we obtain an explicit and complete set of double coset representatives for $S\backslash GL_{2n}(F)/Q$ where $S$ is the Shalika subgroup and $Q$ a maximal parabolic subgroup of the group $GL_{2n}(F)$ of invertible $2n\times 2n$ matrices. We compute the cardinality of $S\backslash GL_{2n}(F)/Q$ and also give an alternate perspective on the double cosets arising intrinsically from certain subgroups which are relevant for applications in representation theory. Finally, if $Q$ is a maximal parabolic subgroup of the type $(r,2n-r),$ we prove that $S\backslash GL_{2n}(F)/Q$ is in one to one correspondence with $ΔS_n\backslash S_{2n}/S_{r}\times S_{2n-r}$ leading to a Bruhat decomposition. The results and proofs discussed in this article are valid over any arbitrary field $F$ even though our motivation is from representation theory of $p$-adic and finite linear groups. |
| title | On a Bruhat decomposition related to the Shalika subgroup of $GL(2n)$ |
| topic | Representation Theory Group Theory 20G05, 22E50 |
| url | https://arxiv.org/abs/2512.24368 |