On rational representations and rational group algebra of $\operatorname{GL}_2(q)$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917556171833344 |
|---|---|
| author | Choudhary, Ram Karan Prajapati, Sunil Kumar |
| author_facet | Choudhary, Ram Karan Prajapati, Sunil Kumar |
| contents | In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $ρ$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $ρ$ affords the character \[ Ω(χ)=m_{\mathbb{Q}}(χ)\sum_{σ\in\operatorname{Gal}(\mathbb{Q}(χ)/\mathbb{Q})}χ^σ, \] for some irreducible complex character $χ$ of $G$, where $m_{\mathbb{Q}}(χ)$ denotes the Schur index of $χ$ over $\mathbb{Q}$, with the converse also holding. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of distinct degrees. Furthermore, we present a method to construct an irreducible rational matrix representation $ρ$ of $G$ affording the character $Ω(χ)$, where $χ$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results from the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02415 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On rational representations and rational group algebra of $\operatorname{GL}_2(q)$ Choudhary, Ram Karan Prajapati, Sunil Kumar Representation Theory Combinatorics Group Theory Rings and Algebras [2020]{primary 20C15, secondary 20C33, 20C05} In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $ρ$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $ρ$ affords the character \[ Ω(χ)=m_{\mathbb{Q}}(χ)\sum_{σ\in\operatorname{Gal}(\mathbb{Q}(χ)/\mathbb{Q})}χ^σ, \] for some irreducible complex character $χ$ of $G$, where $m_{\mathbb{Q}}(χ)$ denotes the Schur index of $χ$ over $\mathbb{Q}$, with the converse also holding. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of distinct degrees. Furthermore, we present a method to construct an irreducible rational matrix representation $ρ$ of $G$ affording the character $Ω(χ)$, where $χ$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results from the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$. |
| title | On rational representations and rational group algebra of $\operatorname{GL}_2(q)$ |
| topic | Representation Theory Combinatorics Group Theory Rings and Algebras [2020]{primary 20C15, secondary 20C33, 20C05} |
| url | https://arxiv.org/abs/2606.02415 |