Carlos A. M. Andr'e, Ana L. Branco Correia, Joao Dias
{"title":"Shintani descent for standard supercharacters of algebra groups","authors":"Carlos A. M. Andr'e, Ana L. Branco Correia, Joao Dias","doi":"10.1142/s0218196724500073","DOIUrl":null,"url":null,"abstract":"Let $\\mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $\\mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+\\mathcal{A}(q)$. On the other hand, let $\\Bbbk$ denote the algebraic closure of $\\mathbb{F}_{q}$, and let $\\mathcal{A} = \\mathcal{A}(q) \\otimes_{\\mathbb{F}_{q}} \\Bbbk$. Then $G = 1+\\mathcal{A}$ is an algebraic group over $\\Bbbk$ equipped with an $\\mathbb{F}_{q}$-rational structure given by the usual Frobenius map $F:G\\to G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{F}$. For every $n \\in \\mathbb{N}$, the $n$th power $F^{n}:G\\to G$ is also a Frobenius map, and $G^{F^{n}}$ identifies with $G(q^{n}) = 1 + \\mathcal{A}(q^{n})$. The Frobenius map restricts to a group automorphism $F:G(q^{n})\\to G(q^{n})$, and hence it acts on the set of irreducible characters of $G(q^{n})$. Shintani descent provides a method to compare $F$-invariant irreducible characters of $G(q^{n})$ and irreducible characters of $G(q)$. In this paper, we show that it also provides a uniform way of studying supercharacters of $G(q^{n})$ for $n \\in \\mathbb{N}$. These groups form an inductive system with respect to the inclusion maps $G(q^{m}) \\to G(q^{n})$ whenever $m \\mid n$, and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group $G$. Indeed, we show that Shintani descent permits the definition of a certain ``superdual algebra'' which encodes information about the supercharacters of $G(q^{n})$ for $n \\in \\mathbb{N}$.","PeriodicalId":0,"journal":{"name":"","volume":"216 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0218196724500073","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let $\mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $\mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+\mathcal{A}(q)$. On the other hand, let $\Bbbk$ denote the algebraic closure of $\mathbb{F}_{q}$, and let $\mathcal{A} = \mathcal{A}(q) \otimes_{\mathbb{F}_{q}} \Bbbk$. Then $G = 1+\mathcal{A}$ is an algebraic group over $\Bbbk$ equipped with an $\mathbb{F}_{q}$-rational structure given by the usual Frobenius map $F:G\to G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{F}$. For every $n \in \mathbb{N}$, the $n$th power $F^{n}:G\to G$ is also a Frobenius map, and $G^{F^{n}}$ identifies with $G(q^{n}) = 1 + \mathcal{A}(q^{n})$. The Frobenius map restricts to a group automorphism $F:G(q^{n})\to G(q^{n})$, and hence it acts on the set of irreducible characters of $G(q^{n})$. Shintani descent provides a method to compare $F$-invariant irreducible characters of $G(q^{n})$ and irreducible characters of $G(q)$. In this paper, we show that it also provides a uniform way of studying supercharacters of $G(q^{n})$ for $n \in \mathbb{N}$. These groups form an inductive system with respect to the inclusion maps $G(q^{m}) \to G(q^{n})$ whenever $m \mid n$, and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group $G$. Indeed, we show that Shintani descent permits the definition of a certain ``superdual algebra'' which encodes information about the supercharacters of $G(q^{n})$ for $n \in \mathbb{N}$.