Shintani descent for standard supercharacters of algebra groups

Pub Date : 2024-01-17 DOI:10.1142/s0218196724500073
Carlos A. M. Andr'e, Ana L. Branco Correia, Joao Dias
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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 $n \in \mathbb{N}$.
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代数群标准超字符的新谷下降
让 $\mathcal{A}(q)$ 是一个有限域 $\mathbb{F}_{q}$ 上有 $q$ 元素的有限维无势代数,并让 $G(q) = 1+\mathcal{A}(q)$.另一方面,让 $\Bbbk$ 表示 $\mathbb{F}_{q}$ 的代数闭包,并让 $\mathcal{A} = \mathcal{A}(q) \otimes_\mathbb{F}_{q}}.\Bbbk$.那么 $G = 1+\mathcal{A}$ 是一个在 $\Bbbk$ 上的代数群,具有由通常的 Frobenius 映射 $F:G\to G$ 给出的 $\mathbb{F}_{q}$ 有理结构,并且 $G(q)$ 可以看作是定点子群 $G^{F}$。对于每一个 $n \in \mathbb{N}$,$n$的幂 $F^{n}:G\to G$ 也是一个弗罗贝尼斯映射,并且 $G^{F^{n}}$ 与 $G(q^{n}) = 1 + \mathcal{A}(q^{n})$相一致。弗罗贝尼斯映射限制了$F:G(q^{n})\to G(q^{n})$的群自变,因此它作用于$G(q^{n})$的不可还原字符集。新谷下降提供了一种比较 $G(q^{n})$ 的 $F$ 不变不可还原字符与 $G(q)$ 的不可还原字符的方法。在本文中,我们证明它也为研究 $n \in \mathbb{N}$ 的 $G(q^{n})$ 的超字符提供了统一的方法。当 $m \mid n$ 时,这些群构成了一个关于包含映射 $G(q^{m}) \to G(q^{n})$ 的归纳系统,这一事实使我们能够同时研究所有超字符理论,建立它们之间的联系,并将它们与代数群 $G$ 联系起来。事实上,我们证明了新谷后裔允许定义某种 "超偶代数",它编码了 $n \in \mathbb{N}$ 的 $G(q^{n})$ 的超字符信息。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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