Toward a Classification of the Supercharacter Theories of $C_p\times C_p$

Supercharacter theories and Schur rings

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring constructions of Leung and Man and of Tamaschke.

Finding Supercharacter Theories on Character Tables

We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the coarsest partition that belongs to a supercharacter theory. Our constructions emphasize the duality between superclasses and supercharacters. An algorithm is presented to find all supercharacter theories on a given character table. The algorithm is used to compute the number of supercharacter theories for some nonabelian simple groups with up to 26 conjugacy cla…

An Enumeration of the Supercharacter Theories of $C_p \times C_2 \times C_2$ for Prime $p$

The supercharacter theories of $C_p \times C_2 \times C_2$ were classified in the language of Schur rings by Evdokimov, Kovács, and Ponomarenko in [EKP16]. It was shown that every nontrivial supercharacter theory of $C_p \times C_2 \times C_2$ can be constructed as a wedge product, a direct product, or is generated by automorphisms. We use this classification to give a precise count of the distinct supercharacter theories of $C_p\times C_2\times C_2$ and describe when a supercharacter theory can be constructed by more than one method. We also present an alternative proof of the classification …

An analog of nilpotence arising from supercharacter theory

The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper and lower annihilator series of $J$ can be described in terms of certain central series for the algebra group $G=1+J$ defined by $\mathsf{S}$, when $\mathsf{S}$ is the algebra group supercharacter …

Supercharacters for parabolic contractions of finite groups of A,B,C,D Lie types

We construct supercharacter theories for the finite groups constructed by parabolic constraction from simple groups of $A,B,C,D$ Lie types. In terms of rook placements in the root systems we classify supecharacters and superclasses.

Upper and Lower Semimodularity of the Supercharacter Theory Lattices of Cyclic Groups

We consider the lattice of supercharacter theories, in the sense of Diaconis and Isaacs, of the cyclic group of order n. We find necessary and sufficient conditions on n for that lattice to be upper or lower semimodular.

关于单项群和Artin猜想的超特征理论推广

On supercharacter theoretic generalizations of monomial groups and Artin's conjecture

本文在超特征理论框架下推广了拟单项群与几乎单项群的概念，并探讨其与Artin $L$-函数全纯性的Artin猜想之间的联系。

FormaTheoria：从数学文献构建大规模Lean理论——迈向有限单群分类定理的形式化

FormaTheoria: Constructing Large-Scale Lean Theories from Mathematical Literature $-$ Toward the Formalization of the Classification of Finite Simple Groups

本研究提出AI辅助工作流FormaTheoria，应用于有限单群分类定理的形式化，构建了经机器检查的Lean开发项目，验证了深度依赖的有限群理论，为推进该定理的形式化奠定基础。