A Schur ring approach to supercharacters of groups associated with finite radical rings

Abstract

We consider the central Schur ring associated with the standard supercharacters of the adjoint group $G(\mathcal{A})$ of a finite radical ring $\mathcal{A}$, and define supercharacters of the subgroup $C_{G(\mathcal{A})}(\sigma )$ consisting of elements fixed by an involution of $G(\mathcal{A})$ naturally defined by an (anti-)involution of $\mathcal{A}$. In particular, we extend known results for unipotent subgroups of the classical finite Chevalley groups.

KEYWORDS:

• Adjoint group of a radical ring
• Involutive adjoint groups
• Schur ring
• Supercharacter theory

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C15
• 20D15
• 05E10
• 05E99

Notes

1 Since there is no danger of ambiguity, we abbreviate the terminology, and refer to a standard superclass simply as a superclass of G.

2 As for superclasses, by a supercharacter of G we understand a standard supercharacter.

Additional information

Funding

This research was made within the activities of the Group for Linear, Algebraic and Combinatorial Structures of the Center for Functional Analysis, Linear Structures and Applications (University of Lisbon, Portugal), and was partially supported by the Portuguese Science Foundation (FCT) through the Strategic Project UID/MAT/04721/2013. The second author was funded by FCT project UID/HIS/00286/2019. The third author was partially supported by the Lisbon Mathematics PhD program (funded by the Portuguese Science Foundation). This work is part of the third author Ph.D. thesis.