A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

TL;DR

This paper develops a supercharacter theory for the Sylow p-subgroups of classical finite groups of types B, C, and D, providing explicit superclasses, supercharacters, and their properties, simplifying representation analysis.

Contribution

It introduces a supercharacter theory for Sylow p-subgroups of classical groups, including superclass definitions, supercharacter evaluations, and orthogonality relations, extending previous work.

Findings

Superclasses are explicitly defined and factorize into elementary superclasses.

Supercharacters are constant on superclasses and their values are evaluated.

The supercharacter table satisfies orthogonality relations similar to irreducible characters.

Abstract

We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of $U$, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).