Supercharacters of the Sylowp-subgroups of the finite symplectic and orthogonal groups

Abstract

We define and study supercharacters of the classical finite unipotent groups of types Bn(q), Cn(q) and Dn(q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.