Supercharacters of Unipotent and Solvable Groups

Abstract

The notion of the supercharacter theory was introduced by P.Diaconis and I.M.Isaaks in 2008. In this paper we review the main statements of the general theory, we observe the construction of supercharacter theory for algebra groups and the theory of basic characters for the unitriangular groups over the finite field. Basing on the previous papers of the author, we construct the supercharacter theory for the finite groups of triangular type. We characterize the structure of Hopf algebra of supercharacters for the triangular group over the finite field.

Two supercharacter theories for the parabolic subgroups in orthogonal and symplectic groups

“…They set up a problem: given a group, to construct a supercharacter theory as close to the theory of irreducible characters as possible. Numerous papers are devoted to this problem and also to solutions of representations theory problems in term of supercharacters (see surveys [10,13]).…”

Towards a Supercharacter Theory of Parabolic Subgroups

“…In the series of papers [17,18,19,20], the author constructed the supercharacter theory for the finite groups of triangular type; a special case of these groups is the triangular group T(n, F q ). This theory provides the better approximation of the theory of irreducible representations than the general construction of supercharacters for semidirect products from the paper [13].…”

“…This theory provides the better approximation of the theory of irreducible representations than the general construction of supercharacters for semidirect products from the paper [13]. In the above papers, we calculate supercharacter values on superclasses for the triangular group [17], research the restriction and induction in the framework of constructed supercharacter theory, prove the Frobenius reciprocity theorem for supercharacters [19], obtain an analog of A.A.Kirillov formula for supercharacters [18], characterize the Hopf algebra of supercharacters of the triangular group as partially symmetric functions in noncommuting variables [20].…”

New Supercharacter Theory for Sylow Subgroups in Orthogonal and Symplectic Groups

“…, X N } is a partition of the system of irreducible characters Irr(G). Moreover, here each character χ i differs from the character σ i = ψ∈X i ψ(1)ψ by a constant factor (see [1,9,10]).…”

“…of right stabilizers of λ ∈ J * . The sets of characters {χ λ } and classes {K(g)}, where λ and g run through the set of representatives of equivalence classes of J * and G respectively, give rise to a supercharacter theory of the algebra group G. Supercharacters χ λ afford the analog of A.A.Krillov formula (see [1,10]):…”