Towards a Supercharacter Theory of Parabolic Subgroups

Abstract

A supercharacter theory is constructed for the parabolic subgroups of the group GL(n, Fq) with blocks of orders less or equal to two. The author formulated the hypotheses on construction of a supercharacter theory for an arbitrary parabolic subgroup in GL(n, Fq).

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

“…Finally, if u and u ′ belong to a common class, then there exists g ∈ G a , such that f (u) = g f (u ′ ). Substituting for (14), we verify that the values of σ D,φ at u and u ′ coincide. ✷…”

“…In our case, G = U a and a classification of (U a × U a )-orbits in u a and (u a ) * is considered an extremely difficult problem (see [10,14,16]). Therefore, we consider the more "coarse" supercharacter theory (see Theorem 3.6) in which superclasses are attached to the (GL a (n) × GL a (n))-orbits in u a , and supercharacters a up to a constant multiple equal to sums of functions ε µ(x) , where µ runs through a (GL a (n) × GL a (n))-orbit in (u a ) * .…”

“…Lemma 4.11 implies that the subgroup G a D = H D ⋉ U a D is abelian modulo U a D . The characters σ D,φ defined by formula ( 14) can be taken modulo U a D (see formula (14)). The formula ξ a (g) = θ(h)σ D,φ (u), g = hu, h ∈ H D , u ∈ U a , defines the character of subgroup G a D .…”

“…To construct a supercharacter theory for the parabolic contraction GL a (n), we use the method of [10] (see Theorem 3.8). The case of Borel contraction is studed in [11], where we apply the other method from [12,13,14] that gives rise a supercharacter theory that is slightly different from the one constructed in the present paper. Also, we classify supercharacters and superclasses for the unipotent radical U a (see Theorems 3.3 and 3.6).…”

Finding Supercharacter Theories on Character Tables

“…In fact, the original motivation of the theory was to construct a supercharacter theory for the unitriangular groups UT(n, q), where the full character table is not available, and Diaconis and Isaacs constructed more generally a supercharacter theory of algebra groups [8]. Since then, similar constructions have been given for many other unipotent groups [2,3,17,18].…”