Supercharacters and pattern subgroups in the upper triangular groups

Abstract

Let Un(q) denote the upper triangular group of degree n over the finite field Fq with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form U k (q) ∩ w U k (q), where w is a monomial matrix in GL k (q) for some k < n.

Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

“…As a consequence all irreducible constituents of CO r A are induced from precisely the irreducible modules of U R in the sum of Wedderburn components attached to the central idempotent ǫ ⊥ A . This result has been observed as well by Tung Le in [14] but was shown there by different methods. In the special case of hook disconnected main conditions it follows immediately from proposition 6.7: A is an irreducible constituent of CO r A .…”

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

“…In addition to the references above, we call attention to a selection of recent papers. In particular, the results of this paper have been generalized to unipotent subgroups of type D [8], there has been some exploration into further decomposing these supercharacters [26,29] and further analysis of their underlying combinatorics [30,31]. In addition to the above results, the American Institute of Mathematics workshop generated several items that might be of interest to those who would like to pursue these thoughts further.…”

Representations of McLain groups

“…This is the perfect analogue for André formula (cf. [A2, Theorem 2]) for the inner product of a basic character with itself (see also [Le2,Corollary 2.10] The most delicate part of our paper deals with the actual decomposition of a basic character. This requires considerably more effort than above.…”

“…This is the perfect analogue for André formula (cf. [A2, Theorem 2]) for the inner product of a basic character with itself (see also [Le2,Corollary 2.10]). Thus,…”