Structure of internal modules and a formula for the spherical vector of minimal representations (original) (raw)

The restriction of minuscule representations to parabolic subgroups

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

Let G be a universal Chevalley group defined over an algebraically closed field F of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of G of the irreducible FG-modules corresponding to minuscule highest weights via a combinatorial scheme of "chamber systems" indexed over certain cosets in the Weyl group.

On some characters of group representations

Чебышевский сборник, 2019

We study realization fields and integrality of characters of discrete and finite subgroups of 2 (C) and related lattices with a focus on on the integrality of characters of finite groups. Theory of characters of finite and infinite groups plays the central role in the group theory and the theory of representations of finite groups and associative algebras. The classical results are related to some arithmetic problems: the description of integral representations are essential for finite groups over rings of integers in number fields, local fields, or, more generally, for Dedekind rings. A substantial part of this paper is devoted to the following question, coming back to W. Burnside: whether every representation over a number field can be made integral. Given a linear representation : → () of finite group over a number field /Q, is it conjugate in () to a representation : → () over the ring of integers ? To study this question, it is possible to translate integrality into the setting of lattices. This question is closely related to globally irreducible representations; the concept introduced by J. G. Thompson and B. Gross, was developed by Pham Huu Tiep and generalized by F. Van Oystaeyen and A.E. Zalesskii, and there are still many open questions. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, splitting fields, and, in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras.

Two Observations on Irreducible Representations of Groups

2002

For an irreducible representation of a connected affine algebraic group G in a vector space V of dimension at least 2, it is shown that the intersection of any orbit π(G)x (with x∈V ) and any hyperplane of V is non-empty. The question is raised to decide whether an analogous fact holds for irreducible continuous representations of connected compact groups, for example of SU(2).

Some Geometries Associated with Parabolic Representations of Groups of Lie Type

Canadian Journal of Mathematics, 1976

Introduction. Suppose (P, A) is an undirected graph without loops or multiple edges. We will denote by A(x) the vertices adjacent to x and x 1 = jx) U A(x). Let (G, P) be a transitive permutation representation of a group G in a, set P, and A be a non-trivial self-paired (i.e. symmetric) orbit for the action of G on P X P. We identify A with the set of all two subsets {x, y) with (x, y) in A. Then we have a graph (P, A) with G ^ Aut(P, A), transitive on both P and A. For x, y an adjacent pair of points we define the (singular) line xy on x ar\à y by xy = n.2 1. It is well known (c.f. [4]) that G is transitive on lines, a line is a clique (i.e. complete subgraph) and if u 7 e v are on xy, then xy = uv. As a result all lines have the same cardinality and also satisfy: If z is a point not on xy and z is adjacent to at least two points of xy, then z is adjacent to every point of xy. If we let L be the set of all such lines we get an incidence structure (P, L) (by this we mean a set of points and a collection of distinguished subsets called lines) with G S Aut (P,L), transitive on both P and L. Generally, (P, L) is trivial in the sense that lines only carry two points. This will certainly be the case if G X A{X) is primitive on A(x) since xy-{x} is a block of imprimitivity for the action of G X A(X) on A(x). Thus the representations of McL and HiS as rank three groups are examples of representations where the associated incidence structures are trivial. The representation of M22 as a rank three group acting on the seventy seven blocks of the Steiner system S(3, 6, 22) also affords trivial structures. However, in this representation a point stabilizer is isomorphic to a semi-direct product Z 2 M 6 , it is faithful on both suborbits, and has a set of imprimivity on one of the suborbits. Therefore the imprimitivity of G X A(X) on A(x) is not sufficient for the existence of thick (i.e. with more than two points) lines. We give some non-trivial examples: (1) Let G c^. 2 3 £, the symmetric group on 3k letters, with k at least two. Let P be the set of all ^-subsets of the 3k letters and A the set of pairs of non