Combinatorial and geometric properties of weight systems of irreducible finite-dimensional representations of simple split Lie algebras over fields of 0 characteristic

Benedict Seifert

Vol. 101 (1982), No. 1, 163–183

DOI: 10.2140/pjm.1982.101.163

Abstract

Let R be a simple split Lie algebra over K, a field of 0 characteristic. Let π = π(λ⁺) be the representation with highest weight λ⁺. Let Wt(λ⁺) be its weight system. Let S be a subset of the root system. We define a graph gr_s(λ⁺) whose set of nodes is Wt(λ⁺) and set of links is given by pairs of weights whose difference is a root in S. In particular taking S = Σ⁰, the system of simple roots, we investigate the properties of representations π(λ⁺) such that gr⁰(λ⁺) is simply connected. We give a complete list of these, for each simple Lie algebra.

We then attach to π(λ⁺) an affine rational lattice L(λ⁺), the root of λ⁺ in the weight lattice modulo the root lattice and a rational polyhedron, C(λ⁺), the rational convex closure of the Weyl group orbit of λ⁺. We give the following geometric characterization of the weight system: Wt(λ⁺) = L(λ⁺) ∩ C(λ⁺)¹.

Mathematical Subject Classification 2000

Primary: 17B10

Milestones

Received: 15 September 1979

Published: 1 July 1982