Vol. 101, No. 1, 1982

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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

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 grs(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
Received: 15 September 1979
Published: 1 July 1982
Benedict Seifert