Maximal Weight Composition Factors for Weyl Modules

Abstract

AbstractFix an irreducible (finite) root system R and a choice of positive roots. For any algebraically closed field k consider the almost simple, simply connected algebraic group Gk over k with root system k. One associates with any dominant weight λ for R two Gk-modules with highest weight λ, the Weyl module V( λ)k and its simple quotient L(µ)k . Let λ and μ be dominant weights with μ < j such that μ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists k such that L(μ)k is a composition factor of V(j)k , and they exhibit an example in type E8 where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs (λ, μ), and another that relies only on the classiûcation of root systems.