Abstract
Fix an irreducible (finite) root system R and a choice of positive roots. For any algebraically closed field κ consider the almost simple, simply connected algebraic group G κ over k with root system κ. One associates with any dominant weight A for R two G κ-modules with highest weight λ, the Weyl module V (λ) κ and its simple quotient L(λ) κ. Let λ and μ be dominant weights with μ > λ such that \i is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists k such that L(μ) κ is a composition factor of V(λ) κ, and they exhibit an example in type Es 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 classification of the possible pairs (λ, μ), and another that relies only on the classification of root systems.
| Original language | English |
|---|---|
| Journal | Canadian Mathematical Bulletin |
| Volume | 60 |
| Issue | 4 |
| Pages (from-to) | 762 - 773 |
| Number of pages | 12 |
| ISSN | 0008-4395 |
| DOIs | |
| Publication status | Published - Dec 2017 |
Keywords
- Algebraic groups
- Represention theory