On complete reducibility of tensor products of simple modules over simple algebraic groups

On Extensions of Simple Modules over Symmetric and Algebraic Groups

The Cohomology of Small Irreducible Modules for Simple Algebraic Groups

LetG be a quasisimple, connected, and simply connected algebraic group defined and split over the field k of characteristic p > 0. In this paper, we are interested in small modules for G; for us, small modules are those with dimension ≤ p. By results of Jantzen [Jan96] one knows that anyGmodule V with dimV ≤ p is semisimple. (We always understand aG-module V to be given by a morphism of algebra...

Unique Decomposition of Tensor Products of Irreducible Representations of Simple Algebraic Groups

We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.

Simple Tensor Products

Let F be the category of finite dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S1⊗· · ·⊗SN of simple objects of F is simple if and only if for any i < j, Si ⊗ Sj is simple.

Products of Conjugacy Classes in Finite and Algebraic Simple Groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad–Herzog co...