Semisimplicity in symmetric rigid tensor categories

TL;DR

The paper establishes conditions under which an object in a symmetric rigid tensor category is semisimple based on the semisimplicity of its Schur functor, extending Serre's theorems and proving a related conjecture.

Contribution

It introduces a finite set of integers F(λ) characterizing when the semisimplicity of Schur functors implies the semisimplicity of objects in tensor categories, extending classical results.

Findings

If mbda V is semisimple and dim(V) not in F(mbda), then V is semisimple.

For each d in F(mbda), there exists a non-semisimple V with dim(V)=d such that mbda V is semisimple.

Theorem extends Serre's results and proves a conjecture of Serre.

Abstract

Let \lambda be a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of integers F(\lambda) and prove that if the Schur functor \mathbb{S}_{\lambda} V of V is semisimple and the dimension of V is not in F(\lambda), then V is semisimple. Moreover, we prove that for each d in F(\lambda) there exist a symmetric rigid tensor category C over k and a non-semisimple object V in C of dimension d such that \mathbb{S}_{\lambda} V is semisimple (which shows that our result is the best possible). In particular, Theorem 4.3 extends two theorems of Serre for C=Rep(G), G is a group, and \mathbb{S}_{\lambda} V is \wedge^n V or Sym^n V, and proves a conjecture of Serre (\cite{s1}).