Deligne categories as limits in rank and characteristic

Abstract

We give new interpretations of the Deligne categories $\underline{Rep}(G{L}_t)$ and $\underline{Rep}(S_t)$ (and their abelian envelopes) over $\mathbb{C}$ in terms of modular representations of general linear and symmetric groups of large rank in large characteristic. In particular we make sense of the sentence "$\underline{Rep}(S_n)$ is the limit of $Rep(S_{p+n})$ over ${\bar{\mathbb{F}}}_p$ as $p$ goes to infinity". We then give examples of how to pass results between these different settings.

10 pages

References in corpus (3)

• Stability and periodicity in the modular representation theory of symmetric groups
• Simple commutative algebras in Deligne's categories Rep($S_t$)
• Semisimple and $G$-equivariant simple algebras over operads

Cited by in corpus (6)

• Monoidal abelian envelopes
• Lectures on Symmetric Tensor Categories
• Harish-Chandra bimodules in the Deligne category $\text{Rep}(G{L}_t)$
• Classification of simple algebras in Deligne category $Rep(S_t)$
• Harish-Chandra bimodules of finite $K$-type in Deligne categories
• Olshanski's Centralizer Construction and Deligne Tensor Categories