Homotopy, homology, and $GL_2$

TL;DR

This paper develops a homological framework using weak 2-categories and operators to analyze the rational representation theory of $GL_2$ over fields of positive characteristic, revealing new symmetries and structures.

Contribution

It introduces a novel categorical approach with operators controlling homological aspects of $GL_2$ representations, including derived equivalences and braid group actions.

Findings

Operators $ extbf{O}_p$ govern homological properties of $GL_2$ representations.

Existence of tight $ extbf{Z}_+$-gradings on Schur algebras $S(2,r)$.

Braid group actions on derived categories of Schur algebra blocks.

Abstract

We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras.