Homotopy, homology, and $G{L}_2$

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 $G{L}_2$, 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.

28 pages

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