# Homotopy, homology, and $GL 2$ - Mathematics > Representation Theory

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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 allhomological aspects of the rational representation theory of the algebraicgroup $GL 2$, over a field of positive characteristic. We prove that when $x$is a Rickard tilting complex, the operators $\mathbb{O} {c,x}$ honour derivedequivalences, in a differential graded setting. We give a number ofrepresentation theoretic corollaries, such as the existence of tight$\mathbb{Z} +$-gradings on Schur algebras $S2,r$, and the existence of braidgroup actions on the derived categories of blocks of these Schur algebras.

Author: ** Vanessa Miemietz, Will Turner**

Source: https://arxiv.org/