### Abstract

We define weak 2-categories of finite-dimensional algebras with bimodules, along with collections of operators ¿(c, x) on these 2-categories. We prove that special examples ¿p of these operators control all homological aspects of the rational representation theory of the algebraic group GL2, over a field of positive characteristic. We prove that when x is a Rickard tilting complex, the operators ¿(c, x) honour derived equivalences in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight 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.

Original language | English |
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Pages (from-to) | 585-606 |

Number of pages | 22 |

Journal | Proceedings of the London Mathematical Society |

Volume | 100 |

Issue number | 2 |

Early online date | 27 Oct 2009 |

DOIs | |

Publication status | Published - Mar 2010 |

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## Cite this

Miemietz, V., & Turner, W. (2010). Homotopy, homology and GL2.

*Proceedings of the London Mathematical Society*,*100*(2), 585-606. https://doi.org/10.1112/plms/pdp040