Homotopy, homology and GL2

Vanessa Miemietz, Will Turner

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)585-606
Number of pages22
JournalProceedings of the London Mathematical Society
Volume100
Issue number2
Early online date27 Oct 2009
DOIs
Publication statusPublished - Mar 2010

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Homotopy
Schur Algebras
Homology
Operator
Derived Equivalence
Derived Category
Tilting
Braid Group
Bimodule
Positive Characteristic
Finite Dimensional Algebra
Grading
Algebraic Groups
Group Action
Representation Theory
Corollary

Cite this

Homotopy, homology and GL2. / Miemietz, Vanessa; Turner, Will.

In: Proceedings of the London Mathematical Society, Vol. 100, No. 2, 03.2010, p. 585-606.

Research output: Contribution to journalArticle

Miemietz, Vanessa ; Turner, Will. / Homotopy, homology and GL2. In: Proceedings of the London Mathematical Society. 2010 ; Vol. 100, No. 2. pp. 585-606.
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