# Graduate Colloquium Winter 2014

McHenry Building - Room 4130

Refreshments served at 3:30 in room 4161

For further information, please contact 831-459-2969

***Time Changed to 2:45pm* March 13, 2014**

p-permutation equivalences between blocks of finite groups

p-permutation equivalences between blocks of finite groups

**Philipp Perepelitsky, University of**

**California, Santa Cruz**

In this talk we describe joint work with Robert Boltje. Let *F* be an algebraically closed field of positive characteristic *p*. Let *G *and *H* be finite groups. Let *A* be a block of *FG* and let *B *be a block of *FH*. A *p-permutation equivalence* between *A* and *B* is an element y in the group of (*A, B*)-*p-*permutation bimodules with twisted diagonal vertices such that y *H* Ÿ y ° = [*A*] and y °*G* y = [*B*]. A *p-*permutation equivalence lies between a splendid Rickard equivalence and an isotypy.

We introduce the notion of a y-Brauer pair, which generalizes the notion of a Brauer pair for a *p-*block of a finite group. The y-Brauer pairs satisfy an appropriate Sylow theorem. Furthermore, each maximal y-Brauer pair identities the defect groups, fusion systems and Külshammer-Puig classes of *A* and *B*. Additionally, the Brauer construction applied to y induces a *p-*permutation equivalence at the local level, and a splendid Morita equivalence between the Brauer correspondents of *A *and *B*.

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