Algebra and Number Theory Seminar Winter 2013

January 17, 2013

p-permutation equivalences between blocks of finite groups

Philipp Perepelitsky, UCSC Mathematics Graduate Student

Let F be an algebraically closed field and G a finite group. There is a unique indecomposable decomposition of the group algebra FG into a direct sum of two-sided ideals, called the blocks of FG. Given two finite groups G and H and blocks A and B of FG and FH respectively, there are two major open questions in representation theory:

1. What is a sufficient condition for A and B to be "equivalent"?
2. Given that A and B are "equivalent", what invariants of A and B must be preserved under this equivalence?

In the case that the characteristic of F is zero, it follows from theorems of Maschke and Wedderburn that A and B are both Morita equivalent to the F-algebra F, and hence to each other. In particular, questions (1) and (2) are answered rather easily in this case.

In the situation that F has positive characteristic p, the situation is much more complicated:

In particular, it is not even clear what the desired notion of "equivalence" in questions (1) and (2) should be. Of course a Morita equivalence is pretty much the nicest thing one can ask for, but there are many known cases of two blocks which are not Morita equivalent, but are equivalent via a so-called isotypy, or even via a form of equivalence that lies between an isotypy and a Morita equivalence called a splendid Rickard equivalence. These two notions of equivalence were introduced in 1990 by M. Brouee and in 1996 by J.Rickard respectively.

In 2008, R. Boltje and B. Xu introduced a new notion of equivalence called a p-permutation equivalence, which they showed lies between an isotypy and a splendid Rickard equivalence. In this talk, I shall describe the notion of a p-permutation equivalence and present joint work with R. Boltje in which we give a partial answer to question (2) for a p-permutation equivalence. In particular, we show that certain fundamental invariants of a block must be preserved under a p-permutation equivalence. One consequence of this is that if two blocks are p-permutation equivalent, then their Brauer correspondents must be Morita equivalent.

January 28, 2013

Geometrizing characters of tori

David Roe, Post Doc, University of Calgary

The passage from functions to sheaves has proven a valuable tool in the geometric Langlands program. In this talk I'll describe a "geometric avatar" for the group of characters of $T(K)$, where $T$ is an algebraic torus over a local field $K$. I will then give some potential applications to the classical Langlands correspondence. This is joint work with Clifton Cunningham.

February 11, 2013

No seminar this week

February 25, 2013

Cameron Franc, Postdoc, UCSC Mathematics

TBA

March 4, 2013

Twisted category algebras and quasi-heredity

Susanne Danz, Department of Mathematics, University of Kaiserslautern

In this talk we shall consider twisted category algebras over fields of characteristic 0. The underlying category will always be finite and will have an additional property, which is called "split". The multiplication in such an algebra is essentially induced by the composition of morphisms in the category. Prominent examples of twisted category algebras are various classes of diagram algebras (for suitable parameters) such as Brauer algebras, Temperley-Lieb algebras, or partition algebras. Twisted category algebras also arise in connection with double Burnside rings and biset functors.

We shall show that a twisted split category algebra in characteristic 0 is quasi-hereditary, that is, the corresponding module category is a highest weight category. Moreover, we shall give an explicit description of its standard modules with respect to a particular partial order on the set of isomorphism classes of simple modules. This provides, in particular, a unified proof of the known fact that the aforementioned diagram algebras are quasi-hereditary.

This is joint work with Robert Boltje.