Darstellungstheorietage and Nikolaus Conference 2013

Speaker: Frank Lübeck (Aachen)

Title: Kazhdan-Lusztig Polynomials and Conjectures by Guralnick and Wall

Abstract: I have computed certain Kazhdan-Lusztig polynomials which were not known before.

It turned out that some of these polynomials provide unexpected examples for two long standing conjectures. One is a conjecture by Guralnick which says that the dimension of the 1-cohomology of any finite group for a finite absolutely irreducible module should be globally bounded. For a long time all known examples of such dimensions were at most 3.

The other conjecture by G.E.Wall (1962) says that a finite group \(G\) has at most \(|G|-1\) maximal subgroups. As a corollary from my computations one can show that there exists a \(\delta > 0\) such that there are infinitely many finite groups which have more than \(|G|^{1+\delta}\) maximal subgroups.

In this talk I will sketch how these topics are related (I learned this from Bob Guralnick and Len Scott).