# Postdoc day, Seminar 3: Topology and representation theory of the frame complex of unitary groups

#### Higher algebraic structures in algebra, topology and geometry

#### 21 January 15:10 - 15:40

#### Kevin Piterman - Universidad de Buenos Aires

For the finite unitary group GU(n,q), we consider the frame complex F(n,q), whose simplices are the sets of pairwise orthogonal non-degenerate and 1-dimensional subspaces of the underlying vector space. In this talk, we will discuss some recent results on the properties of this object: we will characterize its connectivity and the fundamental group, show that it is not Cohen-Macaulay nor a wedge of spheres in general, and apply Garland's method to show that some homology groups vanish if the dimension n is "small enough" with respect to q. Although this complex has dimension n-1, it collapses to a subcomplex of dimension n-2. Thus, I will propose a method to show that the homology group of degree n-2 does not vanish by studying irreducible characters of the unitary groups. A positive answer to the non-zeroness of this homology group would prove a conjecture raised by Aschbacher-Smith, which implies Quillen's conjecture for odd primes. Some of these results were obtained in collaboration with Volkmar Welker.