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

Abstract: 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.