Seminar

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.

 

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Organizers
Gregory Arone
Stockholm University
Tilman Bauer
KTH Royal Institute of Technology
Alexander Berglund
Stockholm University
Søren Galatius
University of Copenhagen
Jesper Grodal,
University of Copenhagen
Thomas Kragh
Uppsala University

Program
Contact

Alexander Berglund

alexb@math.su.se

Søren Galatius

galatius@math.ku.dk

Thomas Kragh

thomas.kragh@math.uu.se

Other
information

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