The McKay conjecture was refined in 2004 by G. Navarro 
taking into account the action of Galois automorphisms on characters. 
This refinement is known as the Galois McKay conjecture, and it is one 
of the last counting conjectures without a reduction theorem. In this 
talk I will present a possible reduction theorem for the Galois McKay 
conjecture to a question on finite simple groups. This is joint work 
with G. Navarro and B. Späth.

 

Let $V$ be a finite-dimensional (complex) vector space, and $Sym(V)$ be the symmetric algebra on this vector space. We can consider the multiplication map $Sym(V) \otimes V \to V$ as a complex of $GL(V)$-representations of length $2$. I this talk, I will describe how tensor powers of the above complex define interesting complexes of representations of the symmetric group $S_n$, which were studied by Deligne in the paper "La Categorie des Representations du Groupe Symetrique $S_t$, lorsque $t$ n’est pas un Entier Naturel". I will then explain how computing the cohomology of these complexes helps establish a relation between the Deligne categories and the representations of $S_{\infty}$, which are two natural settings for studying stabilization in the theory of finite-dimensional representations of the symmetric groups. This is joint work with D. Barter and Th. Heidersdorf