Aaron Chan (Nagoya), Irreducible representations of the symmetric groups from slash homologies of p-complexes.

Abstract:

Working over a field of positive characteristic p, if we remove the signs in the definition of the degeneracy maps in the chain complex associated to a simplex, then one obtains a p-complex, i.e. a graded vector space equipped with a degree 1 endomorphism whose p-th power is zero. This p-complex is also a p-complex of (permutation) modules (associated to two-row compositions) over the group algebra of a symmetric group. We calculate the slash homologies, in the sense of Khovanov-Qi, of this p-complex and confirm a conjecture of Wildon. As a by-product, we obtain a basis of these irreducible representations indexed by the so-called p-standard tableaux, in the sense of Kleshchev, without any use of group representation theory. This is a joint work with William Wong.