Amritanshu Prasad (IMS Chennai)

Odd dimensional representations in Young's graph

Abstract:

The vertices of Young's graph are integer partitions. Two partitions are joined by an edge if their Young diagrams differ by one box. A fundamental invariant of an integer partition is its f-number, which is the number of geodesic paths joining it to the unique partition of 1. This number arises in representation theory as the dimension of an irreducible representation of a symmetric group, and is given by the hook-length formula of Frame-Robinson and Thrall.

In this talk I will discuss the induced subgraph of Young's graph obtained by taking only those partitions which have odd f-number. Results about this subgraph (obtained jointly with Arvind Ayyer and Steven Spallone) were used by Eugenio Giannelli, Alexander Kleshchev, Gabriel Navarro and Pham Huu Tiep to outline a bijective version of the McKay correspondence for the symmetric group. This is a correspondence between the odd dimensional representations of S_n and its 2-Sylow subgroup.