Amritanshu Prasad (IMS Chennai)

Representations of symmetric groups with non-trivial determinant.

Abstract:

In a field of characteristic zero, the irreducible representations of the nth symmetric group are indexed by partitions of n. Given a representation, taking the determinant of the representing matrix gives rise to a multiplicative character. The symmetric group has two multiplicative characters, the trivial character, and the sign character. It is natural to ask how often the determinant of an irreducible representation is trivial and how often it is the sign character. According to Stanley, this problem was first considered by L. Solomon.

Let b(n) denote the number of partitions of n for which the corresponding irreducible representation of the nth symmetric group has non-trivial determinant. We will present a closed formula for b(n). This formula is obtained by characterizing the 2-core towers of such partitions.

This result was obtained jointly with Arvind Ayyer and Steven Spallone.