Tuesday, May 10, 14:00, 7.527
Amritanshu Prasad (IMS Chennai)
Representations of symmetric groups with non-trivial determinant.
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
This result was obtained jointly with Arvind Ayyer and Steven Spallone.