IMSc Webinar

Quasi p-Steinberg Character for Symmetric, Alternating Groups and their Double Covers

Digjoy Paul

IMSc

Given a finite group of Lie type in characteristic p, Steinberg constructed a distinguished ordinary representation of dimension equals to the
cardinality of a Sylow-p-subgroup and whose character, which is now known as p-Steinberg character, vanishes except at p-regular elements.
The following question was raised by W. Feit and was answered by M. R. Darafsheh for the alternating group or the projective special linear group:
"Let G be a finite simple group of order divisible by the prime p, and suppose that G has a p-Steinberg character. Does it follow that G is a
semisimple group of Lie type in characteristic p?"

This motivates us to define Quasi p-Steinberg character for finite groups.
An irreducible character of a finite group G is called quasi p-Steinberg
for a prime p dividing order of G if it is non zero on every p-regular element of G.
In this talk, we discuss the existence of quasi p-Steinberg Characters of Symmetric as well as Alternating groups and their double covers. On the
way, we also answer a question, similar to Feit, asked by Dipendra Prasad.
This is based on ongoing work with Pooja Singla.

References:

1. Humphreys, J. E. The Steinberg representation,1987.

2. W. Feit, Extending Steinberg Characters,1993.

3. M. R. Darafsheh, p-Steinberg Characters of Alternating and Projective Special Linear Groups 1995.