Thursday, July 16 2020
20:30 - 21:30

IMSc Webinar

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

Digjoy Paul


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.


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.

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