Tuesday, June 27, 14:00, 7.527

Geetha Thangavelu (IISER Thiruvananthapuram), On the degrees of representations not divisible by powers of 2.

Abstract:
An interesting problem that arises in the study of finite groups is counting the number of distinct irreducible representations of a group G, where the degree of each representation is not divisible by a given prime number p. This question was initially explored by I. G. Macdonald in a paper focusing on prime numbers, and it served as motivation for the McKay conjecture, which was announced in 1971 and led to various counting conjectures in finite group theory.

Expanding Macdonald's work to encompass all integers poses a significantly more challenging problem. However, driven by inquiries related to chiral representations of wreath products, we will examine a generalization of the aforementioned question for composite numbers of the form 2^k. We will also explore recursive formulas applicable to groups such as S_n (the symmetric group), A_n (the alternating group), and (ℤ/rℤ) ≀ S_n (the wreath product of the cyclic group of order r with S_n).

Despite efforts made to count and describe these representations, even for smaller integers, a comprehensive characterization of irreducible representations with degrees not divisible by a given prime number remains absent in the existing literature. Towards the end of our talk, we will see some special cases where such characterizations are available, along with further open problems in this research direction. This is a joint work with Amrutha P.