Tuesday, June 27, 14:00, 7.527
Geetha Thangavelu (IISER Thiruvananthapuram), On the degrees of
representations not divisible by powers of 2.
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 2k.
We will also
explore recursive formulas applicable to groups such as Sn
group), An (the alternating group), and (ℤ/rℤ) ≀
Sn (the wreath product of the cyclic group of order r with
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