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Characters of odd degree of symmetric groups

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LMS Groups and their applications triangle meeting

Let G be a finite group and let P be a Sylow p-subgroup of G. Denote by Irrp(G) the set consisting of all irreducible characters of G of degree prime to p. The McKay Conjecture asserts that |Irrp(G)|=|Irrp(NG(P))|. Sometimes, we do not only have the above equality, but it is also possible to determine explicit natural bijections (McKay bijections) between Irrp(G) and Irrp(NG(P)).

In the first part of this talk I will describe the construction of McKay bijections for symmetric groups at the prime p=2.

In the second part of the talk I will present a recent joint work with Kleshchev, Navarro and Tiep, concerning the construction of natural bijections between Irrp(G) and Irrp(H) for various classes of finite groups G and corresponding subgroups H of odd index. This includes the case G=Sn and H any maximal subgroup of odd index in Sn, as well as the construction of McKay bijections for solvable and general linear groups.

This talk is part of the Algebra Seminar series.

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