Abstract: In view of the relevance of the McKay Conjecture in the representation theory of finite groups, it is of common interest to investigate how irreducible characters decompose when restricted to Sylow p-subgroups. In this talk I will focus on the symmetric groups and our main object of investigation is the set of irreducible characters of Sn that have a constituent of a given fixed degree in their restriction to Pn, a Sylow p-subgroup of Sn. I will briefly mention the combinatorial theory needed for the study of the irreducible characters. Then I will show some results on the constituents with degree strictly greater than 1, extending the knowledge on linear ones that we have from [GN18], [GL18] and [GL21].
This is joint work with Eugenio Giannelli.
References
[GN18] E. Giannelli and G. Navarro, Restricting irreducible characters to Sylow p-subgroups, Proc. Amer. Math. Soc. 146 (2018), no. 5, 1963–1976.
[GL18] E. Giannelli and S. Law, On permutation characters and Sylow p- subgroups of Sn, J. Algebra 506 (2018), 409–428.
[GL21] E. Giannelli and S. Law, Sylow branching coefficients for symmetric groups, Journal of the London Mathematical Society, (2) 103 (2021), 697–728.