Multiplicity-free induced characters of symmetric groups

Abstract: Let $n$ be a sufficiently large positive integer. A character is multiplicity-free if its irreducible constituents appear with multiplicity one. Wildon in 2009 and independently Godsil and Meagher in 2010 have found all multiplicity-free permutation characters of the symmetric group $S_n$. In this talk, we focus on a significantly more general problem when the permutation characters are replaced by induced characters of the form $\rho\!\uparrow^{S_n}$ with $\rho$ irreducible.

             Despite the nature of the problem, I explain, combining results from group theory, representation theory and combinatorics, why this problem may be feasible and present a close to full answer. I also mention some of my (often surprising) results to questions about conjugate partitions, which naturally arise when solving the problem, and the remarkable complete classification of subgroups $G$ of $S_n$, which have an irreducible character which stays multiplicity-free when induced to $S_n$.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.