Alladi Ramakrishnan Hall

Webinar: Symmetric Group Characters as Symmetric Functions

Mike Zabrocki

York University

Livestream link: us02web.zoom.us/j/86726183427

I will present a basis of the symmetric functions whose evaluations are irreducible
characters of the symmetric group in the same way that the evaluations of Schur
functions are irreducible characters of the general linear group. These symmetric
functions are related to character polynomials (that go back to a paper of
Frobenius in 1904) but they have the advantage that we are able to use the Hopf
structure of the symmetric functions to compute with them. In addition, they
indicate that the combinatorics of Kronecker coefficients is governed by multiset
tableaux. We use this basis to give a combinatorial interpretation for the tensor
products of the form
$$\chi^{(n-|\lambda|,\lambda)} \otimes \chi^{(n-a_1,a_1)}
\otimes \chi^{(n-a_r,a_2)} \otimes \cdots \otimes \chi^{(n-a_r,a_r)}$$
where $\lambda$ is a partition and $a_1, a_2, \ldots, a_r$ are non-negative
integers.

This is joint work with Rosa Orellana.