#### 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.

Done