### CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Thursday, March 3, 2022 - 3:30pm

#### Marino Romero

Penn

Location

University of Pennsylvania

DRL 4C4

We will start with introducing a statistic $Comaj_{R,S}$ on permutations, which depends on a subset $S$ of $\{1,\dots, n-1\}$. Depending on the choice of $R$, $S$ and input, this Comaj statistic specializes to the comaj statistic on permutations and the comaj statistic on semistandard tableaux.  This statistic gives the graded multiplicities of irreducible representation in the $k^{\text{th}}$ tensor power of the Harmonics of $S_n$ (or equivalently, Coinvariants); and it also gives the principal evaluation of Schur functions, or Gessel's Fundamental basis of quasisymmetric functions, at an arbitrary number of indeterminates.

We will finish by describing how principal evaluations are related to the problem of restricting irreducible representations of $GL_n$ to $S_n$, proving stability results about the multiplicities of irreducibles in these restrictions.