Relative Frobenius Formula

TL;DR

This paper generalizes Frobenius's formula for finite groups to a relative setting involving subgroups, enabling new computations of E-polynomials and insights into the Gelfand property and Fock--Goncharov spaces.

Contribution

It introduces a relative Frobenius formula for sums over irreducible representations involving subgroup dimensions, extending classical results.

Findings

Derived a formula for sums involving both group and subgroup representation dimensions.

Computed E-polynomials of Fock--Goncharov spaces using the new formula.

Connected Gelfand property with the geometry of generalized Fock--Goncharov spaces.

Abstract

For a finite group $G$, Frobenius found a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s}$ for even integers $s$, where $\mathrm{Irr} G$ is the set of irreducible representations of $G$. We generalize this formula to the relative case: for a subgroup $H$, we find a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s} (\dim\, \pi ^H)^{-t}$. We apply our results to compute the E-polynomials of Fock--Goncharov spaces and to relate the Gelfand property to the geometry of generalized Fock--Goncharov spaces.