Equivariant version of the characteristic quasi-polynomials of root systems

TL;DR

This paper introduces an equivariant refinement of characteristic quasi-polynomials for root systems, computes them explicitly for all irreducible cases, and explores their relation to folded Dynkin diagrams.

Contribution

It develops an equivariant framework for characteristic quasi-polynomials and provides explicit computations for all irreducible root systems.

Findings

Explicit formulas for equivariant characteristic quasi-polynomials of all irreducible root systems.

Refined properties of characteristic quasi-polynomials under equivariant considerations.

Connections established between folded Dynkin diagrams and equivariant quasi-polynomials.

Abstract

An equivariant characteristic quasi-polynomial is a quasi-polynomial in $q$ consisting of the permutation character on the mod $q$ complement of the corresponding Coxeter arrangement. This concept is a refinement of the conventional characteristic quasi-polynomials of root systems. In this paper, we will show equivariant-theoretic refinements of the some properties of characteristic quasi-polynomials of root systems. Furthermore, we will explicitly compute equivariant characteristic quasi-polynomials of all irreducible reduced root systems and discuss the relationship with root systems constructed by the folding of the extended Dynkin diagrams.