A combinatorial formula expressing periodic $R$-polynomials

TL;DR

This paper provides a new combinatorial formula for periodic R-polynomials, which are linked to important representation theory problems in algebraic groups and Kac-Moody algebras.

Contribution

It introduces a closed combinatorial formula expressing periodic R-polynomials via the doubled Bruhat graph for finite Weyl groups.

Findings

Provides a combinatorial formula for periodic R-polynomials.

Connects periodic R-polynomials to the doubled Bruhat graph.

Facilitates computation of periodic Kazhdan-Lusztig polynomials.

Abstract

In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac-Moody algebra at the critical level. The periodic Kazhdan-Lusztig polynomials can be computed by using another family of polynomials, called the periodic $R$-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic $R$-polynomials in terms of the "doubled" Bruhat graph associated to a finite Weyl group and a finite root system.