Bounded Bruhat intervals and affine Coxeter groups

Abstract

Dyer (1991) proved that, for each fixed length $k$, only finitely many isomorphism types of Bruhat intervals occur in finite Coxeter groups. In this short paper, we prove that this result holds also for affine Coxeter groups, and moreover that this characterizes the affine groups among all infinite Coxeter groups. We also show that the coefficients of $q$ in the Kazhdan-Lusztig polynomials of a Coxeter group are bounded if and only if the group is finite or affine.