Right-angled Coxeter groups, universal graphs, and Eulerian polynomials

We define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W, S). once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G(2) constructed from the unlabeled Coxeter graph G of (W, S). When G = P-n is a path with n vertices, we conjecture that G(2) is n-universal; this property is equivalent to the existence of an action of R and H(R)) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (P-n)(2) is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group U-n, using results on the Kazhdan-Lusztig basis of H(U-n). (C) 2019 Elsevier Ltd. All rights reserved.