Eulerian polynomials of spherical type
Year: 2008
Type: article
Abstract: . The Eulerian polynomial of a finite Coxeter system (W, S) of rank n records, for each k ∈ {1,..., n}, the number of elements w ∈ W with an ascent set {s ∈ S | l(ws)> l(w)} of size k, where l(w) denotes the length of w with respect to S. The classical Eulerian polynomial occurs when the Coxeter group has type An, so W is the symmetric group on n + 1 letters. Victor Reiner gave a formula for arbitrary Eulerian polynomials and showed how to compute them in the classical cases. In this note, we compute the Eulerian polynomial for any spherical type. Let M be a Coxeter matrix of rank n. This means M is a symmetric n × n matrix with entries in N such that Mii = 1 and Mij> 1 if i 6 = j. We also refer to M as a diagram, that is, an edge-labeled graph with nodes {1,..., n} and edge {i, j} labeled Mij whenever Mij> 2. Our setting will involve Coxeter groups as introduced in [3]. Accordingly, we let (W,S) be a Coxeter system of type M. Then {1,..., n} and the set S = {s1,..., sn} of simple reflections are in bijective correspondence and we will often identify the two, so S can be viewed as the set of nodes of M. We also write W (M) instead of W to record the dependence on M. If W (M) is finite, then M is called spherical. The
Source: TU/e Research Portal
Author: AM Arjeh Cohen
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2166197172