The primitive Eulerian polynomial

Abstract: We introduce the Primitive Eulerian polynomial $P_\mathcal{A}(z)$ of a central hyperplane Arrangement $\mathcal{A}$. It is a reparametrization of the cocharacteristic polynomial of the arrangement. Previous work (2021) implicitly showed that this polynomial has nonnegative coefficients in the simplicial case. If $\mathcal{A}$ is the arrangement corresponding to a Coxeter group $W$ of type A or B, then $P_\mathcal{A}(z)$ is the generating function for the (flag)excedance statistic on a particular subset of $W$. No interpretation was found for reflection arrangements of type D. We present an alternative geometric and combinatorial interpretation for the coefficients of $P_\mathcal{A}(z)$ for all simplicial arrangements $\mathcal{A}$. For reflection arrangements of types A, B, and D, we find recursive formulas that mirror those for the Eulerian polynomial of the corresponding type. We also present real-rootedness results and conjectures for $P_\mathcal{A}(z)$. This is joint work with Christophe Hohlweg and Franco Saliola.

combinatoricsrepresentation theory

Audience: researchers in the topic

CRM-Séminaire du LACIM

Series comments: Description: Seminar Series - Combinatorics and Representation Theory

Sur place / On site : Pavillon Président-Kennedy de l’UQAM (PK-4323)

Pour obtenir des détails concernant ce séminaire, veuillez communiquer avec les organisateurs / For details on this seminar, please contact: lacim@uqam.ca