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.
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07
octobre
Vendredi
Jose Bastidas (LACIM, UQAM): The primitive Eulerian polynomial
07 octobre 2022, 11:00
- 07 octobre 2022, 12:00
Détails
Date :
octobre 7, 2022
Heure :
11:00 am - 12:00 pm