# Gamma positivity of the Excedance based Eulerian polynomial in positive elements of Classical Weyl Groups

## TL;DR

This paper investigates gamma positivity properties of Eulerian polynomials associated with positive elements in classical Weyl groups, revealing conditions under which these polynomials are gamma positive and expressing some as sums of gamma positive polynomials.

## Contribution

It establishes gamma positivity for positive Eulerian polynomials in Weyl groups, including type D, and for sums over derangements, extending known results to new algebraic structures.

## Key findings

- Gamma positivity holds for certain n and parity conditions.
- Positive Eulerian polynomials can be expressed as sums of gamma positive polynomials.
- Gamma positivity is shown for polynomials summed over conjugacy classes and derangements.

## Abstract

The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that $\mathsf{AExc^{+}}_n(t)$ is gamma positive iff $n \geq 5$ and $n \equiv 1$ (mod 2). When $n \geq 4$, and $n \equiv 0$ (mod 2) we show that $\mathsf{AExc^{+}}_n(t)$ can be written as a sum of two gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials. Finally, we show gamma positivity results when we sum excedances over derangements with positive and negative sign. Our main resuls is that the polynomial obtained by summing excedance over a conjugacy class indexed by $\lambda$ is gamma positive.