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

## TL;DR

This paper investigates the gamma positivity of positive Eulerian polynomials derived from classical Weyl groups, revealing specific modular conditions under which these polynomials are gamma positive or can be expressed as sums of gamma positive polynomials.

## Contribution

It establishes gamma positivity criteria for positive Eulerian polynomials in classical Weyl groups, including type-A and type-D, based on the congruence class of n modulo 4.

## Key findings

- $A_n^+(t)$ is gamma positive iff n ≡ 0,1 (mod 4)
- For n ≡ 2 (mod 4), $A_n^+(t)$ is a sum of two gamma positive polynomials
- For n ≡ 3 (mod 4), $A_n^+(t)$ is a sum of three gamma positive polynomials

## Abstract

The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $A_n^+(t)$ as the polynomial obtained when we sum descents over the alternating group. We show that $A_n^+(t)$ is gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2$ (mod 4) we show that $A_n^+(t)$ can be written as a sum of two gamma positive polynomials while if $n \equiv 3$ (mod 4), we show that $A_n^+(t)$ can be written as a sum of three gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials.