Gamma-positivity of generalized W-Eulerian polynomials

Prove that the generating polynomials of the two-sided W-Eulerian numbers N_{ij} are gamma-positive for finite Coxeter groups W, thereby implying that each diagonal of the matrix N(W) is unimodal.

Background

The two-sided W-Eulerian numbers N_{ij} form a matrix indexed by the numbers of left and right ascents of elements of a finite Coxeter group. The paper notes that T. K. Peterson’s generalized Gessel’s conjecture concerns the generating polynomials of these numbers.

Gamma-positivity is a strengthening of unimodality. In this setting, the conjectured gamma-positivity would imply that every diagonal of the matrix N(W) is unimodal, which the paper describes as a numerical shadow of the hard Lefschetz theorem.

References

Note that T. K. Peterson’s generalized Gessel’s conjecture [P, Conjecture 16] asserts that the generating polynomials of the W -Eulerian numbers Nij are γ-positive, which easily implies that each diagonal of the matrix N(W ) is unimodal, a numerical shadow of the hard Lefschetz theorem.

— On the Total Positivity of Contingency Metamatrices  (2503.02213 - Wang et al., 4 Mar 2025) in Remark 3.1.3, Section 3.1, p. 9