Existence of a projective variety realizing two-sided Eulerian numbers as Hodge numbers

Construct, for each finite Coxeter group W, a smooth projective variety X(W) whose Hodge numbers are given by the two-sided W-Eulerian numbers N_{ij}, thereby realizing the matrix N(W) as a Hodge diamond after a rotation of 45 degrees.

Background

For a finite Coxeter group (W,S), the paper defines the two-sided W-Eulerian numbers N_{ij} as the numbers of elements w in W having prescribed cardinalities of their left and right ascent sets. These numbers form the matrix N(W), which satisfies symmetry relations analogous to Hodge symmetry and Serre duality.

Tao Gui’s unpublished conjecture proposes that these numerical symmetries arise geometrically: after a 45-degree rotation, N(W) would be the Hodge diamond of a smooth projective variety X(W). The paper explicitly notes that this conjecture remains unresolved even for Coxeter groups of type A.

References

There is an interesting conjecture by Tao Gui (unpublished) saying that there should be a smooth projective variety X(W ) whose Hodge numbers are given by those Nij ’s. That is, after a rotation of 45 degrees, the matrix N(W ) gives the Hodge diamond of X(W ). If this conjecture is true, then the first kind of symmetry in (3.1.3.1) comes from the Hodge symmetry whereas the second kind of symmetry in (3.1.3.1) comes from the Serre duality. 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. As far as we know, Gui’s conjecture is currently unsolved even for type A.

— 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