Determinants of incidence and Hessian matrices arising from the vector space lattice

TL;DR

This paper explicitly computes the Hessian determinant of a dual generator related to the lattice of subspaces over a finite field and links it to incidence matrices, advancing understanding of algebraic and combinatorial properties.

Contribution

It provides an explicit calculation of the Hessian determinant for the Gorenstein algebra associated with the subspace lattice and connects it to incidence matrices, revealing new algebraic-combinatorial relationships.

Findings

Hessian determinant expressed in terms of incidence matrix determinants

Explicit formulas for the Hessian at the symmetric point

Insights into the Sperner and Lefschetz properties for the lattice

Abstract

Let $\mathcal{V}=\bigsqcup_{i=0}^n\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and let $\mathcal{A}$ be the graded Gorenstein algebra defined over $\mathbb{Q}$ which has $\mathcal{V}$ as a $\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\mathcal{A}$. We compute explicitly the Hessian determinant $|\frac{\partial ^2F}{\partial X_i \partial X_j}|$ evaluated at the point $X_1 = X_2 = \cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\mathcal{V}_1$ and $\mathcal{V}_{n-1}$. Our exploration is motivated by the fact that both of these matrices arise naturally in the study of the Sperner property of the lattice and the Lefschetz property for the graded Artinian Gorenstein algebra associated to it.