  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.
