DETERMINANTS OF INCIDENCE AND HESSIAN MATRICES ARISING FROM THE VECTOR SPACE LATTICE

Let V = (SIC)(i=0)(n) V-i be the lattice of subspaces of the n-dimensional vector space over the finite field F-q, and let A be the graded Gorenstein algebra defined over Q which has V as a Q basis. Let F be the Macaulay dual generator for A. We explicitly compute the Hessian determinant vertical bar partial derivative F-2/partial derivative X-i partial derivative X-j vertical bar, evaluated at the point X-1 = X-2 = ... = X-N = 1, and relate it to the determinant of the incidence matrix between V-1 and Vn-1.Our exploration is motivated by the fact that both of these matrices naturally arise in the study of the Sperner property of the lattice and the Lefschetz property for the graded Artinian Gorenstein algebra associated to it.