Determinants of Incidence and Hessian Matrices Arising From the Vector Space Lattice

Authors

Saeed Nasseh, Georgia Southern UniversityFollow
Alexandra Seceleanu, University of Nebraska - LincolnFollow
Junzo Watanabe, Tokai UniversityFollow

Date of this Version

Spring 2019

Citation

2019 Rocky Mountain Mathematics Consortium

Comments

S. NASSEH, A. SECELEANU AND J. WATANABE DOI:10.1216/JCA-2019-11-1-131

Abstract

Let V = ⁿi= _o V^I bethe lattice of subspaces of the n-dimensional vector space over the finite field Fq, 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 j 2F= Xi Xj j, evaluated at the point X1 = X2 = ... = XN = 1, and relate it to the determinant of the incidence matrix between V1 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.