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2019 Determinants of incidence and Hessian matrices arising from the vector space lattice
Saeed Nasseh, Alexandra Seceleanu, Junzo Watanabe
J. Commut. Algebra 11(1): 131-154 (2019). DOI: 10.1216/JCA-2019-11-1-131

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 explicitly compute the Hessian determinant $|{\partial ^2F}/{\partial X_i \partial X_j}|$, 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 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.

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Saeed Nasseh. Alexandra Seceleanu. Junzo Watanabe. "Determinants of incidence and Hessian matrices arising from the vector space lattice." J. Commut. Algebra 11 (1) 131 - 154, 2019. https://doi.org/10.1216/JCA-2019-11-1-131

Information

Published: 2019
First available in Project Euclid: 13 March 2019

zbMATH: 07037591
MathSciNet: MR3923368
Digital Object Identifier: 10.1216/JCA-2019-11-1-131

Subjects:
Primary: 05B20 , 05B25 , 51D25
Secondary: 13A02.

Keywords: Finite geometry , Gorenstein algebras , Hessian , Incidence matrix , strong Lefschetz property , Vector space lattice

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.11 • No. 1 • 2019
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