## Journal of Commutative Algebra

- J. Commut. Algebra
- Volume 11, Number 1 (2019), 131-154.

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

Saeed Nasseh, Alexandra Seceleanu, and Junzo Watanabe

#### 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.

#### Article information

**Source**

J. Commut. Algebra, Volume 11, Number 1 (2019), 131-154.

**Dates**

First available in Project Euclid: 13 March 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.jca/1552464135

**Digital Object Identifier**

doi:10.1216/JCA-2019-11-1-131

**Mathematical Reviews number (MathSciNet)**

MR3923368

**Zentralblatt MATH identifier**

07037591

**Subjects**

Primary: 05B20: Matrices (incidence, Hadamard, etc.) 05B25: Finite geometries [See also 51D20, 51Exx] 51D25: Lattices of subspaces [See also 05B35]

Secondary: 13A02.

**Keywords**

Vector space lattice incidence matrix Hessian strong Lefschetz property Gorenstein algebras finite geometry

#### Citation

Nasseh, Saeed; Seceleanu, Alexandra; Watanabe, Junzo. Determinants of incidence and Hessian matrices arising from the vector space lattice. J. Commut. Algebra 11 (2019), no. 1, 131--154. doi:10.1216/JCA-2019-11-1-131. https://projecteuclid.org/euclid.jca/1552464135