## Abstract

Let V = ∐ ^{n} _{i=0} V _{i} be the lattice of sub- spaces of the n-dimensional vector space over the finite field F _{q} , and let A be the graded Gorenstein algebra de- fined over Q which has V as a Q basis. Let F be the Macaulay dual generator for A. We explicitly compute the Hessian determinant |∂ ^{2} /∂X _{i} ∂X _{j} |, 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 V _{n-1} . Our explo- ration 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.

Original language | English |
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Pages (from-to) | 131-154 |

Number of pages | 24 |

Journal | Journal of Commutative Algebra |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

## Keywords

- Finite geometry
- Gorenstein algebras
- Hessian
- Incidence matrix
- Strong Lefschetz property
- Vector space lattice