A Gorenstein simplicial complex for symmetric minors

TL;DR

This paper demonstrates that the ideal generated by the (n-2) minors of a generic symmetric matrix has an initial ideal corresponding to a Stanley-Reisner ideal of a simplicial polytope boundary, sharing Betti numbers.

Contribution

It establishes a connection between algebraic ideals of symmetric minors and combinatorial structures of simplicial polytopes, revealing new geometric insights.

Findings

Initial ideal is Stanley-Reisner ideal of a simplicial polytope boundary

The ideal shares the same Betti numbers as the polytope boundary complex

Provides a geometric interpretation of symmetric minors' algebraic properties

Abstract

We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.