Invariants of toric double determinantal rings

TL;DR

This paper investigates the algebraic and combinatorial properties of a class of double determinantal ideals, revealing their structure as Hibi rings, computing key invariants, and characterizing Gorenstein cases.

Contribution

It provides a comprehensive analysis of $I_{mn}^r$, including explicit formulas for invariants, a new proof of dimension, and a characterization of when the associated ring is Gorenstein.

Findings

Computed minimal generators, multiplicity, regularity, a-invariant, Hilbert function, and h-polynomial.

Characterized Gorenstein property of the rings $R/I_{mn}^r$.

Described facets of the Stanley-Reisner complex of initial ideals.

Abstract

We study a class of double determinantal ideals denoted $I_{mn}^r$, which are generated by minors of size 2, and show that they are equal to the Hibi rings of certain finite distributive lattices. We compute the number of minimal generators of $I_{mn}^r$, as well as the multiplicity, regularity, a-invariant, Hilbert function, and $h$-polynomial of the ring $R/I_{mn}^r$, and we give a new proof of the dimension of $R/I_{mn}^r$. We also characterize when the ring $R/I_{mn}^r$ is Gorenstein, thereby answering a question of Li in the toric case. Finally, we give combinatorial descriptions of the facets of the Stanley-Reisner complex of the initial ideal of $I_{mn}^r$ with respect to a diagonal term order.