Constructing a Gr\"obner basis of Griffin's ideal

TL;DR

This paper develops a recursive method to construct a Gr"obner basis for Griffin's ideals, unifying several important algebraic structures and confirming their monomial bases.

Contribution

It provides the first explicit recursive construction of a Gr"obner basis for Griffin's ideals, linking them to known algebraic objects.

Findings

Gr"obner basis constructed recursively

Monomial basis confirmed as standard basis

Coefficients are integers with leading coefficient one

Abstract

In his Ph.D. thesis, Sean Griffin introduced a family of ideals and found monomial bases for their quotient rings. These rings simultaneously generalize the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We recursively construct a Gr\"{o}bner basis of Griffin's ideals with respect to the graded reverse lexicographical order. Consequently, Griffin's monomial basis is the standard monomial basis. Coefficients of polynomials in our Gr\"{o}bner basis are integers and leading coefficients are one.