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

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.