Ordered set partitions, Garsia-Procesi modules, and rank varieties.

We introduce a family of ideals In,λ,s in Q[x1, . . . , xn] for λ a partition of k ≤ n and an integer s ≥ l (λ). This family contains both the Tanisaki ideals Iλ and the ideals In,k of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings Rn,λ,s as symmetric group modules. When n = k and s is arbitrary, we recover the Garsia-Procesi modules, and when λ = (1k) and s = k, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for Rn,λ, s in terms of (n,λ,s)-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the Sn-module structure of Rn,λ,s in terms of an action on (n, λ, s)-ordered set partitions. We find a formula for the Hilbert series of Rn,λ,s in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with (n, λ, s)-ordered set partitions. Furthermore, we prove an expansion of the graded Frobenius characteristic of our rings into Gessel's fundamental quasisymmetric basis. We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on Rn,λ,s, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices. [ABSTRACT FROM AUTHOR]