We give a monomial basis for Rn,λ,s in terms of (n,λmbda,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 (n,λ,s)-ordered set partitions. Furthermore, we give an expansion of the graded Frobenius characteristic of our rings in terms of Gessel's fundamental basis and in terms of dual Hall-Littlewood symmetric functions.
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.
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