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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Ordered set partitions, Garsia-Procesi modules, and rank varieties
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by Sean T. Griffin PDF
Trans. Amer. Math. Soc. 374 (2021), 2609-2660 Request permission


We introduce a family of ideals $I_{n,\lambda , s}$ in $\mathbb {Q}[x_1,\dots , x_n]$ for $\lambda$ a partition of $k\leq n$ and an integer $s \geq \ell (\lambda )$. This family contains both the Tanisaki ideals $I_\lambda$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings $R_{n,\lambda , s}$ as symmetric group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi modules, and when $\lambda =(1^k)$ and $s=k$, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono.

We give a monomial basis for $R_{n,\lambda , s}$ in terms of $(n,\lambda , s)$-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the $S_n$-module structure of $R_{n,\lambda , s}$ in terms of an action on $(n,\lambda , s)$-ordered set partitions. We find a formula for the Hilbert series of $R_{n,\lambda , s}$ in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with $(n,\lambda , s)$-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 $R_{n,\lambda , s}$, 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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Additional Information
  • Sean T. Griffin
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • MR Author ID: 1228593
  • Email:
  • Received by editor(s): April 3, 2020
  • Received by editor(s) in revised form: June 9, 2020
  • Published electronically: February 2, 2021
  • Additional Notes: The author was supported in part by NSF Grant DMS-1764012.
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 2609-2660
  • MSC (2020): Primary 05E05, 20C30, 05E10; Secondary 05A19, 05A18, 13D40
  • DOI:
  • MathSciNet review: 4223028