Higher Specht Bases for Generalizations of the Coinvariant Ring


The classical coinvariant ring \(R_n\) is defined as the quotient of a polynomial ring in n variables by the positive-degree \(S_n\)-invariants. It has a known basis that respects the decomposition of \(R_n\) into irreducible \(S_n\)-modules, consisting of the higher Specht polynomials due to Ariki, Terasoma, and Yamada (Hiroshima Math J 27(1):177–188, 1997). We provide an extension of the higher Specht basis to the generalized coinvariant rings \(R_{n,k}\) introduced in Haglund et al. (Adv Math 329:851–915, 2018). We also give a conjectured higher Specht basis for the Garsia–Procesi modules \(R_\mu \), and we provide a proof of the conjecture in the case of two-row partition shapes \(\mu \). We then combine these results to give a higher Specht basis for an infinite subfamily of the modules \(R_{n,k,\mu }\) recently defined by Griffin (Trans Amer Math Soc, to appear, 2020), which are a common generalization of \(R_{n,k}\) and \(R_{\mu }\).

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We thank Nicolas Thiéry for several illuminating conversations about higher Specht polynomials, and for sharing some helpful code. Thanks also to Sean Griffin for many discussions regarding his new \(S_n\)-modules \(R_{n,k,\mu }\). We used the open source mathematics software Sage extensively in testing conjectures and generating examples for this work.

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Gillespie, M., Rhoades, B. Higher Specht Bases for Generalizations of the Coinvariant Ring. Ann. Comb. (2021). https://doi.org/10.1007/s00026-020-00516-1

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Mathematics Subject Classification

  • 05E05
  • 05E10
  • 05E40
  • 20C30


  • Young tableaux
  • Representation theory of \(S_n\)
  • Symmetric functions
  • Polynomial ideals
  • Coinvariants