Higher Specht Bases for Generalizations of the Coinvariant Ring

Abstract

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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Notes

  1. 1.

    In [1], the terminology used is ‘charge’, but we use ‘cocharge’ to be consistent with the original notation of Lascoux and Schutzenberger [14].

  2. 2.

    It is straightforward to verify that the inequality in (9) is equivalent to the one stated in [4], and we omit the proof for brevity.

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Acknowledgements

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

Keywords

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