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 $R_{\mu }$.