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 \(R_{\mu }\).

将经典协变环\（R_n \）定义为n个变量中多项式环的商乘以正度\（S_n \）-不变式。它具有已知的依据，可以将\（R_n \）分解为不可约的\（S_n \）-模块，该模块由Ariki，Terasoma和Yamada产生的较高Specht多项式组成（广岛数学J 27（1）：177– 188，1997）。我们对Haglund等人引入的广义协变环\（R_ {n，k} \）提供了更高的Specht基础的扩展。（Adv Math 329：851–915，2018）。我们还为Garsia–Procesi模块\（R_ \ mu \）提供了一个较高的Specht基础。，并且在两行分区形状\（\ mu \）的情况下，我们提供了猜想的证明。然后，我们将这些结果结合起来，为Griffin最近定义的模块\（R_ {n，k，\ mu} \）的无限子族提供更高的Specht基础（Trans Amer Math Soc，将于2020年出现）。\（R_ {n，k} \）和\（R _ {\ mu} \）的通用概括。