Representation Theoretic Bases for the $\hat{I}”$-Springer Module

Abstract

We give a descent monomial basis of $\hat{I}”$-Springer modules $R_{n,\hat{I}»,s}$, first defined by Griffin. Our construction simultaneously generalizes the descent basis for the Garsia-Procesi module $R_{\hat{I}}»$ studied by Carlsson-Chou and Hanada, as well as the descent basis for the generalized coinvariant algebras $R_{n,k}$ studied by Haglund-Rhoades-Shimozono. This basis is deeply connected with a combinatorial object called battery-powered tableaux, introduced by Gillespie-Griffin. We highlight the representation theoretic properties of this monomial basis by using it to give a direct combinatorial proof of the graded Frobenius character of $R_{n,\hat{I}»,s}$ in terms of battery-powered tableaux, a fact which has only the geometric proof of Gillespie-Griffin. We also conjecture a higher Specht basis of $R_{n,\hat{I}»,s}$, generalizing the higher Specht basis of the coinvariant ring defined in Ariki-Terasoma-Yamada. This construction coincides with the Gillespie-Rhoades higher Specht basis for $R_{n,k}$. We give a proof for when $\hat{I}»=(\hat{I}{»}_1,\hat{I}{»}_2)$ is a partition of two rows.

49 pages