Representation Theoretic Bases for the $\Delta$-Springer Module

We give a descent monomial basis of $\Delta$-Springer modules $R_{n,\lambda,s}$, first defined by Griffin. Our construction simultaneously generalizes the descent basis for the Garsia-Procesi module $R_\lambda$ 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 $R_{n,\lambda,s}$ 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 $R_{n,\lambda,s}$, 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 $\lambda = (\lambda_1,\lambda_2)$ is a partition of two rows.