UC Davis

We define a combinatorial construction that gives a natural subset of the Garsia-Stanton descentmonomials whose images under the canonical projection $R_n \to R_\lambda$ form a vector space basis of the Garsia-Procesi module $R_\lambda$. As a consequence, our indexing set yields a new formula for the modified Hall-Littlewood polynomials. Our work was discovered whilst searching for a basis of the Garsia- Haiman module, and we discuss partial results in this direction, as well as other connections with the modified Macdonald polynomials $\widetilde{H}_\lambda(X;q,t)$.