A basis for a quotient of symmetric polynomials

Consider the ring $\mathcal{S}$ of symmetric polynomials in $k$ variables over an arbitrary base ring $\mathbf{k}$. Fix $k$ scalars $a_1,a_2,\dots ,a_k\in \mathbf{k}$. Let $I$ be the ideal of $\mathcal{S}$ generated by $h_{n-k+1}-a_1,h_{n-k+2}-a_2,\dots ,h_n-a_k$, where $h_i$ is the $i$-th complete homogeneous symmetric polynomial. The quotient ring $\mathcal{S}\mathrm{/}I$ generalizes both the usual and the quantum cohomology of the Grassmannian. We show that $\mathcal{S}\mathrm{/}I$ has a $\mathbf{k}$-module basis consisting of (residue classes of) Schur polynomials fitting into an $\left( n-k\right) \times k$-rectangle; and that its multiplicative structure constants satisfy the same $S_3$-symmetry as those of the Grassmannian cohomology. We prove a Pieri rule and a “rim hook algorithm”, and conjecture a positivity property generalizing that of Gromov-Witten invariants. We construct two further bases of $\mathcal{S}\mathrm{/}I$ as well. We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same $k$ polynomials as $I$.