Darij Grinberg: A quotient of the ring of symmetric functions generalizing quantum cohomology

Abstract: Let \(\mathcal{S}\) be the ring of symmetric polynomials in k variables over an arbitrary base ring \(\mathbf{k}\). Fix k scalars \(a_1, a_2, \ldots, a_k \in \mathbf{k}\). Let I be the ideal of \(\mathcal{S}\) generated by \(h_{n-k+1}-a_1, h_{n-k+2}-a_2, \ldots, h_n-a_k\), where \(h_i\) is the i-th complete homogeneous symmetric polynomial.

We study the quotient ring \(\mathcal{S}/I\), which generalizes both the classical and the quantum cohomology of the Grassmannian.

We show that \(\mathcal{S}/I\) has a \(\mathbf{k}\)-module basis consisting of (residue classes of) Schur polynomials fitting into a k by (n-k) rectangle; a Pieri rule (for multiplication by \(h_j\)); a "rim hook algorithm"; and that its multiplicative structure constants satisfy the same \(S_3\)-symmetry as those of the Grassmannian cohomology. These constants also appear to satisfy a (sign-alternating) positivity property generalizing that of Gromov-Witten invariants; if true, this suggests a hidden geometric meaning.