Séminaire Lotharingien de Combinatoire, 82B.86 (2019), 12 pp.

Darij Grinberg

A quotient of the ring of symmetric functions generalizing quantum cohomology

Abstract. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a1, a2, ..., ak in k. Let I be the ideal of S generated by hn-k+1-a1, hn-k+1-a2, ..., hn-k+1-ak, where hi is the i-th complete homogeneous symmetric polynomial.

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k x (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S3-symmetry as those of the Grassmannian cohomology. We conjecture the existence of a Pieri rule (proven in two particular cases) and a positivity property generalizing that of Gromov-Witten invariants.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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