Equivariant quantum cohomology of the odd symplectic Grassmannian

Abstract

The odd symplectic Grassmannian IG :" IGpk, 2n`1q parametrizes k dimensional subspaces of C 2n`1 which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IGpk, 2n`2q. We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k " 2, and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

Conjecture O holds for the odd symplectic Grassmannian

“…In this section, we briefly introduce the odd symplectic Grassmanian and some basic properties of its cohomology ring. We refer to for details; we follow closely the exposition from .…”

“…Unlike the homogeneous case, these numbers might be negative in general. For instance, using Pech's quantum Pieri rule to multiply in the case of the odd symplectic Grassmannian of lines , one obtains in , For arbitrary , the second and third authors found a Chevalley formula calculating in the equivariant quantum cohomology ring, and proved that this formula gives a recursive algorithm to calculate all the other structure constants (see ). The nonequivariant rule was also found in the recent paper .…”

“…Notice also that is an ample divisor in (it is simply the restriction of the Schubert divisor from ), therefore is a Fano variety. We refer to either or for details. To describe the quantum multiplication by , we need to recall the ordinary Chevalley formula in proved in .…”

“…Fix and let be the odd symplectic Grassmannian. This is a smooth Fano algebraic variety parametrizing dimensional linear subspaces , which are isotropic with respect to a skew‐symmetric, bilinear form with kernel of dimension 1 (see ). The purpose of this paper is to prove Galkin et al.…”

“…However, the positivity does not hold for the odd symplectic Grassmannian (see, for example, below), and it is still unknown whether analogous symmetries exist. We circumvent this problem by making heavy use of the combinatorics of the recently found quantum Chevalley formula in , which governs the quantum multiplication by .…”