Total positivity for the Lagrangian Grassmannian

Sign variation, the Grassmannian, and total positivity

Twist Positivity for Lagrangian Symmetries *

We prove twist positivity and positivity of the pair correlation function for combined spatial and internal symmetries of free bosonic Lagrangians. We work in a general setting, extending the results obtained in Twist Positivity [1]. Work supported in part by the Department of Energy under Grant DE-FG02-94ER-25228. This research was carried out in part for the Clay Mathematics Institute.

Quantum Cohomology of the Lagrangian Grassmannian

Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V . We give a presentation for the (small) quantum cohomology ring QH∗(LG) and show that its multiplicative structure is determined by the ring of Q̃-polynomials. We formulate a ‘quantum Schubert calculus’ which includes quantum Pieri and Giambelli formulas, as well as algor...

Quasimaps, Straightening Laws, and Quantum Cohomology for the Lagrangian Grassmannian

The Drinfel’d Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the canonical embedding of the Lagrangian Grassmannian. We show that the defining ideal of any Schubert subvariety of the Drinfel’d Lagrangian Grassmannian is generated by polynomials which gi...

Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ = (λ1, . . . , λk) with λ1 ≤ n there is a Schubert variety X(λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T -equivariant cohomology of LGn and the T -equivariant fundamental class...