Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Details Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

2005-08-05

arxiv.org/abs/math/0508110v3

Mathematics

Algebraic Geometry

Scientific paper

Let $LG_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $\lambda=(\lambda_1,...,\lambda_k)$ with $\lambda_1\leq n$ there is a Schubert variety $X(\lambda).$ Let $T$ denote a maximal torus of the symplectic group acting on $LG_n.$ Consider the $T$-equivariant cohomology of $LG_n$ and the $T$-equivariant fundamental class $\sigma(\lambda)$ of $X(\lambda).$ The main result of the present paper is an explicit formula for the restriction of the class $\sigma(\lambda)$ to any torus fixed point. The formula is written in terms of factorial analogue of the Schur $Q$-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for $LG_n.$ As another consequence of the main result, we obtained a presentation of the ring $H_T^*(LG_n).$

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