Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

TL;DR

This paper links the cohomology classes of conormal bundles of Schubert varieties in Grassmannians to Yangian weight functions, revealing new orthogonality relations and basis transformations via the Yangian R-matrix.

Contribution

It expresses conormal bundle classes as sums of Yangian weight functions and establishes their orthogonality and basis relations in equivariant cohomology.

Findings

Fundamental classes are expressed as sums of Yangian weight functions.

Orthogonality relations for modified classes are established.

Basis transformations are described by the Yangian R-matrix.

Abstract

We consider the conormal bundle of a Schubert variety $S_I$ in the cotangent bundle $T^* Gr$ of the Grassmannian $Gr$ of $k$-planes in $C^n$. This conormal bundle has a fundamental class ${\kappa_I}$ in the equivariant cohomology $H^*_{T}(T^* Gr)$. Here $T=(C^*)^n\times C^*$. The torus $(C^*)^n$ acts on $T^* Gr$ in the standard way and the last factor $C^*$ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum $Y_I$ of the Yangian $Y(gl_2)$ weight functions $(W_J)_J$. We describe a relation of $Y_I$ with the double Schur polynomial $[S_I]$. A modified version of the $\kappa_I$ classes, named $\kappa'_I$, satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold $T^* Gr$. This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on $Gr$. The classes $(\kappa'_I)_I$ form a basis in the suitably localized equivariant cohomology $H^*_{T}(T^* Gr)$. This basis depends on the choice of the coordinate flag in $C^n$. We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.