Positivity of Equivariant Schubert Classes Through Moment Map Degeneration

TL;DR

This paper provides a positive, combinatorial formula for equivariant Schubert class restrictions on flag manifolds of types A, B, and C, using moment map degeneration and co-adjoint orbit identification.

Contribution

It introduces a new positive formula for computing equivariant Schubert classes via moment map degeneration, extending known results and providing explicit combinatorial descriptions.

Findings

The formula is valid for types A, B, and C.

In type A, the formula matches Billey's known formula.

The approach uses degeneration of the moment map and co-adjoint orbit identification.

Abstract

For a flag manifold $M=G/B$ with the canonical torus action, the $T-$equivariant cohomology is generated by equivariant Schubert classes, with one class $\tau_u$ for every element $u$ of the Weyl group $W$. These classes are determined by their restrictions to the fixed point set $M^T \simeq W$, and the restrictions are polynomials with nonnegative integer coefficients in the simple roots. The main result of this article is a positive formula for computing $\tau_u(v)$ in types A, B, and C. To obtain this formula we identify $G/B$ with a generic co-adjoint orbit and use a result of Goldin and Tolman to compute $\tau_u(v)$ in terms of the induced moment map. Our formula, given as a sum of contributions of certain maximal ascending chains from $u$ to $v$, follows from a systematic degeneration of the moment map, corresponding to degenerating the co-adjoint orbit. In type A we prove that our formula is manifestly equivalent to the formula announced by Billey in \cite{Bi}, but in type C, the two formulas are not equivalent.