Tangent spaces and T-invariant curves of Schubert varieties

TL;DR

This paper investigates the structure of tangent spaces and T-invariant curves in Schubert varieties, establishing relationships between their weights and providing criteria for smoothness using equivariant K-theory.

Contribution

It proves that tangent space weights are contained in the rational cone generated by T-invariant curve weights and offers conditions for equality and smoothness criteria.

Findings

Weights of tangent spaces are in the rational cone of T-invariant curve weights.

In simply laced types, tangent space weights are in the integral cone.

Provides smoothness criteria based on root decomposability.

Abstract

The set of T-invariant curves in a Schubert variety through a T-fixed point is relatively easy to characterize in terms of its weights, but the tangent space is more difficult. We prove that the weights of the tangent space are contained in the rational cone generated by the weights of the T-invariant curves. In simply laced types, this remains true if "rational" is replaced by "integral". We also obtain conditions under which every weight of the tangent space is the weight of a T-invariant curve, as well as a smoothness criterion. The results rely on equivariant K-theory, as well as the study of different notions of decomposability of roots.