Invariant Spinors on Flag Manifolds

TL;DR

This paper characterizes when non-trivial invariant spinors exist on maximal flag manifolds linked to complex simple Lie algebras, using combinatorial properties of positive roots, and provides bounds on their dimensions.

Contribution

It offers a new combinatorial criterion for the existence of invariant spinors on flag manifolds and estimates their dimension.

Findings

Invariant spinors exist under specific combinatorial conditions.

Bounds are provided for the dimension of invariant spinor spaces.

The characterization links root system properties to spinor invariance.

Abstract

In this note, we characterise the existence of non-trivial invariant spinors on maximal flag manifolds associated to complex simple Lie algebras. This characterisation is based on the combinatorial properties of their set of positive roots. We also give some bounds for the dimension of the space of invariant spinors in each case.