The decomposition of the spinor bundle of Grassmann manifolds

TL;DR

This paper decomposes the spinor bundle of Grassmann manifolds into irreducible components, providing a universal construction and analyzing the Dirac operator's spectrum and eigenspaces.

Contribution

It introduces a universal method for decomposing the spinor bundle of specific Grassmann manifolds and proves the general case for certain dimensions.

Findings

Decomposition of spinor bundles into irreducible representations.

Analysis of the Dirac operator's spectrum and eigenspaces.

Universal construction applicable to multiple Grassmann manifolds.

Abstract

The decomposition of the spinor bundle of the spin Grassmann manifolds $G_{m,n}=SO(m+n)/SO(m)\times SO(n)$ into irreducible representations of $\mathfrak{so}(m)\oplus\mathfrak{so}(n)$ is presented. A universal construction is developed and the general statement is proven for $G_{2k+1,3}$, $G_{2k,4}$, and $G_{2k+1,5}$ for all $k$. The decomposition is used to discuss properties of the spectrum and the eigenspaces of the Dirac operator.