The decomposition of the spinor bundle of 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.

arXiv comment30 pages. v2 coincides with published version

References in corpus (3)

• The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces
• The point spectrum of the Dirac operator on noncompact symmetric spaces
• A formula for the First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

Cited by in corpus (2)

• A note on the regularity of matrices with uniform polynomial entries
• The First Eigenvalue of the Dirac Operator on Compact Outer Spin Symmetric Spaces