$Sp(n)$-orbits of isoclinic subspaces in the real Grassmannians

TL;DR

This paper classifies $Sp(n)$-orbits of isoclinic subspaces in real Grassmannians using invariants derived from quaternionic structures, providing a detailed description of their geometric and algebraic properties.

Contribution

It introduces new invariants for isoclinic subspaces and characterizes their $Sp(n)$-orbits in real Grassmannians, extending previous classifications.

Findings

Invariants $(\xi,\chi,\eta)$ and $(\Gamma,\Delta)$ classify orbits.

Angles of isoclinicity determine the orbit when combined with invariants.

Special cases reduce invariants to pairs $(\xi= ext{±1}, ext ext{±1})$.

Abstract

In the framework of the study of the $Sp(n)$-orbits in the real Grassmannian $G^\R(k,4n)$ of $k$-dimensional non oriented subspaces of a real $4n$-dimensional vector space $V$, here we consider the case of the isoclinic subspaces whose set we indicate with $\mathcal{IC}$. Endowed $V$ with an Hermitian quaternionic structure $(\mathcal{Q},<,>)$, a subspace $U$ is isoclinic if for any compatible complex structure $A \in \mathcal{Q}$ the principal angles of the pair $(U,AU)$ are all the same, say $\theta^A$. We will show that, fixed an admissible hypercomplex basis $(I,J,K)$, to any such subspace $U$ we can associate two set of invariants, namely a triple $(\xi,\chi,\eta)$ and a pair $(\Gamma, \Delta)$ where $\Gamma$ itself is a function of $(\xi,\chi,\eta)$. We prove that the angles of isoclinicity $(\theta^I,\theta^J,\theta^K)$ together with $(\xi,\chi,\eta, \Delta)$ determine its $Sp(n)$-orbit. In particular if $\dim U= 8k+2$ or $\dim U= 8k+6$ with $k \geq 0$ the last set reduce to the pair $(\xi= \pm 1,\chi= \pm 1)$.