Universal subspaces for compact Lie groups

TL;DR

This paper establishes a criterion for subspaces to intersect all orbits in representations of compact Lie groups, generalizing classical results like Schur's triangularization theorem using cohomology and invariant theory.

Contribution

It provides a new sufficient and sometimes necessary condition for subspaces to meet all group orbits, extending classical matrix decomposition theorems.

Findings

Derived a criterion for subspace orbit intersection in Lie group representations

Applied the criterion to matrix conjugation actions of classical groups

Generalized Schur's triangularization theorem

Abstract

For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also necessary in certain cases. The proof makes use of the cohomology of flag manifolds and the invariant theory of Weyl groups. Then we apply our condition to the conjugation representations of U(n), Sp(n), and SO(n) in the space of $n\times n$ matrices over C, H, and R, respectively. In particular, we obtain an interesting generalization of Schur's triangularization theorem.