The Ize Conjecture Redux: A Parity Criterion for Global Equivariant Bifurcation Guarantees

Abstract

The Ize Conjecture proposed that every absolutely irreducible representation of a compact Lie group admits a maximal isotropy subgroup with an odd-dimensional fixed-point space, which would provide a universal bifurcation guarantee via the equivariant degree. Its disproof by Lauterbach and Matthews necessitates a more targeted criterion. We introduce Ize pairs -- pairs $(G, V)$ for which some maximal isotropy subgroup $H$ satisfies $\dim V^H - \dim V^G \equiv 1 \pmod{2}$ -- and prove that this dimension-parity condition completely captures the algebraic obstruction to a non-trivial equivariant degree change at maximal orbit types. Integrating this criterion with a mod-2 equivariant spectral flow yields local and global bifurcation guarantees without recourse to Burnside ring arithmetic. As an application, we establish unbounded branches of non-stationary periodic solutions in a $\Gamma$-symmetric coupled oscillator network, where the bifurcation guarantees follow entirely from the crossing parity of the linearization at the boundary of a regular parameter window.