The uniqueness of Lyapunov rank among symmetric cones
Michael Orlitzky, Giovanni Barbarino
TL;DR
The paper investigates whether the Lyapunov rank $\beta(K)$ uniquely identifies symmetric cones up to isomorphism. It frames the problem via signatures $\sigma(K)=(\dim K,\beta(K))$ and the notion of simulacra, then provides a complete irreducible-class result: among irreducible symmetric cones, the only ones with no simulacra are the Lorentz cones $\mathcal{L}^n_+$ and the cone $\mathcal{H}^3_+(\mathbb{C}^1)$, with $\mathcal{L}^n_+$ having no simulacra for all $n$ and $\mathcal{H}^3_+(\mathbb{C}^1)$ admitting simulacra for a specific set of sizes. The reducible case is handled through a combination of bottom-up/top-down reasoning and an improved subproblem reduction, showing that, under suitable bounds on $n$ and the dimensions, simulacra rarely occur and are tied to the presence of certain irreducible factors; a single small counterexample ($n=4$) and detailed classifications of sums of Lorentz cones are provided. Overall, the work sharpens the isomorphism problem for symmetric cones and informs how Lyapunov-rank-based identifiers can (or cannot) distinguish cones in symmetric-cone optimization.
Abstract
The Lyapunov rank of a cone is the dimension of the Lie algebra of its automorphism group. It is invariant under linear isomorphism and in general not unique - two or more non-isomorphic cones can share the same Lyapunov rank. It is therefore not possible in general to identify cones using Lyapunov rank. But suppose we look only among symmetric cones. Are there any that can be uniquely identified (up to isomorphism) by their Lyapunov ranks? We provide a complete answer for irreducible cones and make some progress in the general case.