Spectral cones in Euclidean Jordan algebras

Abstract A spectral cone in a Euclidean Jordan algebra V of rank n is of the form K = λ − 1 ( Q ) , where Q is a permutation invariant convex cone in R n and λ : V → R n is the eigenvalue map (which takes x to λ ( x ) , the vector of eigenvalues of x with entries written in the decreasing order). In this paper, we describe some properties of spectral cones. We show, for example, that spectral cones are invariant under automorphisms of V , that the dual of a spectral cone is a spectral cone when V is simple or carries the canonical inner product, and characterize the pointedness/solidness of a spectral cone. We also show that for any spectral cone K in V , dim ⁡ ( K ) ∈ { 0 , 1 , m − 1 , m } , where dim ⁡ ( K ) denotes the dimension of K and m is the dimension of V .