Realization of Homogeneous Cones Through Oriented Graphs

Abstract

Abstract. In this paper, we realize any homogeneous cone by assembling uniquely determined subcones. These subcones are realized in the cones of positive-definite real symmetric matrices of minimal possible sizes. The subcones are found through the oriented graphs drawn by using the data of the given homogeneous cones. We also exhibit several interesting examples of our realizations of homogeneous cones. These are of rank 5, of dimension 19, of dimension 11 of continuously many inequivalent homogeneous cones, and some of the low-dimensional homogeneous cones.

A Characterization of Symmetric Cones by the Degrees of Basic Relative Invariants

“…In particular, we list up the technical terms from graph theory which we borrow in this paper. These technical terms are very useful to clarify our argument similarly to the previous paper [18]. In Section 3, two important properties of basic relative invariants are presented.…”

“…In the previous paper [18], we borrowed some terminology from graph theory. In this paper also, terminology from graph theory is helpful to clarify the discussion.…”

“…This follows from the fact that σ permutes the 'components' of the clan V as noted in Remark 4.15 in [18]. To be more convinced, we add the prime symbol for those which arise from the normal decomposition (3.3).…”

“…In Section 3, two important properties of basic relative invariants are presented. The first property established in the previous paper [18,Proposition 4.2] shows in particular that the actual variables in the basic relative invariant k (x) come only from the out-neighbor subclan V out [k]. This observation will be used several times in this paper.…”

Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones

“…The cone P n of positive definite n × n real symmetric matrices is a typical example of homogeneous cones. It is known [12][13][14][15][16] that every homogeneous cone is linearly isomorphic to a cone P n ∩ Z with an appropriate subspace Z of the vector space Sym(n, R) of all n × n real symmetric matrices, where Z admits a specific block decomposition. Based on such results, our matrix realization method [15,17,18] has been developed for the purpose of the efficient study of homogeneous cones.…”