A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps
type: Item
instance of: scholarly article
title: A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps
zbMATH Open document ID: 1167.32015
author: Chifune Kai
published in: Journal of the Mathematical Society of Japan
publication date: 2009-01-30T00:00:00Z
review text: The author establishes a new criterion which characterizes the symmetric cones among the homogeneous convex ones. An open convex cone \(\Omega \) in a real vector space \(V\) is said to be homogeneous if the group \(G(\Omega )\) of linear automorphisms of \(\Omega \) acts on \(\Omega \) transitively. Moreover if there exists on \(V\) an inner product such that the cone \(\Omega \) is equal to the dual cone \(\Omega ^*\), then \(\Omega \) is said to be symmetric. Let us recall the definition of the Vinberg map \(x\mapsto x^*\), from a homogeneous cone \(\Omega \) onto its dual \(\Omega ^*\). One defines first the characteristic function \(\varphi \) of \(\Omega \): \[ \varphi (x)=\int _{\Omega ^*} e^{-\langle x,f\rangle } \,df. \] Then \(x^*=\text{grad\;log}\, \varphi (x)\). One considers on \(\Omega \) a partial ordering: \(x\geq y\) if \(x-y\in \overline{\Omega }\), and similarly on \(\Omega ^*\). The author proves that the cone \(\Omega \) is symmetric if and only if it satisfies the following property: for \(x,y\in \Omega \), \(x\geq y\) if and only if \(y^*\geq x^*\). In fact the author proves a more general result for which the characteristic function \(\varphi \) is replaced by a semi-invariant function on \(\Omega \) with respect to a solvable subgroup \(H\) of \(G(\Omega )\) acting transitively on \(\Omega \). The proofs use criterions for a homogeneous convex cone to be symmetric involving the generalized Peirce decomposition of \(V\) related to the clan structure on \(V\) associated to the homogeneous cone \(\Omega \) and the subgroup \(H\).
reviewed by: Jacques Faraut
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 5500753
zbMATH Keywords: homogeneous convex cone
zbMATH Keywords: symmetric cone
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06041107
OpenAlex ID: W1978658330
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DOI: 10.2969/JMSJ/06041107
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description: scientific article; zbMATH DE number 5500753