A characterization of symmetric cones through pseudoinverse maps (Q1775420)

Let us first recall the following classical fact about the cone \(\Omega \) of positive definite symmetric matrices in the vector space \(V= \text{Sym}(n,{\mathbb R})\). Let \(z\) be a symmetric complex matrix. Then \(\Re (z)\) is positive definite if and only if \(z\) is invertible and \(\Re (z^{-1})\) is positive definite. This property can be written \(I(\Omega +iV)=\Omega +iV\), where \(I\) is the inverse map, \(I:z\mapsto z^{-1}\). This property is shared by all symmetric cones, i.e. homogeneous and selfadjoint cones, with respect to the inverse in the Jordan algebra sense. In this paper the authors show that this property charaterizes symmetric cones among homogeneous ones. Let \(\Omega \) be a homogeneous regular open convex cone in a finite dimensional real vector space \(V\). Then, as it has been shown by Vinberg, the vector space \(V\) has the structure of a clan, a non-associative algebra. The product is denoted as \(x\Delta y\), and the left multiplication by \(x\) as \(L_x\). Then \(\langle x|y\rangle = \operatorname{Tr} L_{x\Delta y}\) defines a Euclidean inner product. Let \(\varphi \) be the characteristic function of the cone \(\Omega\): \[ \varphi (x)=\int _{\Omega ^*}e^{-\langle x |y\rangle }\,dy\quad (x\in \Omega ), \] where \(\Omega ^*\) is the dual cone. Then the Vinberg's map \(I\) is defined by \(I(x)=-\text{grad} \log \varphi (x)\). It extends as a homomorphic \(V\)-valued function on the tube \(\Omega +iV\). The main result of the paper is: Assume \(\Omega \) irreducible. Then \(\Omega \) is selfadjoint if and only if \(I(\Omega +iV)=\Omega ^*+iV\). In fact the authors prove a more general result. They consider a family of Euclidean inner products \(\langle x|y\rangle _{\mathbf s}\) on \(V\), parametrized by systems \({\mathbf s}=(s_1,\ldots ,s_r)\) of positive numbers. For each \(\mathbf s\) one defines a pseudoinverse \(I_{\mathbf s}\) and a dual cone \(\Omega ^{\mathbf s}\). The general result is stated as: Assume \(\Omega \) irreducible, and fix \(\mathbf s\). The following properties are equivalent: (A) \(I_{\mathbf s}(\Omega +iV)=\Omega ^{\mathbf s}+iV\), (B) \(\mathbf s\) is a positive multiple of \(\mathbf d\) and \(\Omega \) is a symmetric cone, (C) \(\mathbf s\) is a positive multiple of \(\mathbf d\) and \(\Omega =\Omega ^{\mathbf s}\). The system \({\mathbf d}=(d_1,\ldots ,d_r)\) is related to the normal decomposition of the clan \(V\), \[ V=\sum _{1\leq j\leq k\leq r}V_{jk}. \] One of the key steps in the proof is to show that, if (A) holds, then the multiplicities \(n_{jk}=\dim V_{jk}\) (\(j<k\)) are independent of \(j\) and \(k\). By a result of Vinberg, this implies that the cone \(\Omega \) is symmetric.