According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the identity matrix $I_n$, or more generally of any positive definite matrix $P$.
Gårding's statement means that the differential operator $$L:=\det\left(\frac{\partial}{\partial a_{ij}}\right),$$ where $a_{ij}\equiv a_{ji}$, is hyperbolic. In other words, the Cauchy problem $Lf(A)=0$, with initial data given on the hyperplane $\operatorname{Tr} A=0$ (for instance), is well-posed.
My question is whether this differential operator occurs somewhere, apart from Gårding's work. If $n=2$, $L$ is equivalent to the wave operator in $3$ variables (two space dimensions). But in general, $L$ has order $n$. Mind that an initial condition consists of $n$ arbitrary functions on the initial hyperplane.
Edit. As commented by Mariano, $L$ is the restriction to $\mathbf{Sym}_n$ of Cayley's $\Omega$ process. However this does not answer my question, because $\Omega$ is not hyperbolic. The hyperbolicity makes $L$ somewhat specific in the context of differential operators.
