First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..


**Problem S.** Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following properties.


**1.** $V$ is a finite dimensional complex vector space equipped with a Hermitian metric. $\DeclareMathOperator{\Sym}{Sym}$ We denote by $\Sym(V)$ the space of symmetric complex linear operators $V\to V$.


**2.** $\newcommand{\bR}{\mathbb{R}}$ $S$ is a linear map $S:\bR^n\to\Sym(V)$ such that for any $\xi\in\bR^n\setminus 0$ the symmetric operator $S(\xi)$ is invertible.


Readers familiar with the basics of p.d.e.-s will surely recognize $S(\xi)$ as the principal **symbol** of an elliptic, first order partial differential operator with constant coefficients that acts on $C^\infty(\bR^n, V)$. That explains the letter $S$ in the name of the problem.


We denote by $\newcommand{\eS}{\mathscr{S}}$ $\eS\_n$ the space of solutions of **Problem S** for a given positive integer $n$.


Observe that $\eS\_n$ is equipped with two basic algebraic operations $\oplus,\otimes$ $\newcommand{\one}{\boldsymbol{1}}$


$$(V\_1, S\_1)\oplus (V\_2, S\_2):= ( V\_1\oplus V\_2, S\_1\oplus S\_2), $$


$$ (V\_1, S\_1)\otimes (V\_2, S\_2):= ( V\_1\otimes V\_2, S\_1\otimes\one\_{V\_2}+ \one\_{V\_1}\otimes S\_2). $$


The group $\DeclareMathOperator{\GL}{GL}$ $\GL(n,\bR)$ acts in an obvious way on $\eS\_n$. More precisely if $S:\bR^n\to\Sym(V)$ is a solution $S\in\eS\_n$, and $T\in \GL(n,\bR)$, then $S\circ T\in \eS\_n$.


Let us also observe that for each $n$, the set $\eS\_n$ is not empty. We can obtain maps $S: \bR^n\to\Sym(V)$ with the desired properties by using complex representations of the Clifford algebra generated by an Euclidean inner product on the space $\bR^n$. I will refer to such examples as Clifford examples and I will denote by $\newcommand{\eC}{\mathscr{C}}$ $\eC\_n$ the subset of $\eS\_n$ constructed as above using representations of Clifford algebras. Observe that $\eC\_n$ is also closed under the operations $\oplus,\otimes$ and invariant under the above action of $\GL(n,\bR)$


**Main Question.** Fix $n$ Are there non Clifford solutions to Problem S? In other words, is the set $\eS\_n\setminus \eC\_n$ non-empty?


**Addendum.** Apparently this question is related to a classical question discussed by Porteous in his book *Topological Geometry*. For a given *real* vector space $V$ find the *largest* $n$ find the maximal subspaces $\DeclareMathOperator{\Endo}{End}$ $S\subset \Endo(V)$ such that $S\setminus 0 \subset \GL(V)$. The answer has to do with Radon-Hurwitz numbers, and it basically says that if $S$ is such a subspace, maximal or not, then $V$ s a module over the Clifford algebra generated by an inner product on $S$. 



Symbols of elliptic operators
Maybe I'm misunderstanding something, but it seems that the answer is probably 'no', at least if $d = \dim\_\mathbb{C} V$ is large enough. 


What really matters is the $n$-dimensional real subspace $\mathsf{S}=\mathrm{Im}(S)\subset\mathrm{Sym}(V)$. What you need is that this space not meet the cone of singular matrices in $\mathrm{Sym}(V)$ anywhere but at the origin. This is an open condition on the element $\mathsf{S}\in\mathrm{Gr}\_n\bigl(\mathrm{Sym}(V),\mathbb{R}\bigr)$, and this latter space has dimension $n(d^2{-}n)$. Thus, the set of such subspaces with your property has this latter dimension, but the group of unitary transformations on $V$ (which, I gather, is the space of symmetries of the problem) is only of dimension $d^2$, so there must be at least an $(n{-}1)d^2-n^2$ parameter family of 'inequivalent' subspaces that meet your criteria. 


However, there are only a finite number of inequivalent $d$-dimensional complex representations of the Clifford algebra on $\mathbb{R}^n$ endowed with a definite inner product.


