Is it possible to find matrix solutions to the following :
$$\left(\sum\_1^m M\_k x\_k\right)^n=\left(\sum\_1^m x\_k^n\right)I\_d$$
where $M\_k$ are the desired $d \times d$ matrices (no restriction on $d$) and $x\_i$ are indeterminate variables;
For n=2 the gamma matrices satisfing $M\_i M\_j + M\_j M\_i = 2\delta\_{ij}$ work; so in a way this is a generalization of these to larger $n$.



a generalization of gamma matrices
Yes, these are called Generalized Clifford Algebras. The earliest reference I could find was [an article by Yamazaki](http://hdl.handle.net/2261/6042) from 1964.


An explicit construction is given by [Morris](http://qjmath.oxfordjournals.org/content/18/1/7)


For example, with $m=2$ and $n=3$, we find as a solution $M\_1 = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{pmatrix}$ and $M\_2 = \begin{pmatrix} 1 & 0 & 0\\ 0 & e^{2 \pi i/3} & 0\\ 0 & 0 & e^{4\pi i /3}\end{pmatrix}$


