TITLE: a generalization of gamma matrices
QUESTION [9 upvotes]: 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 $n$.

REPLY [3 votes]: Yes, these are called Generalized Clifford Algebras. The earliest reference I could find was an article by Yamazaki from 1964.
An explicit construction is given by Morris
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 $M_2 = \begin{pmatrix} 1 & 0 & 0\\ 0 & e^{2 \pi i/3} & 0\\ 0 & 0 & e^{4\pi i /3}\end{pmatrix}$