Is it possible to find matrix solutions to the following :
$$(\sum_1^m M_k x_k)^n=\sum_1^m x_k^n$$
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 $n$.