On families of anticommuting matrices

Abstract

Let $e_1,\dots ,e_k$ be complex $n\times n$ matrices such that $e_i{e}_j=-e_j{e}_i$ whenever $i\ne j$. We conjecture that $rk(e_1^2)+rk(e_2^2)+\cdots +rk(e_k^2)\le O(n\log n)$, and prove some results in this direction.