On families of anticommuting matrices

Let e1,…,ek be complex n×n matrices such that eiej=−ejei whenever i≠j. We conjecture that rk(e12)+rk(e22)+⋯+rk(ek2)≤O(nlog⁡n). We show that:(i).rk(e1n)+rk(e2n)+⋯+rk(ekn)≤O(nlog⁡n),(ii).if e12,…,ek2≠0 then k≤O(n),(iii).if e1,…,ek have full rank, or at least n−O(n/log⁡n), then k≤O(log⁡n). (i) implies that the conjecture holds if e12,…,ek2 are diagonalisable (or if e1,…,ek are). (ii) and (iii) show it holds when their rank is sufficiently large or sufficiently small.