Anti-commuting varieties

We study the anti-commuting variety which consists of pairs of anti-commuting n\times n matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT (geometric invariant theory) quotient of the anti-commuting variety with respect to the conjugation action of GL_n is shown to be of pure dimension n. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension n^2 and describe its irreducible components.