Exact dimension of the variety of idempotent matrices

Determine the exact dimension, as a function of n, of the affine algebraic variety of idempotent matrices in M_n(K) over an algebraically closed field K; specifically, compute dim({E in M_n(K) | E^2 = E}).

Background

The paper views the set of idempotent n×n matrices over an algebraically closed field K as an affine algebraic variety and proves non-irreducibility, along with general upper and lower bounds on its dimension. In particular, they show n−1 ≤ dim ≤ n^2−2 for n ≥ 2 and compute small-n cases computationally.

Despite these bounds and examples, the exact dimension formula remains unsettled, motivating a direct question about determining the precise dimension for general n.