Fixed coordinate subspace (${\mathbb{R}}_k^p=\{\theta :{\theta }_j=0(j>k)\}$)

The fixed coordinate subspace of dimension $k$ in ${\mathbb{R}}^p$ consists of vectors whose coordinates after the first $k$ vanish. Restricting a quadratic form to it uses only the leading $k$ by $k$ matrix block. It specifies one fixed support; allowing every support of size $k$ would introduce another combinatorial counting step.