Affine subspaces of antisymmetric matrices with constant rank.

For every $ n \in \mathbb {N} $ n ∈ N and every field K, let $ A(n,K) $ A (n , K) be the vector space of the antisymmetric $ (n \times n) $ (n × n) -matrices over K. We say that an affine subspace S of $ A(n,K) $ A (n , K) has constant rank r if every matrix of S has rank r. Define $$\begin{align*} {\mathcal{A}}_{antisym}^K(n;r)& = \{ S \;| \; S \; {\rm affine \; subspace \; of } \; A(n,K) \; {\rm of \; constant \; rank } \; r \}, \\ a_{antisym}^K(n;r) & = \max \{\dim S \mid S \in {\mathcal{A}}_{antisym}^K(n;r) \}. \end{align*}$$ A antisym K (n ; r) = { S | S affine subspace of A (n , K) of constant rank r } , a antisym K (n ; r) = max { dim ⁡ S ∣ S ∈ A antisym K (n ; r) }. In this paper, we prove the following formulas: for $ n \geq ~2r +2 $ n ≥ 2 r + 2 \[ a_{antisym}^{\mathbb{R}}(n; 2r) = (n-r-1) r ; \] a antisym R (n ; 2 r) = (n − r − 1) r ; for n = 2r \[ a_{antisym}^{\mathbb{R}}(n; 2r) =r(r-1) ; \] a antisym R (n ; 2 r) = r (r − 1) ; for n = 2r + 1 \[ a_{antisym}^{\mathbb{R}}(n; 2r) = r(r+1). \] a antisym R (n ; 2 r) = r (r + 1). [ABSTRACT FROM AUTHOR]