Classification of skew symmetric matrices

Abstract:

The group ${\text {GL(}}d,\mathbb {Z}{\text {) = Aut(}}{\mathbb {Z}^d}{\text {)}}$ acts on the $\mathbb {Z}$-module $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z}){\text {by}}\varphi \to \varphi {\text {(}}\alpha \Lambda \alpha {\text {)}}\quad {\text {(}}\alpha \in {\text {Aut}}{\mathbb {Z}^d}{\text {)}}$. Associated with each $\varphi$ in $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z})$ is a finite set of invariants completely describing the orbit of $\varphi$ under this action. The result holds with $\mathbb {Z}$ replaced by an arbitrary commutative principal ideal domain.