Steven H. Weintraub
(Lehigh University)
Spreads of non-singular pairs in symplectic vector spaces
Abstract [ps] [pdf]: Let $V$ be a vector space of dimension $2n$ over a field $\mathbf{F}$ equipped with a non-singular symplectic form $<,>$.
Def. A non-singular pair (or ns-pair) in $V$ is $\Delta = \{\delta,\delta^\perp\}$ consisting of a pair of $n$-dimensional subspaces, mutually orthogonal, whose direct sum is $V$, and such that the restriction of $<,>$ to each of them is non-singular.
Def. A spread of non-singular pairs or nsp-spread is a set of ns-pairs $\sigma = \{\Delta_i\}$ whose non-zero elements partition $V-\{0\}$ (regarding $\Delta$ as the union of the elements in $\delta$ and $\delta^\perp$).
Theorem. For any even $n$, any field $\mathbf{F}$ that is either a finite field of odd characteristic or an algebraic number field, and any vector space $V$ of dimension $2n$ over $\mathbf{F}$, as above, there exist nsp-spreads in $V$.
$\Sigma = \{\mbox{nsp-spreads } \sigma\ \mbox{ in } V\}$ is an interesting algebraic/combinatorial object, which we discuss. This theorem is new except in case $n=2$ and $\mathbf{F}=\mathbf{F}_3$, in which case (as Coble knew in 1908) the $27$ elements of $\Sigma$ correspond to the $27$ lines on the non-singular cubic surface.