Orbits under Dual Symplectic Transvections

Abstract

Consider an arbitrary field $K$ and a finite-dimensional vector space $X$ over $K$ equipped with a, possibly degenerate, symplectic form $\omega$. Given a spanning subset $S$ of $X$, for each $k$ in $K$ and each vector $s$ in $S$, consider the symplectic transvection mapping a vector $x$ to $x+k\omega(x,s)s$. The group generated by these transvections has been extensively studied, and its orbit structure is known. In this paper, we obtain corresponding results for the orbits of the dual action on $X^\ast$. As for the non-dual case, the analysis gets harder when the field contains only two elements. For that field, the dual transvection group is equivalent to a game known as the lit-only sigma game, played on a graph. Our results provide a complete solution to the reachability problem of that game, previously solved only for some special cases.