2023/12/06 by Jonas Sjöstrand, Sjöstrand, Jonas
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #05C25 #05C50 #05C57 #05E18 #15A63 #20F10 #Combinatorics (math.CO) #Economic theories and models #FOS: Mathematics #Game Theory and Applications #Mathematical Dynamics and Fractals #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2312.03933
openalex publication_date 2023/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an arbitrary field K and a finite-dimensional vector space X over K equipped with a, possibly degenerate, symplectic form ω. 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+kω(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^∗. 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.