Groups Generated by Symplectic Transvections over Local Rings

Abstract

Let R be a commutative local ring with the unique maximal ideal A. Let V be a free module of rank n over R. And let Spn(V) be the symplectic group on V with an alternating bilinear form f: V×V→R. We study the generation of a subgroup TR(M) of Spn(V), where M={x∈V|f(x,V)=R} and TR(M) is defined as the subgroup generated by all symplectic transvections with axis x⊥ for x∈M.Our main goal is to get a nice necessary and sufficient condition for any subset N⊆M satisfying TR(N)=TR(M), where TR(N) is the group generated by all symplectic transvections with axis x⊥ for x∈N. In particular, if f is nonsingular we have TR(M)=Spn(V), and therefore our necessary and sufficient condition gives us a criterion for an arbitrarily given N⊆M satisfying TR(N)=Spn(V).Also we shall investigate the TR(N) orbit of each x∈M, find some small sets of generators of TR(M) consisting of transvections in TR(N), and as a result solve the author's conjecture in “Generators and Relations in Groups and Geometries” (A. Barlotti et al., Eds.), pp. 47–67, Proc. NATO ASI (C).