Matroid Automorphisms And Symmetry Groups

For a subgroup W of the hyperoctahedral group On which is generated by reflections, we consider the linear dependence matroid MW on the column vectors corresponding to the reflections in W . We determine all possible automorphism groups of MW and determine when W ∼= Aut(MW ). This allows us to connect combinatorial and geometric symmetry. Applications to zonotopes are also considered. Signed graphs are used as a tool for constructing the automorphisms.