The affine preservers of non-singular matrices
arXiv:1003.5675 · doi:10.1007/s00013-010-0173-6
Abstract
When K is an arbitrary field, we study the affine automorphisms of M_n(K) that stabilize GL_n(K). Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n>2 or #K>2. We include a short new proof of the more general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank. We also find that the group of affine transformations of M_2(F_2) that stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of quadratic forms over F_2, we construct explicit isomorphisms between it, the symplectic group Sp_4(F_2) and the symmetric group S_6.
13 pages, very minor corrections from the first version