Polytopal Linear Groups
نویسندگان
چکیده
The main objects of this paper are the graded automorphisms of w x polytopal semigroup rings, i.e., semigroup rings k S where k is a field P Ž and S is the semigroup associated with a lattice polytope P Bruns, P w x. w x Gubeladze, and Trung BGT . The generators of k S correspond bijecP tively to the lattice points in P, and their relations are the binomials representing the affine dependencies of the lattice points. The simplest examples of such rings are the polynomial rings w x k X , . . . , X with the standard grading. They are associated to the unit 1 n Ž . w x simplices D . The graded automorphism group GL k of k X , . . . , X ny1 n 1 n is generated by diagonal and elementary automorphisms. Our main result is a generalization of this classical fact to the graded automorphism group Ž . G P of an arbitrary polytopal semigroup ring: It says that each automork Ž . Ž . phism g g G P has a non-unique normal form as a composition of toric k and elementary automorphisms and symmetries of the underlying polyŽ . tope. In view of this analogy we call the groups G P polytopal linear k Ž . groups. As in the case of GL k , the elementary and toric automorphisms n Ž . Ž . generate G P if and only if this group is connected. The elementary k automorphisms are defined in terms of so-called column structures on P.
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