Hilbert Series of Group Representations and Grr Obner Bases for Generic Modules

Each matrix representation : G ?! GL n (K) of a nite group G over a eld K induces an action of G on the module A n over the polynomial algebra A = Kx 1 ; ; x n ]. The graded A-submodule M() of A n generated by the orbit of (x 1 ; ; x n) is studied. A decomposition of M() into generic modules is given. Relations between the numerical invariants of and those of M(), the later being eeciently computable by Grr obner bases methods, are examined. It is shown that if is multiplicity-free, then the dimensions of the irreducible constituents of can be read oo from the Hilbert series of M(). It is proved that determinantal relations form Grr obner bases for the syzygies on generic matrices with respect to any lexicographic order. Grr obner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M() is obtained for an arbitrary representation.