Hilbert Series of Group Representations and Grbner Bases for Generic Modules

Each matrix representation P: G —> GLn(K) of a finite Group G over a field K induces an action of G on the module A over the polynomial algebra A = K [ x 1 , . . . , xn]. The graded A-submodule M(P) of A generated by the orbit of (x1, ..., xn) is studied. A decomposition of MO) into generic modules is given. Relations between the numerical invariants of P and those of M(P), the latter being efficiently computable by Grobner bases methods, are examined. It is shown that if P is multiplicity-free, then the dimensions of the irreducible constituents of P can be read off from the Hilbert series of M(P). It is proved that determinantal relations form Grobner bases for the syzygies on generic matrices with respect to any lexicographic order. Grobner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M(P) is obtained for an arbitrary representation.