|Title||Hilbert Series of Group Representations and Gröbner Bases for Generic Modules|
|Author(s)|| Shmuel Onn|
|Type||Technical Report, Misc|
|Abstract||Each matrix representation pi : G --> GL_n (K) of a finite group G|
over a field K induces an action of G on the module A^n over the polynomial algebra A = K[x_1, ... , x_n]. The graded A-submodule M(pi) of A^n generated by the orbit of (x_1, ... , x_n) is studied.
A decomposition of M(pi) into generic modules is given. Relations
between the numerical invariants of pi and those of M(pi), the later being efficiently computable by Groebner bases methods, are examined. It is shown that if pi is multiplicity-free, then the dimensions of
the irreducible constituents of pi can be read off from the Hilbert
series of M(pi). It is proved that determinantal relations form Groebner bases for the syzygies on generic matrices with respect to any lexicographic order. Groebner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently,
the Hilbert series of M(pi) is obtained for an arbitrary representation.