ABSTRACT:  It is possible to compute the Hilbert series of an invariant ring a priori, i.e., without knowing generators of the invariant ring. For finite groups one can use Molien's theorem and for connected reductive groups one can compute the Hilbert series of the invariant ring by (iterated) use of the Residue Theorem. Dixmier made a conjecture about the Hilbert series of the invariant ring for binary forms. This conjecture can be verified for binary forms up to a large degree. If R is a finitely generated graded ring (R_0 = K, the base field and R_d = 0 for d < 0) and M is a finitely generated graded module then the denominator of the Hilbert series of M does not necessarily divide the denominator of the Hilbert series of R. This is why I define the universal denominator of M as the least common multiple of the denominators of the Hilbert series of all graded submodules of M. There are many interesting interpretations of this universal denominator. Explicit formulas for universal denominators of invariant rings of finite groups or tori can be given in terms of geometry. For binary forms, the universal denominator of the ring of invariants is equal to the denominator conjectured by Dixmier. This notion of universal denominator can also be used to shed new light on a result of Jerzy Weyman and myself that Littlewood-Richardson coefficients depend polynomially on the partitions.

For further information visit http://www.northeastern.edu/martsinkovsky/p/rtrt.html  or contact Alex Martsinkovsky <alexmart@neu.edu>