The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let ρ be a linear representation of a finite group over a field of characteristic 0. Further, let Rρ be the corresponding algebra of invariants, and let Pρ(t) be its Hilbert–Poincaré series. Then the series Pρ(t) represents a rational function Ψ(t)/Θ(t). If Rρ is a complete intersection, then Ψ(t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where ρ is an “almost regular” (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.