The Hilbert-Poincare series for some algebras of invariants of cyclic groups

N. L. Gordeev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 7, Tome 272 (2000), pp. 144-160

Résumé

Let $\rho$ be a linear representation of a finite group over a field of characteristic 0. Further, let $R_{\rho}$ be the corresponding algebra of invariants and let $P_{\rho}(t)$ be its Hilbert-Poincare series. Then the series $P_{\rho}(t)$ presents a rational function $\Psi(t)/\Theta(t)$. If $R_{\rho}$ is a complete intersection then $\Psi(t)$ is a product of cyclotomic polynomials. Here we prove the inverse statement for the case when $\rho$ is “almost regular” (in particular, regular) representation of a cyclic group. It gives the answer to a question of R. Stanley in this very particular case.