Computing noncommutative Hilbert series

Abstract

We propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the Hilbert series have a rational sum. Efficient variants of the methods are also developed for the truncations of infinite-dimensional algebras which provide approximations of possibly irrational Hilbert series. Moreover, we provide a characterization of the finite-dimensional algebras in terms of the nilpotency of a key matrix involved in the computations. Finally, we present a well-tested and complete implementation for the computation of graded and multigraded Hilbert series which has been developed in the kernel of the computer algebra system Singular (for the details, see preprint[1]).