Multigraded Hilbert Series of noncommutative modules

TL;DR

This paper introduces methods for computing multigraded Hilbert series of noncommutative modules, providing conditions for rationality, characterizations of finite-dimensional algebras, and efficient algorithms with implementation in Singular.

Contribution

It develops new algorithms for computing multigraded Hilbert series of noncommutative algebras, including effective methods, characterizations, and implementations.

Findings

Methods are effective when the algebra's associated matrix is nilpotent.

The Hilbert series can be expressed as rational functions under certain conditions.

The implementation in Singular enables comprehensive testing and practical computation.

Abstract

In this paper, 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 sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.