Monomial right ideals and the Hilbert series of noncommutative modules

TL;DR

This paper introduces a new method for computing the Hilbert series of finitely generated monomial right modules over free associative algebras, utilizing automata theory to ensure termination and efficiency.

Contribution

It presents the first general computational procedure for noncommutative Hilbert series based on automata theory and colon right ideal operations.

Findings

Procedure terminates when cyclic submodules are annihilated by regular language monomial ideals.

Automata theory guarantees minimal iterations in the computation process.

Implementation in Maple demonstrates practical efficiency for noncommutative Hilbert series.

Abstract

In this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module $N$ over the free associative algebra $K\langle x_1,\ldots,x_n \rangle$. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing $N$ are annihilated by monomial right ideals whose monomials define regular formal languages. The method is based on the iterative application of the colon right ideal operation to monomial ideals which are given by an eventual infinite basis. By using automata theory, we prove that the number of these iterations is a minimal one. In fact, we have experimented efficient computations with an implementation of the procedure in Maple which is the first general one for noncommutative Hilbert series.