Primitivity Testing in Free Group Algebras via Duality
Matan Seidel, Danielle Ernst-West, Doron Puder
TL;DR
This work addresses primitivity and related extension problems in free group algebras by introducing a duality, called $Q$-duality, between extensions of column and row spaces over a free ideal ring. The double $Q$-dual yields the algebraic closure, providing concrete criteria to decide freeness versus algebraicity and enabling explicit algorithms for computing algebraic closures, testing algebraicity, and testing primitivity-like properties for extensions of free $K\left[F\right]$-modules and, via augmentation ideals, for extensions of free groups. Central technical advances include the invariant $\phi_{I,J}(N)$ under duality and a natural rank-preserving involution in the $(w-1)$ case, together with a practical algorithmic toolkit built on Rosenmann’s algorithm for right ideals. The framework also delivers an intersection algorithm for free modules and translates to efficient procedures for free-group extensions, connecting to word measures and highlighting potential computational applications in algorithmic group theory and representation-theoretic contexts.
Abstract
Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces.