A gluing construction for polynomial invariants

TL;DR

This paper introduces a polynomial gluing method to construct larger groups from smaller ones, ensuring their invariant rings are tensor products, thus enabling the creation of many groups with polynomial invariants.

Contribution

It presents a novel gluing construction that produces groups with polynomial invariant rings, extending known classes including p-groups and sparsity pattern groups.

Findings

Constructed groups with polynomial rings of invariants.

Extended classes of groups with polynomial invariants.

Unified various known examples under a common framework.

Abstract

We give a polynomial gluing construction of two groups $G_X\subseteq GL(\ell,\mathbb F)$ and $G_Y\subseteq GL(m,\mathbb F)$ which results in a group $G\subseteq GL(\ell+m,\mathbb F)$ whose ring of invariants is isomorphic to the tensor product of the rings of invariants of $G_X$ and $G_Y$. In particular, this result allows us to obtain many groups with polynomial rings of invariants, including all $p$-groups whose rings of invariants are polynomial over $\mathbb F_p$, and the finite subgroups of $GL(n,\mathbb F)$ defined by sparsity patterns, which generalize many known examples.