A gluing construction for polynomial invariants

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.

arXiv comment10 pages, to appear in Journal of algebra