A gluing construction for polynomial invariants

We give a polynomial gluing construction of two groups GX⊆GL(ℓ,F) and GY⊆GL(m,F) which results in a group G⊆GL(ℓ+m,F) whose ring of invariants is isomorphic to the tensor product of the rings of invariants of GX and GY. 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 Fp, and the finite subgroups of GL(n,F) defined by sparsity patterns, which generalize many known examples.