A gluing construction for polynomial invariants

Abstract

We give a polynomial gluing construction of two groups G X ⊆ GL ( ℓ , F ) and G Y ⊆ 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 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 F p , and the finite subgroups of GL ( n , F ) defined by sparsity patterns, which generalize many known examples.