The first fundamental theorem of coinvariant theory for the quantum general linear group

Abstract

We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum general and special linear groups on tensor products of quantum matrix algebras. More precisely, let m,n,t be arbitrary positive integers, let A and B be the quantum coordinate rings of the $t \times t$ general and special linear groups over an arbitrary field K, and let $C_{m,t}$ and $C_{t,n}$ denote the quantum coordinate rings of $m \times t$ and $t \times n$ matrices over K. We first prove that the set of coinvariants for the coaction of A on $C_{m,t} \otimes C_{t,n}$ equals the image of the natural K-algebra map from the quantum coordinate ring of $m \times n$ matrices to $C_{m,t} \otimes C_{t,n}$ induced by comultiplication. The set of coinvariants for the coaction of B on $C_{m,t} \otimes C_{t,n}$ is shown to be the subalgebra generated by the above image together with a tensor product of two algebras generated by $t \times t$ quantum minors.