The cellular second fundamental theorem of invariant theory for classical groups

TL;DR

This paper establishes a cellular algebra framework for the centralizer algebras of classical groups acting on tensor spaces, extending invariant theory to symplectic, orthogonal, and general linear groups.

Contribution

It introduces an axiomatic framework for quotient towers of diagram algebras, providing new insights into the second fundamental theorem of invariant theory for classical groups.

Findings

Centralizer algebras are cellular over the integers.

Framework applies to symplectic, orthogonal, and general linear groups.

Descriptions of kernels of algebra homomorphisms related to invariant theory.

Abstract

We prove that the centralizer algebras of the symplectic and orthogonal group acting on tensor space are cellular algebras over the integers. We do this by providing an axiomatic framework for studying quotient towers for towers of diagram algebras. Our framework covers the centralizer algebras of the general linear group acting on mixed tensor space as well. We also provide descriptions of the kernels of the homomorphisms from the diagrammatic versions of the algebras to the centralizer algebras on tensor space, which constitute versions of the second fundamental theorem of invariant theory.