An integral second fundamental theorem of invariant theory for partition algebras

TL;DR

This paper establishes an invariant theory analogue for partition algebras, showing their centraliser algebras are cellular, extending classical results to a broader algebraic context.

Contribution

It proves the kernel of the Weyl group action on tensor space forms a cell ideal, demonstrating cellularity of partition algebra centralisers over any commutative ring.

Findings

Kernel of Weyl group action is a cell ideal

Partition algebra centralisers are cellular

Results extend invariant theory to partition algebras

Abstract

We prove that the kernel of the action the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.