A Centraliser Analogue to the Farahat-Higman Algebra

TL;DR

This paper introduces a family of algebras generalising the Farahat-Higman algebra by replacing the center with centraliser algebras, and proves their isomorphism to a tensor product involving the degenerate affine Hecke algebra.

Contribution

It defines and analyses a new class of algebras mathsf{FH}_m, extending the Farahat-Higman algebra concept with centraliser algebras and establishes their isomorphism to a tensor product of known algebraic structures.

Findings

mathsf{FH}_m is isomorphic to the tensor product of the degenerate affine Hecke algebra and symmetric functions.

Marked cycle shapes generalise proper integer partitions and govern the properties of mathsf{FH}_m.

The algebra mathsf{FH}_m extends the classical Farahat-Higman algebra with new combinatorial and algebraic structures.

Abstract

We define a family of algebras \mathsf{FH}_{m} which generalise the Farahat-Higman algebra introduced in [FH59] by replacing the role of the center of the group algebra of the symmetric groups with centraliser algebras of symmetric groups. These algebras have a basis indexed by marked cycle shapes, combinatorial objects which generalise proper integer partitions. We analyse properties of marked cycle shapes and of the algebras \mathsf{FH}_{m}, demonstrating that some of the former govern the latter. The main theorem of the paper proves that the algebra \mathsf{FH}_{m} is isomorphic to the tensor product of the degenerate affine Hecke algebra with the algebra of symmetric functions.