A Centraliser Analogue to the Farahat-Higman Algebra

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.