A centralizer analogue to the Farahat–Higman algebra

We define a family of algebras FH m which generalise the Farahat–Higman algebra introduced in [4] 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 FH m , demonstrating that some of the former govern the latter. The main theorem of the paper proves that the algebra FH m is isomorphic to the tensor product of the degenerate affine Hecke algebra with the algebra of symmetric functions.