The centers of spin symmetric group algebras and Catalan numbers

TL;DR

This paper explores the structure of centers in spin symmetric group superalgebras, introduces spin FH-algebras, and uncovers a link between Jucys-Murphy elements and Catalan numbers, advancing algebraic understanding.

Contribution

It generalizes Farahat-Higman theory to spin symmetric groups and establishes a novel connection between algebra generators and Catalan numbers.

Findings

Description of even centers Z_n of spin superalgebras

Introduction of spin FH-algebras as universal structures

Connection between Jucys-Murphy elements and Catalan numbers

Abstract

Generalizing the work of Farahat-Higman on symmetric groups, we describe the structures of the even centers Z_n of integral spin symmetric group superalgebras, which lead to universal algebras termed as the spin FH-algebras. A connection between the odd Jucys-Murphy elements and the Catalan numbers is developed and then used to determine the algebra generators of the spin FH-algebras and of the even centers Z_n.