Monomial bases for the centres of the group algebra and Iwahori--Hecke algebra of S_4

TL;DR

This paper characterizes the infinitely many monomial symmetric polynomial bases for the center of the group algebra of S_4 and explores similar bases for the Iwahori--Hecke algebra, advancing understanding of algebraic structures.

Contribution

It provides a complete characterization of monomial symmetric polynomial bases for the center of S_4's group algebra and extends the analysis to the Iwahori--Hecke algebra.

Findings

Infinitely many bases for the center of S_4's group algebra are characterized.

Closed-form formulas for class sum coefficients are established.

Several integral bases for the Iwahori--Hecke algebra of S_4 are identified.

Abstract

G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys--Murphy elements. While we have shown in earlier work that the centre of the group algebra of S_3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S_4 has infinitely many bases consisting of monomial symmetric polynomials in Jucys--Murphy elements, which we characterize completely. The proof of this result involves establishing closed forms for coefficients of class sums in the monomial symmetric polynomials in Jucys--Murphy elements, and solving several resulting exponential Diophantine equations with the aid of a computer. Our initial motivation was in finding integral bases for the centre of the Iwahori--Hecke algebra, and we address this question also, by finding several integral bases of monomial symmetric polynomials in Jucys--Murphy elements for the centre of the Iwahori--Hecke algebra of S_4.