Monomial Bases for the Centres of the Group Algebra and Iwahori–hecke Algebra
نویسندگان
چکیده
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 S3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S4 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 S4.
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