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arXiv · 0709.0326

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

Andrew Francis·Lenny Jones
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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.

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Andrew Francis, Lenny Jones. 2007-11-07. Monomial bases for the centres of the group algebra and Iwahori--Hecke algebra of S_4. https://arxiv.org/abs/0709.0326

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