J. Aust. Math. Soc.  82 (2007), 209-220

On the square root of the centre of the Hecke algebra of type A

Andrew Francis   School of Computing and Mathematics   University of Western Sydney   NSW 1797   Australia   a.francis@uws.edu.au and Lenny Jones   Department of Mathematics   Shippensburg University   Pennsylvania   USA   lkjone@ship.edu

Abstract

In this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre, but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements.

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