J. Aust. Math. Soc.
82 (2007), 209220

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



Lenny Jones
Department of Mathematics
Shippensburg University
Pennsylvania
USA
lkjone@ship.edu



Abstract

In this paper we investigate noncentral
elements of the IwahoriHecke algebra of the
symmetric group whose squares are central. In
particular, we describe a commutative subalgebra
generated by certain noncentral square roots of
central elements, and the generic existence of a
rankthree 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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