  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.
