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

TL;DR

This paper explores non-central elements in the Hecke algebra of type A whose squares are central, identifying a commutative subalgebra generated by such elements and revealing new algebraic structures.

Contribution

It introduces a new commutative subalgebra generated by non-central square roots of central elements in the Hecke algebra of type A, with explicit expressions for their squares.

Findings

Existence of a rank-three submodule in the Hecke algebra

Construction of a commutative subalgebra from non-central square roots

Explicit formulas for squares of generators in terms of Murphy elements

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.