SKEIN THEORY AND THE MURPHY OPERATORS

Abstract

The Murphy operators in the Hecke algebra Hn of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of Hn. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, [Formula: see text]. Symmetric functions of the Murphy operators are also central in Hn. I define geometrically a homomorphism from [Formula: see text] to the centre of each algebra Hn, and find an element in [Formula: see text], independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of [Formula: see text] in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.