Skein theory and the Murphy operators

Morton, HR ORCID: 0000-0002-8524-2695 (2002) Skein theory and the Murphy operators. JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, 11 (4). pp. 475-492.

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Abstract

The Murphy operators in the Hecke algebra H_n 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 H_n. 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, C. Symmetric functions of the Murphy operators are also central in H_n. I define geometrically a homomorphism from C to the centre of each algebra H_n, and find an element in C, independent of n, whose image is the m-th power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of C in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

Item Type:	Article
Additional Information:	22 pages, 34 interspersed figures. Submitted for the proceedings of Knot 2000 conference, Korea
Uncontrolled Keywords:	skein theory, Murphy operators, power sums, symmetric functions, annulus, Hecke algebras
Depositing User:	Symplectic Admin
Date Deposited:	22 Apr 2016 14:10
Last Modified:	07 Feb 2023 05:02
DOI:	10.1142/S0218216502001767
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3000533