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Skein theory and the Murphy operators | | Hugh R. Morton
; | | Date: |
13 Feb 2001 | | Journal: | J. Knot Theory Ramif. 11 (2002), 475-492 | | Subject: | Geometric Topology; Quantum Algebra; Rings and Algebras MSC-class: 57M25, 20C08 | math.GT math.QA math.RA | | 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. | | Source: | arXiv, math.GT/0102098 | | Services: | Forum | Review | PDF | Favorites | |
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