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A. K. Aiston

Publications and source records attributed to A. K. Aiston.

3 recordsLinked to original sources
arXiv2019

Idempotents of Hecke algebras of type A

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group SU(N,q).

q-alg↗
arXiv1997

Adams operators and knot decorations

We use an explicit isomorphism from the representation ring of the quantum group U_q(sl(N)) to the Homfly skein of the annulus, to determine an element of the skein which is the image of the mth Adams operator, ψ_m, on the fundamental representation, c_1. This element is a linear combination of m very simple m-string braids. Using this skein element, we show that the Vassiliev invariant of degree n in the power series expansion of the U_q(sl(N)) quantum invariant of a knot coloured by ψ_m(c_1) is the canonical Vassiliev invariant with weight system W_nψ_m^{(n)} where W_n is the weight system for the Vassiliev invariant of degree n in the expansion of the quantum invariant of the knot coloured by c_1 and ψ_m^{(n)} is the Adams operator on n-chord diagrams defined by Bar-Natan.

q-alg↗