A basis for the full Homfly skein of the annulus

Abstract

A basis, denoted {Q λ,µ }, for the full Homfly skein of the annulus C was introduced in [18], where λ and µ are partitions of integers n and p into k and k * parts respectively. The basis consists of eigenvectors of the two meridian maps on C; these maps are the linear endomorphisms of C induced by the insertion of a meridian loop with either orientation around a diagram in the annulus.In this paper we give an explicit expression for each Q λ,µ as the determinant of a (k * + k) × (k * + k) matrix whose entries are simple elements h n , h * n in the skein C. In the case p = 0 (µ = φ) the determinant gives the Jacobi-Trudy formula for the Schur function s λ of N variables as a polynomial in the complete symmetric functions h n of the variables, [12]. The Jacobi-Trudy determinants have previously been used by Kawagoe [6] and Lukac [10] in discussing the elements in the skein of the annulus represented by closed braids in which all strings are oriented in the same direction. Our results and techniques here form a natural extension of the work of Lukac.

Geometrical relations and plethysms in the Homfly skein of the annulus

“…It is shown in [5] that the eigenvectors of ϕ have no repeated eigenvalues, and that there is a basis Q λ,μ of C consisting of these eigenvectors, where λ and μ run through all pairs of partitions. The subspace C m has a basis Q λ,φ = Q λ , where λ runs through partitions of m.…”

“…. , λ l ) (see [5,8]). Since h i = h i [10, Lemma 3.7], the elements Q λ are not affected by the mirror map.…”

“…Remark 21. Since the basis vectors Q λ,μ for C are also eigenvectors of the framing change map [5], we could define τ n/m on the whole of C in a similar way, using the fact that Q λ,μ has eigenvalue τ λ,μ = v k s r with k = |μ| − |λ| and r = 2 x∈λ c(x) − 2 x∈μ c(x).…”

“…We know that Q λ,μ is an eigenvector for any Δ Y , by [5], since Δ Y commutes with ϕ and all the eigenvalues of ϕ have multiplicity 1. Then (τ (a|b) ) n/m = (τ (a|b) ) N/M = v −N s N (a−b) , giving the result.…”

“…where the sum runs over cells x ∈ λ and c(x) = j − i is the content of the cell x in position (i, j), which can be deduced from[5, Theorem 3.4].For λ = (a|b) m we havex∈λ s 2c(x) = s m − s −m s − s −1 s a−b ,…”