Let \({\mathcal C}\) be the framed Homfly skein of the annulus. The subalgebra \({\mathcal C}^+ \subset {\mathcal C}\) in which all the strings are oriented in the same direction is identified with the algebra of symmetric functions in infinitely many variables. Under this identification, the complete symmetric function of degree \(n\) corresponds to the closure \(h_n\) of one of the two basic idempotent elements of the Hecke algebra \(H_n\), and the Schur function \(s_{\lambda}\), \(\lambda\) a partition of \(n\), corresponds to the eigenvector \(Q_{\lambda }\) of the meridian map \(\varphi\) of \({\mathcal C}\). In other words, \(Q_{\lambda }\) can be expressed as a polynomial in the \(\{ h_n \}\) via the Jacobi-Trudy formula [see S. G. Lukac, Math. Proc. Camb. Philos. Soc. 138, No. 1, 79–96 (2005; Zbl 1083.20002)].
The present paper is a natural extension of this work: a basis \(\{ Q_{\lambda , \mu }\}\) for the full (both directions) framed Homfly skein of the annulus (not only for \({\mathcal C}^+\)) is provided, where the elements \(Q_{\lambda , \mu }\) are eigenvectors of the meridian map. Moreover, these elements can be expressed as polynomials in the elements \(h_n, h_n^*\). Here \(^*\) denotes the involution of \({\mathcal C}\) which rotates a diagram in the annulus by \(\pi \) about the horizontal axis. In spite of the generalization, the suitable chosen notation makes the calculations clearer and more concise.