A basis for the full Homfly skein of the annulus

A basis, denoted $\{Q_{\lambda,\mu\}$, for the full Homfly skein of the annulus $\mathcal{C}$ has been introduced by the authors, where $\lambda$ and $\mu$ are partitions of integers $n$ and $p$ into $k$ and $k^*$ parts respectively. The basis consists of eigenvectors of the two meridian maps on $\mathcal{C}$; these maps are the linear endomorphisms of $\mathcal{C}$ induced by the insertion of a meridian loop with either orientation around a diagram in the annulus. Here we present an explicit expression for each $Q_{\lambda,\mu}$ as the determinant of a $(k^*+k)\times(k^*+k)$ matrix whose entries are simple elements $h_n, h_n^*$ in the skein $\mathcal{C}$. In the case $p=0$ ($\mu=\phi$) the determinant gives the Jacobi–Trudy formula for the Schur function $s_{\lambda}$ of $N$ variables as a polynomial in the complete symmetric functions $h_n$ of the variables. The Jacobi–Trudy determinants have previously been used by Kawagoe and Lukac 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 form a natural extension of the work of Lukac.