Question

This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See here for example.)

The minimal idempotents of this category are indexed by pairs $(\lambda_+, \lambda_-)$ of Young diagrams. (The sizes of the diagrams are arbitrary and need not be the same. The diagram $\lambda_+$ corresponds to upward oriented strands, while $\lambda_-$ corresponds to downward oriented strands.) Consequently one can define numerical invariants of oriented links whose components are labeled by pairs of Young diagrams. This is the "colored" HOMFLY-PT polynomial.

Of fundamental importance in this subject are the invariants $S_{\lambda_+\lambda_-,\mu_+\mu_-}$ of the Hopf link with its components labeled by pairs of Young diagrams (i.e. idempotents) $(\lambda_+, \lambda_-)$ and $(\mu_+, \mu_-)$. In TQFT language, this is the "S-matrix" of the theory.

My Question:

Has the S-matrix for the HOMFLY-PT category been calculated and published? If not, are partial results in this direction known?

I am aware of this paper by Morton and Lukac, which does the case where $\lambda_-$ and $\mu_-$ are both empty (i.e. all strands oriented the same direction). This paper by Morton and Hadji is also related. Are there other relevant papers that I have missed?

See also the BMW version of this question here.

Answer 1

The $S$-matrix is given by \begin{equation} \frac{S_{ij}}{S_{00}}=S_{R_i}(q^{\rho})S_{R_j}(q^{\rho+R_i}) \end{equation} where $S_{R}(x_1,\cdots,x_N)$ is the Schur polynomial with highest weight $R$, $S_{R}(q^{\rho})=S_{R}(q^{\rho_{1}},...,q^{\rho_{N}})$ and $\rho$ is the Weyl vector. Furthermore, the paper by Aganagic and Shakirov propoesed the refinement (categorification) of the $S$-matrix \begin{equation} \frac{S_{ij}}{S_{00}}=M_{R_i}(t^{\rho})M_{R_j}(t^{\rho}q^{R_i}) \end{equation} where $M_{R}(x_1,\cdots,x_N;q,t)$ is the Macdonald polynomial with highest weight $R$ and $M_{R}(t^{\rho}q^{R})=M_{R}(t^{\rho_{1}}q^{R_{1}},...,t^{\rho_{N}}q^{R_{n}};q,t)$. It reduces to the above equation for $q=t$. By using the refined topological vertex, Iqbal and Kozcaz showed that the Khovanov-Rozansky polynomial of the Hopf link is actually proportional to the refined $S$-matrix \begin{equation} KhR_{ij}({\rm Hopf},q,t)\propto M_{R_i}(t^{\rho})M_{R_j}(t^{\rho}q^{R_i}) \end{equation} See Eq.(4.10) and appendix B in the paper.