S-matrix for the HOMFLY/Hecke category

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.

Answer 2

This isn't a complete answer, but it might make some partial progress. First I'll slightly restate the question. If we decompose $S^3$ into two solid tori and write $C$ and $C^{op}$ for the Homfly skein modules of these tori, then there is a pairing $$\langle -,-\rangle: C\otimes_A C^{op} \to k$$ where $k$ is the base ring (which is isomorphic to the skein module of $S^3$) and $A$ is the Homfly skein algebra of the torus. As you say, both $C$ and $C^{op}$ have bases indexed by pairs of partitions, and if I understand right, the entries of the $S$ matrix are evaluations of this pairing.

One brute-force approach for computing $\langle (\lambda, \mu), (\lambda',\mu')\rangle$ is to "move the left hand side to the right using the action of $A$ and then evaluate." An explicit presentation of $A$ is given here http://arxiv.org/abs/1410.0859. There are generators $P(a,b)$ for $a,b \in \mathbb(Z)$ (which are "the $gcd(a,b)$ power sum on slope $b/a$"), and they satisfy the commutation relations $$[P(x), P(y)] = (q^d-q^{-d}) P(x+y),\quad x,y \in \mathbb Z^2,\quad d = det[x\,\,y]$$ (If $x$ and $y$ are primitive and $d=1$ this is just the skein relation.) The paper also gives explicit formulas for the action of the generators $P(a,b)$ in the basis $(\lambda,\mu)$ that you mentioned, and these formulas actually aren't too bad (e.g. if $a,b \not= 0$ then the entries in the matrix are monomials).

One difficulty in completing the answer is that expressing the basis elements $(\lambda,\mu)$ in terms of power sums isn't too easy. (Morton does have a determinental formulas in terms of complete symmetric functions in one of his papers.) Also, I'm not sure if the evaluations $((\emptyset, \emptyset),(\lambda,\mu)\rangle$ have actually been computed.

(By the way, this algebra $A$ actually has a lot of different realization which are not obviously the same. One other realization is as (a specialization of) the Hall algebra of coherent sheaves on an elliptic curve over a finite field (this is described in the paper above). Also, if you've found these numbers since the question was asked I'd be curious to hear.)