S-matrix for the HOMFLY/Hecke category - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:05:00Z http://mathoverflow.net/feeds/question/100882 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100882/s-matrix-for-the-homfly-hecke-category S-matrix for the HOMFLY/Hecke category Kevin Walker 2012-06-28T17:37:27Z 2012-08-19T23:11:42Z <p>This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See <a href="http://arxiv.org/pdf/q-alg/9702017.pdf" rel="nofollow">here</a> for example.) </p> <p>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.</p> <p>Of fundamental importance in this subject are the invariants $S_{\lambda_+\lambda_-,\mu_+\mu_-}$ of the <a href="http://mathoverflow.net/questions/100881/s-matrix-for-the-bmw-category" rel="nofollow">Hopf link</a> 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.</p> <p>My Question:</p> <blockquote> <p>Has the S-matrix for the HOMFLY-PT category been calculated and published? If not, are partial results in this direction known?</p> </blockquote> <p>I am aware of <a href="http://arxiv.org/pdf/math.GT/0108011.pdf" rel="nofollow">this paper</a> by Morton and Lukac, which does the case where $\lambda_-$ and $\mu_-$ are both empty (i.e. all strands oriented the same direction). <a href="http://arxiv.org/pdf/math.GT/0106207.pdf" rel="nofollow">This paper</a> by Morton and Hadji is also related. Are there other relevant papers that I have missed?</p> <p>See also the BMW version of this question <a href="http://mathoverflow.net/questions/100881/s-matrix-for-the-bmw-category" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/100882/s-matrix-for-the-homfly-hecke-category/105054#105054 Answer by Satoshi Nawata for S-matrix for the HOMFLY/Hecke category Satoshi Nawata 2012-08-19T20:45:39Z 2012-08-19T23:11:42Z <p>The $S$-matrix is given by $$\frac{S_{ij}}{S_{00}}=S_{R_i}(q^{\rho})S_{R_j}(q^{\rho+R_i})$$ 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 <a href="http://arxiv.org/abs/1105.5117" rel="nofollow">paper</a> by Aganagic and Shakirov propoesed the refinement (categorification) of the $S$-matrix $$\frac{S_{ij}}{S_{00}}=M_{R_i}(t^{\rho})M_{R_j}(t^{\rho}q^{R_i})$$ 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 $$KhR_{ij}({\rm Hopf},q,t)\propto M_{R_i}(t^{\rho})M_{R_j}(t^{\rho}q^{R_i})$$ See Eq.(4.10) and appendix B in the <a href="http://arxiv.org/abs/1111.0525v1" rel="nofollow">paper</a>.</p>