The centralizer algebras of mixed tensor representations of \({\mathcal U}_ q (gl_ n)\) and the HOMFLY polynomial of links (Q1201501)

We construct an algebra \(H_{N-1,M-1}(a,q)\) with complex parameters \(a\) and \(q\). The centralizer algebra of a mixed tensor representation of \({\mathcal U}_ q(gl_ n)\) is a quotient of it. The HOMFLY polynomial of links in \(S^ 3\) is equal to a trace of \(H_{N-1,M-1}(a,q)\). Each irreducible character of it corresponds to an invariant of links in a solid torus. As an application, we get a formula for the HOMFLY polynomial of satellite links. The details will be published elsewhere.