University of Sydney Algebra Seminar

Naihuan Jing

Friday 24 April, 12-1pm, in Carslaw 175

Quantum Littlewood correspondences

In the 1940s Littlewood formulated three fundamental correspondences parallel to the Schur-Weyl duality. I will talk about how to introduce the quantum version of quantum immanants and the quantum Littlewood correspondences between quantum immanants and Schur functions for the quantum general linear group and the Heck algebra. Via the correspondences, we have found an exact relationship between the Gelfand-Tsetlin basis of \(U_q(gl(n))\) and Young's orthonormal basis for the Hecke algebra. This leads to a trace formula for the quantum immanants that has settled the generalization problem of \(q\)-analog of Kostant's formula for \(\lambda\)-immanants. As applications, we also derive general \(q\)-Littlewood-Merris-Watkins identities and \(q\)-Goulden-Jackson identities as special cases of the quantum Littlewood correspondence III. This is joint work with Jian Zhang.