Quantum Littlewood correspondences

In the 1940s Littlewood formulated three fundamental correspondences for the immanants and Schur symmetric functions on the general linear group, which establish deep connections between representation theory of the symmetric group and the general linear group parallel to the Schur-Weyl duality. In this paper, we introduce the notion of quantum immanants in the quantum coordinate algebra using primitive idempotents of the Hecke algebra. By employing $R$-matrix techniques, we establish the quantum analog of Littlewood correspondences between quantum immanants and Schur functions for the quantum coordinate algebra. In the setting of the Schur-Weyl-Jimbo duality, we construct an exact correspondence between the Gelfand-Tsetlin bases of the irreducible representations of the quantum enveloping algebra $U_q(\mathfrak{gl}(n))$ and Young's orthonormal basis of an irreducible representation of the Hecke algebra $\mathcal H_m$. This isomorphism leads to our trace formula for the quantum immanants, which settled the generalization problem of $q$-analog of Kostant's formular 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.