The q-Immanants and Higher Quantum Capelli Identities

Abstract

We construct polynomials \(\mathbb {S}_{\mu }(z)\) parameterized by Young diagrams \(\mu \), whose coefficients are central elements of the quantized enveloping algebra \(\textrm{U}_q(\mathfrak {gl}_n)\). Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of z, we get q-analogues of Okounkov’s quantum immanants for \(\mathfrak {gl}_n\). We show that the Harish-Chandra image of \(\mathbb {S}_{\mu }(z)\) is a factorial Schur polynomial. We derive quantum analogues of the higher Capelli identities by calculating the images of the q-immanants in the braided Weyl algebra. We also give a symmetric function interpretation and new proof of the Newton identities of Gurevich, Pyatov and Saponov.