The $q$-immanants and higher quantum Capelli identities

TL;DR

This paper introduces new polynomials linked to Young diagrams within quantum groups, establishing their properties, connections to quantum immanants, and proving quantum analogues of classical identities.

Contribution

It constructs parameterized polynomials with central elements in quantum groups, linking them to quantum immanants and proving higher quantum Capelli identities.

Findings

Constant terms match central elements of quantum groups.

At specific parameters, recover q-analogues of quantum immanants.

Prove quantum versions of classical 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 ${\rm U}_q({\mathfrak{gl}}_n)$. 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 also prove quantum analogues of the higher Capelli identities and derive Newton-type identities.