The $q$-immanants and higher quantum Capelli identities

TL;DR

The paper constructs $q$-immanants $S_mu(z)$ in the center of $U_q(gl_n)$, parameterized by Young diagrams, and shows their constant terms match Drinfeld–Reshetikhin central elements while their Harish-Chandra images are factorial Schur polynomials. For a special choice of $z$, these provide $q$-analogues of Okounkov's quantum immanants, and their eigenvalues are computed on irreducible modules, linking to factorial Schur polynomials; the $q$-immanants generate the center for fixed $z$ and yield a quantum analogue of higher Capelli identities via a braided Weyl-algebra realization. The work also derives Newton-type identities by leveraging the quantum Liouville formula in the quantum loop algebra, connecting traces of powers of the loop generator to central symmetric functions, with the classical limit recovering known Capelli and Newton results. Overall, the paper extends classical invariant theory to a robust quantum setting, providing central generators, eigenstructure, and Capelli-type identities in the quantized gl_n framework.

Abstract

We construct polynomials ${\mathbb{S}}_μ(z)$ parameterized by Young diagrams $μ$, 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 ${\mathbb{S}}_μ(z)$ is a factorial Schur polynomial. We also prove quantum analogues of the higher Capelli identities and derive Newton-type identities.

Theorems & Definitions (16)

• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5
• Corollary 3.6
• proof