The quantum Casimir operators of $\Uq$ and their eigenvalues

TL;DR

This paper demonstrates that certain quantum Casimir operators and an additional central element generate the entire center of the quantum linear group U_q, providing new formulas for their eigenvalues linked to classical representation characters.

Contribution

It establishes the generating set for the center of U_q and derives novel eigenvalue formulas connected to classical general linear algebra representations.

Findings

Quantum Casimir operators and a central element generate the entire center of U_q.

New eigenvalue formulas expressed via characters of finite-dimensional irreducible representations.

The results connect quantum group invariants with classical representation theory.

Abstract

We show that the quantum Casimir operators of the quantum linear group constructed in early work of Bracken, Gould and Zhang together with one extra central element generate the entire center of $\Uq$. As a by product of the proof, we obtain intriguing new formulae for eigenvalues of these quantum Casimir operators, which are expressed in terms of the characters of a class of finite dimensional irreducible representations of the classical general linear algebra.