Representations of Spectrum of GL(m) type Quantum Matrices
Dimitry Gurevich, Pavel Saponov, Mikhail Zaitsev
TL;DR
The paper develops a universal framework to study spectrum-valued characters in reflection equation algebras ${\cal L}(R)$ associated with even Hecke symmetries. It builds spectral parameterizations through a quantum Cayley–Hamilton identity, derives quantum Newton relations, and computes explicit characters for spectral values in GL($m$)-type representations labeled by partitions. The main contributions are closed-form expressions for $\chi_{\lambda}(\mu_i)$ and $\chi^*_{\lambda}(\mu_i)$, plus explicit formulas for $p_n(\mu)$ and $t_n(\mu)$ in terms of $\mu_i$, which recover classical results in the $q\to1$ limit and align with known quantum-group cases (JLM, PP). The work provides a universal method applicable to arbitrary Hecke symmetries and clarifies the link between the RE algebra, its center, and symmetric functions of quantum eigenvalues.
Abstract
In the present paper we are dealing with reflection equation algebras ${\cal L}(R)$ corresponding to even skew-invertible Hecke symmetries. Our main result consists in computing the characters of the spectral values of the generating matrix $L$ of ${\cal L}(R)$ in finite-dimensional representations labeled by partitions of integers. As is known, the spectral values belong to an algebraic extension of the center of the reflection equation algebra and elements of the center can be presented as symmetric functions in spectral values. As an application of our approach, we calculate the characters of the power sums $\mathrm{Tr}_R(L^n)$ in the mentioned finite dimensional representations. In a particular case of the Drinfeld-Jimbo $R$-matrix the enveloping algebra $U(gl(N))$ can be obtained as a specific limit of the reflection equation algebra. In this limit our results for power sums coincide with the those obtained in [PP].