Representations of Spectrum of GL(m) type Quantum Matrices

TL;DR

This paper computes characters of spectral values in reflection equation algebras related to quantum matrices, providing explicit formulas and connecting to classical limits like the enveloping algebra of gl(N).

Contribution

It introduces a method to compute characters of spectral values in reflection equation algebras associated with Hecke symmetries, extending understanding of quantum matrix representations.

Findings

Characters of spectral values are explicitly computed.

Power sums of spectral values are calculated in finite-dimensional representations.

Results recover classical limits, such as the enveloping algebra of gl(N).

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].