Quantum Schur-Weyl duality and q-Frobenius formula related to Reflection Equation algebras

TL;DR

This paper develops a quantum analogue of Schur-Weyl duality linking Hecke algebras and Reflection Equation algebras, introducing new symmetric functions and a Frobenius-type formula within this framework.

Contribution

It establishes a q-version of Schur-Weyl duality involving Reflection Equation algebras and defines new symmetric functions with a Frobenius-like relation.

Findings

Established a q-Schur-Weyl duality with Hecke algebra and Reflection Equation algebra.

Defined analogues of Schur polynomials and power sums in Reflection Equation algebras.

Derived a Frobenius-type formula connecting Schur polynomials and power sums via Hecke algebra characters.

Abstract

We establish a q-version of the Schur-Weyl duality, in which the role of the symmetric group is played by the Hecke algebra and the role of the enveloping algebra U(gl(N)) is played by the Reflection Equation algebra, associated with any skew-invertible Hecke symmetry. Also, in each Reflection Equation algebra we define analogues of the Schur polynomials and power sums in two forms: as polynomials in generators of a given Reflection Equation algebra and in terms of the so-called eigenvalues of the generating matrix $L$, defined by means of the Cayley-Hamilton identity. It is shown that on any Reflection Equation algebra there exists a formula, which brings into correlation the Schur polynomials and power sums by means of the characters of the Hecke algebras in the spirit of the classical Frobenius formula.