Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2307.06677

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

Dimitry Gurevich·Pavel Saponov
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.QA

Keep this discovery

BibTeXRIS

Dimitry Gurevich, Pavel Saponov. 2023-07-13. Quantum Schur-Weyl duality and q-Frobenius formula related to Reflection Equation algebras. https://arxiv.org/abs/2307.06677

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Diagrammatic Categorification of Verma Modules I: Braiding

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

math.QA↗
arXiv2026

Some finite dimensional representations of shifted quantum affine algebras of type A

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

math.QA↗