Canonical bases of a coideal subalgebra in $U_q(\mathfrak{sl}_2)$

TL;DR

This paper studies the structure of dual canonical bases in tensor products of representations of a coideal subalgebra in quantum sl2, providing explicit formulas, positivity properties, and eigenanalysis with combinatorial conjectures.

Contribution

It offers explicit diagrammatic expressions for dual canonical bases, proves their positivity, and analyzes the eigenstructure of the coideal subalgebra's generator.

Findings

Explicit dual canonical bases formulas derived

Proved positivity and integrality of tensor product decompositions

Eigenvalues and eigenfunctions explicitly characterized

Abstract

We consider tensor products of finite-dimensional representations of a coideal subalgebra in $U_{q}(\mathfrak{sl}_2)$. We present an explicit expression for the dual of the canonical bases through a diagrammatic presentation. We show that the decomposition of tensor products of dual canonical bases and the action of the coideal subalgebra have integral and positive properties. As an application, we consider the eigensystem of the generator of the coideal subalgebra on the dual canonical bases. We provide all the eigenvalues and obtain an explicit expression of the eigenfunction for the largest eigenvalue. The sum of the components of this eigenfunction is conjectured to be equal to the total number of arrangements of bishops with a certain symmetry.