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

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.

arXiv comment34 pages

References in corpus (7)

• Quantum symmetric Kac-Moody pairs
• A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs
• Quantum symmetric spaces and related q-orthogonal polynomials
• A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
• A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products
• Kazhdan-Lusztig polynomials for the Hermitian symmetric pair $(B_N,A_{N-1})$
• Quantum Symmetric Pairs and Their Zonal Spherical Functions

Cited by in corpus (2)

• Positivity vs negativity of canonical bases
• A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian