A graphical categorification of the two-variable Chebyshev polynomials of the second kind

Abstract

We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q({\mathfrak{sl}}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.

18 pages, TikZ pictures

References in corpus (2)

• Orthogonal polynomials of compact simple Lie groups: Branching rules for polynomials
• q-Chebyshev polynomials