The Dual Canonical Basis in the Spin Representation via the Temperley-Lieb Algebra

TL;DR

This paper introduces a simplified, diagrammatic construction of the dual canonical basis in the spin representation, extending Khovanov's work and providing explicit formulas and alternative axioms.

Contribution

It generalizes Khovanov's diagrammatic approach to the entire dual canonical basis and offers explicit formulas and axiomatic definitions.

Findings

Provided explicit formulas for the dual canonical basis.

Reproved Khovanov's results using a new perspective.

Demonstrated the duality between canonical and dual canonical bases.

Abstract

The spin representation $(\mathbb C^2)^{\otimes n}$ has a dual canonical basis introduced by Lusztig that is important in many areas of algebra, geometry, and physics. Khovanov observed that a portion of the dual canonical basis can be viewed diagrammatically through the Temperley-Lieb algebra. We provide a simpler construction that we generalize to the entire dual canonical basis, and write explicit formulas to compute the dual canonical basis, and thus the canonical basis of the spherical module, as a byproduct. We reprove some of Khovanov's results using our new perspective. Furthermore, we use the Hecke algebra to reprove the fact that the canonical basis is indeed dual to the dual canonical basis, leading to similar results about the canonical basis in $\mathcal M^*$ and $\mathcal N^*$, the dual spaces to the spherical and aspherical modules, as a byproduct. Finally, we present an alternative axiomatic definition of the canonical basis using diagrams.