Web bases for sl(3) are not dual canonical

Abstract

We compare two natural bases for the invariant space of a tensor product of irreducible representations of A2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A2 spider. The other basis is the dual canonical basis, the dual of the basis defined by Lusztig and Kashiwara. For sl(2) or A1, the web bases have been discovered many times and were recently shown to be dual canonical by Frenkel and Khovanov. We prove that for sl(3), the two bases eventually diverge even though they agree in many small cases. The first disagreement comes in the invariant space Inv((V+ ⊗ V+ ⊗ V- ⊗ V-)⊗3), where V+ and V- are the two 3-dimensional representations of sl(3); if the tensor factors are listed in the indicated order, only 511 of the 512 invariant basis vectors coincide.