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.
