ar X iv : q - a lg / 9 71 20 46 v 2 2 8 Se p 19 98 Web bases for sl ( 3 ) are not dual canonical

or its invariant space Inv(V1 ⊗ V2 ⊗ . . .⊗ Vn). The quantum group Uq(g) has representations and vector spaces of invariants which generalize these, and one can also study their bases, with or without the intention of specializing to q = 1. (For simplicity, we will usually consider Uq(g) as an algebra over C(q ), and we will only occassionally mention Z[q] as a ground ring.) Lusztig’s remarkable canonical bases [6], which are the same as Kashiwara’s crystal bases [2], extend to bases of these spaces and have many important properties. When g = sl(2), the Temperley-Lieb category [1, 3] gives another set of bases for the invariant spaces. It was recently established that these bases are dual canonical, i.e., dual in the sense of linear algebra to canonical bases [1]. The Temperley-Lieb category gives a particularly explicit, simple, and useful definition of the dual canonical bases of invariants (dual canonical invariants) which establishes further natural properties of these bases. Reference 5 defines generalizations of the Temperley-Lieb category to the three rank two Lie algebras A2 ∼= sl(3), B2 ∼= sp(4) ∼= so(5), and G2. These generalizations are called combinatorial rank two spiders. The bases they yield are called web bases and their individual basis vectors are called webs. One may conjecture that these bases are also dual canonical. As evidence for the conjecture, consider the following properties which the A2 web bases share with dual canonical invariants: