Cellular Algebra of $A_2$ webs

Abstract

We introduce a diagram algebra ${\mathcal{K}}_L$ based on Kuperberg's $A_2$-webs and prove a Schur--Weyl duality ${\mathcal{K}}_L\cong {\mathrm{End}}_{U_q({\mathfrak{sl}}_3)}((V^{+}\oplus V^{-}{)}^{\otimes L})$, where $V^{\pm }$ are the fundamental representations of $U_q({\mathfrak{sl}}_3)$. Thus, ${\mathcal{K}}_L$ is a natural $A_2$ analogue of the Temperley--Lieb algebra. We construct an explicit cellular structure for ${\mathcal{K}}_L$, with the web basis as its cellular basis and cell modules indexed by dominant $U_q({\mathfrak{sl}}_3)$-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of ${\mathcal{K}}_L$ and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate ${\mathcal{K}}_L$, and identify an additional local generator. Finally, we derive a bimodule decomposition of $(V^{+}\oplus V^{-}{)}^{\otimes L}$, yielding a complete set of irreducible ${\mathcal{K}}_L$-modules for formal or generic $q$.

61 pages