Elijah Bodish (Oregon)
Abstract
In 1932 Rumor-Teller-Weyl observed that endomorphism algebras of tensor products of the vector representation for sl_2 can be described by (linear combinations of) crossingless matching diagrams. This is now well-known under the moniker “Temperley-Lieb algebra”. The Jones-Wenzl projectors are linear combinations of crossingless matching diagrams which describe the idempotent projecting to the symmetric powers. These projectors satisfy recursive formulas which can aid in their computation, which have proved useful in representation theory, knot theory, the study of subfactors, and categorification.
In his 1996 paper “Rank 2 spiders for Lie algebras", Kuperberg defined an analogue of Tempeley-Lieb algebras for each rank 2 simple Lie algebra (i.e. sl_3, sp_4, and g_2). He also proved that the analogues of the Jones-Wenzl projectors exist for rank 2 as well. Then D. Kim and later B. Elias found recursive descriptions of these projectors in the case of sl_3, and Elias gave a conjecture about how these recursions may look for sl_n. The most interesting aspect of this conjecture is that the coefficients in the projectors, which are by definition solutions to some complicated recursive formula, are actually described compactly by formulas analogous to the Weyl dimension formula.
In my talk I will review the above background material and then discuss my work on finding recursive descriptions of the projectors in the case of sp_4 and g_2 (sp_4 appears in arxiv:2102.05186 and g_2 is joint work in progress with Haihan Wu from UC Davis). I will also discuss how the rank 2 projectors fit into the framework of Elias’s conjecture in type A, and suggest how the whole story may generalize to other Lie algebras.