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Webs, Ladders, and Clasps

Algebra & Discrete Mathematics

Speaker: Ben Elias, University of Oregon
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Location: 1147 MSB
Start time: Mon, Oct 24 2016, 4:10PM

Let g be a complex semisimple lie algebra and Rep g denote its category of finite dimensional representations. From a naive perspective this category is easy to work with: it is semisimple. However, Rep g is also a monoidal category, and computing the monoidal product on its morphism spaces can be quite interesting. The approach pioneered by Greg Kuperberg in rank 2 is to consider the combinatorial subcategory Fund g, consisting of tensor products of fundamental representations, and to describe it by generators and relations using diagrams called "webs." In type A, the correct diagrammatic calculus (of so-called "sl_n webs") was produced by Cautis-Kamnitzer-Morrison in 2014.

We give a new basis for sl_n webs, called the "double ladders basis." It is motivated by a philosophy we plan to advertise, which combines the monoidal and semisimple structures to produce a nice cellular basis. This basis gives a new proof that sl_n webs are the correct algebra, a proof which also works over integral forms.

Clasps are the morphisms between tensor products of fundamental representations which project to their highest weight summand. The same philosophy leading to the double ladders basis also provides a recursive construction of clasps involving some unknown coefficients, and a recursive (though difficult) algorithm to compute these coefficients. We provide a mysterious conjecture as to the coefficients in these clasps, which we have proven for sl_n, n <= 4.