SL2 Seminar

A basic technique in algebra, when describing a difficult-to-approach category like Tilt, is to choose a projective generator and study its endomorphism ring. The modern twist on this technique is to choose the projective generator with great naivety: rather than attempting the difficult task of studying the minimal projective generator, one studies a combinatorial and highly redundant projective generator. For example, indecomposable tilting modules over an algebraic group can be hard to access, but iterated tensor products of fundamental representations (which form a subcategory Fund) are fairly straightforward. Morphisms in Fund can be described by generators and relations using so-called web algebras (in some types). Since every indecomposable tilting module is a direct summand of an object of Fund, one can recover the category of tilting modules by taking the Karoubi envelope of the web category, which formally adds all images of idempotents as new objects.This approach makes the study of tilting modules tractable, by separating the complexity of tilting modules from the process of finding a presentation, and instead putting this complexity in the formal process of taking the Karoubi envelope (i.e. finding idempotents). Thankfully, most of the numerical behavior of tilting modules can be computed within Fund, using basic techniques like the cellular filtration and the local intersection pairing. The purpose of this talk is to introduce web algebras, local intersection pairings, and techniques of idempotent computation. These computations are what helped me most to internalize why tilting modules exhibit fractal, generational behavior. Since most of the techniques are basic algebra, it should be accessible to all, and I expect the discussion to be either yawningly familiar or fairly novel depending on the audience member. With the Riche-Williamson theorem in place, there is a more efficient way to compute multiplicities in the trivial block Tilt$_0$ than computing within Tilt itself: one computes instead with the antispherical module $M$ over the affine Hecke category. Though the setting is different ($M$ rather than Tilt) the techniques are the same. One finds a combinatorial subcategory (iterated wall-bouncing functors applied to the trivial representation), describes it by generators and relations (Soergel diagrammatics), and uses local intersection pairings to compute idempotents and multiplicities. Time permitting, I will begin to describe this setting.