The two-color Soergel calculus

Abstract

Abstract. We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W . The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is infinite, the parameter q of the Temperley-Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for sl2. When W is finite, q must be specialized to an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soergel bimodule for the longest element of W .

Classification problems in 2-representation theory

“…5.7, suggested in these cases possibility of existence of simple transitive 2-representations of S W which are neither cell 2-representations nor the ones constructed in Theorem 19(i). These 2-representations were constructed in [39], based on [15], using diagrammatic calculus. Under the additional assumption of gradeability, it was shown that, together with cell 2-representations and the 2-representations constructed in Theorem 19(i), these exhaust all simple transitive 2-representations of S W .…”

“…This direction of study was subsequently called higher representation theory or, alternatively, just 2-representation theory to emphasize that, so far, it directs only at this second level of general higher categories. However, one has to note that these and many further papers like [15,29] and other mainly study special examples of 2-categories which originate in topologically motivated diagrammatic calculus.…”

Soergel calculus

“…This operation extends to an algebra homomorphism from 2T L to the degree zero part of an endomorphism ring inside D. For example, the fact that a red circle in a blue region evaluates to −x can be shown using (5.2), (5.1), and (5.5) until the scalar ∂ t (α s ) = a st appears. What happens to Jones-Wenzl projectors can be seen in [Elid,Example 5.16].…”

Schubert calculus and torsion explosion

“…Recently Elias and the author found a presentation for the monoidal category of Soergel bimodules by generators and relations [EW], building on the work of Libedinsky [Lib10], Elias-Khovanov [EK10] and Elias [Eli16]. One of the applications of this theory is that one can decide whether a given intersection cohomology complex has p-torsion in its stalks or costalks (the bridge between intersection cohomology and Soergel bimodules is provided by the theory of parity sheaves).…”