The Hodge Theory of Soergel Bimodules

On the computation of torus link homology

“…One reason why is more natural that is that any Rouquier complex always has a unique copy of which appears in homological degree and internal degree , so that this copy of appears with shift , where is the braid exponent. It was proven in [EW14] that Rouquier complexes for reduced expressions are perverse (when one works in characteristic zero). A complex is perverse if each indecomposable bimodule in the complex appears with a grading shift equal to its homological degree, or equivalently, that the grading and homological shifts are described only as powers of .…”

Finitary birepresentations of finitary bicategories

“…The one-object bicategory S can be defined over the polynomial algebra, as in e.g. [EW3], or over the coinvariant algebra, as in e.g. [So1].…”

“…Based on results in [EW3], Lusztig [Lu,§18.5] associated with each two-sided cell J of W a semisimple one-object bicategory A J , called the asymptotic limit or the asymptotic bicategory, which categorifies the direct summand of the asymptotic Hecke algebra corresponding to J (or, in Lusztig's terminology, the J-ring associated with J ). By [EW1, Section 5], the monoidal category S is pivotal for any W , and so is A J for any J of W .…”

“…(ii) Recall that for any finite Coxeter group, the one-object bicategory of Soergel bimodules S is finitary when it is defined over the coinvariant algebra. The (left, right, or two-sided, respectively) cells and cell orders in S correspond to the Kazhdan-Lusztig [KL] (left, right, or two-sided, respectively) cells and orders of W by the Soergel-Elias-Williamson categorification theorem [EW3,Theorem 1.1]. This remains true when S is defined over the polynomial algebra and/or the Coxeter group is non-finite.…”

Analogues of Centralizer Subalgebras for Fiat 2-Categories and Their 2-Representations

“…The unique object in S is denoted i. For w ∈ W , we denote by θ w the unique (up to isomorphism) indecomposable Soergel bimodule which corresponds to w. Then the split Grothendieck ring of S is isomorphic to Z[W ] and this isomorphism sends the class of θ w to the corresponding element H w of the Kazhdan-Lusztig basis in Z[W ], see [8,19]. Consequently, the left, right and two-sided cells in S are given by the corresponding left, right and two-sided Kazhdan-Lusztig cells in W .…”