The Hodge theory of Soergel bimodules

By Ben Elias, Geordie Williamson. 2012

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We prove Soergel’s conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coecients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/db941257b12da27478630ce7f211df611b5f711f

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Citations in the corpus

The two-color Soergel calculus (2013, background): “In a realization over R, however, 6 BENELIAS the positivity properties of quantum numbers depend strongly on the choice of primitive root of unity. These positivity properties will play a key role in [12], and are discussed further in that paper. 1.4. Connections with Temperley-Lieb theory. The Temp”
Schubert calculus and torsion explosion (2013, background): “f Ix,w,0⊗ZQ and of Ix,w,0 ⊗Z kagree. Proof. Soergel’stheorem [Soe01, Lemma 5] implies that the indecomposableSoergel bimodules categorify the Kazhdan-Lusztig basis if k is of characteristic zero (see [EW14] for an algebraic proof of this fact). Now Lemma 4.1 says that (2) holds if and only if Bw dec”
The $p$-Canonical Basis for Hecke Algebras (2015, methodology): “what basis of the Hecke algebra the classes of the self-dual indecomposable objects in H correspond to. The answer is given for k=R by Soergel’s conjecture which Elias and the second author proved in [EW14]. Theorem 3.1 (Elias-W. 2013). ch([B w])=H w for all w∈W. This illustrates that the basis of s”
COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS (2017, background): “1 For example, A(X) may be the cohomology of real (q; q)-forms on a compact Kähler manifold (Gromov [1990]), the ring of algebraic cycles modulo homological equivalence on a smooth projective variety (Grothendieck [1969]), McMullen’s algebra generated by the Minkowski summands of a simple convex pol”
On the computation of torus link homology (2016, background): “) always has a unique copy of Rwhich appears in homological degree e() and internal degree e(), so that this copy of Rappears with shift t12e(), where e() is the braid exponent. It was proven in [EW14] that Rouquier complexes for reduced expressions are perverse (when one works in characteristic zer”
Tilting modules and the p-canonical basis (2015, result): “here ℓis the order of the root of unity under consideration, assuming that ℓą n; the existence of such an action is suggested in [CR, Remark 7.27].) Combining these approaches with the main result of [EW1], in this way one can obtain direct algebraic proofs of the Kazhdan–Lusztig conjecture and of L”