Soergel bimodule
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| Soergel bimodule | |
|---|---|
| Name | Soergel bimodule |
| Introduced by | Wolfgang Soergel |
| Year | 1990s |
| Field | Representation theory, category theory, algebraic geometry |
Soergel bimodule
Soergel bimodule is an object in modern algebraic representation theory linking algebraic, geometric, and categorical structures. Originating in work by Wolfgang Soergel, these bimodules provide a bridge between Coxeter groups, Hecke algebras, and geometry associated to flag varieties and category O. They play a central role in categorification programs involving Kazhdan–Lusztig theory and link homology.
Definition and basic properties
A Soergel bimodule is a graded bimodule over a polynomial ring attached to a reflection representation of a Coxeter group such as Weyl group, Coxeter group, Symmetric group, Affine Weyl group, or Dihedral group that satisfies specific homological and combinatorial conditions introduced by Wolfgang Soergel and studied in contexts including Bernstein–Gelfand–Gelfand category O, Kazhdan–Lusztig polynomial, Perverse sheaf, Intersection cohomology, and Equivariant cohomology. Its basic properties include gradings compatible with the action of the Hecke algebra, a behavior under tensor product analogous to convolution in Derived category, and a decomposition pattern governed by indecomposable summands that mirror elements of the Bruhat order, Weyl group element, Longest element (Coxeter group), and Parabolic subgroup. Soergel bimodules satisfy homological dualities connected to Verdier duality and categorical phenomena observed in Koszul duality, Ringel duality, and Serre duality.
Construction and examples
Constructions begin with a reflection faithful representation of a Coxeter system as in examples from Type A, Type B, Type D, Type E6, Type E7, Type E8, Type F4, Type G2, and Affine type A; one attaches the coordinate ring (a polynomial algebra) with grading studied in relation to Schubert variety, Flag variety, Grassmannian, Partial flag variety, and Nilpotent cone. Basic examples include Bott–Samelson bimodules analogous to Bott–Samelson resolutions used by Bott, Samelson, and in geometric constructions by Demazure, Borel, and Grothendieck. Indecomposable Soergel bimodules correspond in examples to intersection cohomology sheaves on Schubert varieties appearing in the work of Beilinson, Bernstein, Deligne, Ginzburg, and Finkelberg, and to projective objects in the BGG category O studied by Bernstein, Gelfand, and Gelfand.
Categorification and connection to Hecke algebras
Soergel bimodules categorify the Hecke algebra of a Coxeter system via a split Grothendieck group isomorphism sending bimodule classes to Kazhdan–Lusztig basis elements studied by Kazhdan and Lusztig. This categorification provides a geometric underpinning for the Kazhdan–Lusztig conjectures settled by methods of Beilinson–Bernstein localization and Brylinski–Kashiwara techniques and relates to Rouquier complex constructions of Raphaël Rouquier and link invariants discovered by Khovanov and Rozansky. The relationship also connects to categorifications in quantum topology such as Khovanov homology, HOMFLY-PT polynomial, and constructions by Cautis and Kamnitzer that tie to the representation theory of Quantum group like Drinfeld and Jimbo's work.
Structure theory and classification
Structure theory classifies indecomposable Soergel bimodules via combinatorics of reduced expressions in Coxeter systems and the geometry of Schubert varieties associated to Bruhat decomposition, Kazhdan–Lusztig basis, and Bruhat order. Key classification results include Soergel’s categorification theorem, Elias–Williamson’s proof of the Soergel conjecture, and connections with parity sheaves studied by Juteau, Mautner, and Williamson; the latter involve counterexamples to expectations in modular settings similar to phenomena found by James Humphreys and Geordie Williamson. The module category exhibits graded multiplicities and decomposition numbers analogous to those in Modular representation theory of Algebraic groups and reflects phenomena in Lusztig’s conjectures and Tilting module theory studied by Donkin and Andersen.
Diagrammatics and graphical calculus
Diagrammatic approaches to Soergel bimodules were developed by Ben Elias and Geordie Williamson who provided a presentation by generators and relations via planar diagrams influenced by graphical calculi in Temperley–Lieb algebra, Brauer algebra, Khovanov–Lauda–Rouquier algebras, and Webster’s diagrammatics. These diagrammatic categories enable computations parallel to those in Categorical representation theory and relate to braid group actions studied by Artin, Brieskorn, and Deligne. Graphical calculus simplifies proofs of positivity for Kazhdan–Lusztig polynomials and interacts with planar diagram techniques from Jones polynomial and constructions by Freyd, Yetter, and Turaev in topological quantum field theory.
Applications in representation theory and geometry
Soergel bimodules apply to computations of decomposition numbers in Category O, to modular representation problems for Reductive groups and Affine Lie algebras, and to categorical actions on derived categories of coherent sheaves on Flag varietys and Quiver varietys studied by Nakajima and Lusztig. They underpin modern approaches to link homology theories by Khovanov–Rozansky and provide tools for studying perverse sheaves in the work of Beilinson, Bernstein, and Deligne. Further applications connect to geometric representation programs of Geometric Langlands program proponents like Drinfeld and Laumon, to categorified quantum invariants explored by Witten, and to combinatorial representation theory advanced by Stanley and Macdonald.