Soergel Bimodules Overview

• Soergel bimodules are graded bimodules over polynomial rings associated with Coxeter systems that categorify the Hecke algebra.
• Their construction features Bott–Samelson generation, diagrammatic calculus, and Hodge-theoretic methods to analyze decomposition and morphism spaces.
• Variants such as singular, odd, and topological extensions expand their applications in representation theory, geometry, and knot homology.

Soergel bimodules are graded bimodules over polynomial algebras associated with Coxeter systems, and they form a monoidal Karoubian category whose split Grothendieck group recovers the Hecke algebra. In the standard setup, one fixes a Coxeter system $(W,S)$, a reflection faithful real realization $\mathfrak h$, and the graded polynomial ring $R=\operatorname{Sym}(\mathfrak h^*)$ with $\deg(\mathfrak h^*)=2$. For each simple reflection $s\in S$, the basic bimodule is

$B_s:=R\otimes_{R^s}R(1),$

and the category of Soergel bimodules is the full additive monoidal Karoubian subcategory of graded $R$-bimodules generated by the $B_s$’s. In this sense, Soergel bimodules are a combinatorial and algebraic categorification of the Hecke algebra attached to a Coxeter system (Elias et al., 2012, Libedinsky, 2017).

1. Algebraic definition and Bott–Samelson generation

The basic construction proceeds from the invariant-theoretic pair $(R,R^s)$. For each $s\in S$, the bimodule $\mathfrak h$0 is free as a left and right $\mathfrak h$1-module. Given an expression $\mathfrak h$2, the associated Bott–Samelson bimodule is

$\mathfrak h$3

and indecomposable Soergel bimodules are the indecomposable direct summands of such Bott–Samelson bimodules. For each $\mathfrak h$4, Soergel’s theorem gives a unique indecomposable summand $\mathfrak h$5 of $\mathfrak h$6 that does not appear in shorter Bott–Samelson objects; these $\mathfrak h$7 classify indecomposable Soergel bimodules up to shift (Elias et al., 2012).

This construction admits several equivalent presentations. In the diagrammatic approach, Bott–Samelson objects are encoded by words in the simple reflections, and the full additive monoidal subcategory generated by them is then idempotent-completed to obtain the Soergel category (Elias et al., 2013). In type $\mathfrak h$8, one also introduces generalized Bott–Samelson bimodules attached to parabolic subsets $\mathfrak h$9,

$R=\operatorname{Sym}(\mathfrak h^*)$0

where $R=\operatorname{Sym}(\mathfrak h^*)$1; these are direct summands of ordinary Bott–Samelson bimodules and become explicit objects in the thick calculus (Elias, 2010).

The tensor product over $R=\operatorname{Sym}(\mathfrak h^*)$2 is the monoidal structure, and grading shifts encode the Hecke parameter. A basic structural decomposition is

$R=\operatorname{Sym}(\mathfrak h^*)$3

which mirrors the quadratic relation in the Hecke algebra. In low rank, further decompositions such as

$R=\operatorname{Sym}(\mathfrak h^*)$4

make the categorified braid relations concrete (Libedinsky, 2017).

2. Characters, indecomposables, and Hecke categorification

The split Grothendieck group $R=\operatorname{Sym}(\mathfrak h^*)$5 of the Soergel category is canonically isomorphic to the Hecke algebra $R=\operatorname{Sym}(\mathfrak h^*)$6, with

$R=\operatorname{Sym}(\mathfrak h^*)$7

This is Soergel’s categorification theorem (Elias et al., 2012). The category therefore provides a many-object, graded realization of Hecke multiplication via tensor product of bimodules.

A key invariant is the character of a Soergel bimodule. Using filtrations by support on graphs $R=\operatorname{Sym}(\mathfrak h^*)$8, one defines

$R=\operatorname{Sym}(\mathfrak h^*)$9

The character map is an inverse to the categorification isomorphism, and Soergel’s hom formula identifies graded morphism ranks with the Hecke pairing: $\deg(\mathfrak h^*)=2$0 In particular, once the character of indecomposables is known, one obtains strong consequences such as $\deg(\mathfrak h^*)=2$1 (Elias et al., 2012).

The decisive character statement is Soergel’s conjecture: $\deg(\mathfrak h^*)=2$2 This identifies the indecomposable Soergel bimodule $\deg(\mathfrak h^*)=2$3 with the Kazhdan–Lusztig basis element $\deg(\mathfrak h^*)=2$4. Writing

$\deg(\mathfrak h^*)=2$5

one obtains positivity: $\deg(\mathfrak h^*)=2$6 where the $\deg(\mathfrak h^*)=2$7 are the structure constants in the Kazhdan–Lusztig basis (Elias et al., 2012).

Relative Hodge-theoretic refinements strengthen these consequences. If

$\deg(\mathfrak h^*)=2$8

then the relative hard Lefschetz theorem implies that the structure constants $\deg(\mathfrak h^*)=2$9 are unimodal; equivalently, they decompose as sums of quantum integers with nonnegative coefficients (Elias et al., 2016).

3. Diagrammatic calculus and morphism bases

The monoidal category of Soergel bimodules admits a presentation by generators and relations using planar diagrammatics. In the Soergel calculus, objects are sequences of simple reflections, and morphisms are generated by polynomial boxes, dots, trivalent vertices, and $s\in S$0-valent vertices, modulo local relations encoding Frobenius structure, polynomial sliding, and rank-two braid data (Elias et al., 2013).

The diagrammatic category is equivalent to the Bott–Samelson category. In this presentation, Libedinsky’s light leaves morphisms provide a combinatorial control of Hom spaces. For a fixed expression $s\in S$1 and subexpression $s\in S$2, one constructs a morphism

$s\in S$3

whose degree is the Deodhar defect of $s\in S$4. Composing a light leaf with the upside-down version of another produces a double leaves morphism, and the set of all such double leaves morphisms is an $s\in S$5-basis for

$s\in S$6

This yields a basis theorem for morphism spaces and a diagrammatic proof of Soergel’s classification theorem (Elias et al., 2013).

In type $s\in S$7, the calculus extends to a thick version in which generalized Bott–Samelson bimodules $s\in S$8 are represented by thick strands labeled by parabolic subsets. The technical core is an explicit idempotent splitting off $s\in S$9 from a Bott–Samelson object attached to a reduced expression of $B_s:=R\otimes_{R^s}R(1),$0, with the projector constructed from the reduced expression graph of the longest element and the Manin–Schechtman semi-orientation (Elias, 2010).

For dihedral groups, the two-color Soergel calculus gives a complete planar presentation. The degree $B_s:=R\otimes_{R^s}R(1),$1 morphisms between color-alternating objects form a copy of the two-colored Temperley–Lieb category, and the indecomposable Soergel bimodules are the images of Jones–Wenzl projectors. When the dihedral group is infinite, the parameter $B_s:=R\otimes_{R^s}R(1),$2 may be generic; when it is finite, $B_s:=R\otimes_{R^s}R(1),$3 is specialized to a root of unity, and the negligible Jones–Wenzl projector yields the Soergel bimodule for the longest element (Elias, 2013).

4. Hodge theory, local theory, and Lefschetz phenomena

A major structural development is the Hodge theory of Soergel bimodules. Fix a degree $B_s:=R\otimes_{R^s}R(1),$4 element $B_s:=R\otimes_{R^s}R(1),$5 strictly positive on all simple coroots, and write

$B_s:=R\otimes_{R^s}R(1),$6

Hard Lefschetz asserts that multiplication by $B_s:=R\otimes_{R^s}R(1),$7 satisfies

$B_s:=R\otimes_{R^s}R(1),$8

Each indecomposable $B_s:=R\otimes_{R^s}R(1),$9 also carries, up to positive scalar, a canonical invariant symmetric form, the intersection form, and the Hodge–Riemann bilinear relations state that the Lefschetz form on primitive subspaces has the expected alternating definiteness (Elias et al., 2012).

These Hodge-theoretic statements yield the proof of Soergel’s conjecture for arbitrary Coxeter systems. The argument uses local intersection forms, an embedding theorem that identifies them with primitive pieces of global Lefschetz forms, a one-parameter family of Lefschetz operators

$R$0

and Rouquier complexes as an algebraic substitute for weak Lefschetz (Elias et al., 2012).

There are also local and relative versions. In the local theory, one studies the canonical map $R$1 between costalk and stalk pieces and the induced local intersection form. For a dominant regular $R$2, the associated specialized module satisfies local hard Lefschetz, and the corresponding primitive spaces satisfy local Hodge–Riemann bilinear relations. For $R$3, the canonical local class $R$4 satisfies

$R$5

relating the local form to equivariant multiplicity (Williamson, 2014).

The relative theory concerns tensor products $R$6. For a dominant regular $R$7, the degree $R$8 operator $R$9 on $B_s$0 induces relative hard Lefschetz and relative Hodge–Riemann bilinear relations on the perverse cohomology spaces. Besides implying unimodality of Kazhdan–Lusztig structure constants, the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any 2-sided cell in a Coxeter group is rigid and pivotal (Elias et al., 2016).

5. Singular, odd, and module-theoretic variants

Singular Soergel bimodules generalize the ordinary theory by allowing different parabolic invariant rings on the left and right. For finitary $B_s$1, one works with graded $B_s$2-bimodules generated by iterated translation onto and out of walls. Indecomposable singular Soergel bimodules are parametrized, up to shift, by double cosets

$B_s$3

and the split Grothendieck group of the resulting 2-category is isomorphic to the Schur algebroid. In characteristic zero, Soergel’s conjecture for ordinary bimodules implies the corresponding character formula for singular bimodules (Williamson, 2010).

This singular theory can be recovered from the ordinary one by higher idempotent completion. In particular, singular Soergel bimodules are obtained from ordinary Soergel bimodules by partial 2-categorical idempotent completion, and in type $B_s$4 they assemble into a semistrict monoidal 2-category related to progressive $B_s$5-foam 2-categories and deformed colored $B_s$6 link homology (Recio, 1 Aug 2025).

A different direction is the odd theory. In the two-variable odd analogue, one replaces the commutative polynomial ring by the supercommutative algebra

$B_s$7

with the $B_s$8-action $B_s$9. The odd two-variable Soergel category is generated by the odd generating bimodule

$(R,R^s)$0

and the transposition bimodule $(R,R^s)$1 with twisted right action. In this setting the transposition bimodule cannot be merged into the generating Soergel bimodule, so the monoidal category has a larger Grothendieck ring than the even type-$(R,R^s)$2 category. Odd Rouquier complexes are nevertheless invertible in the homotopy category, while in three variables the absence of a direct sum decomposition obstructs the Reidemeister III relation (Khovanov et al., 2022).

From the viewpoint of Soergel modules, the structure algebra of the Bruhat graph or moment graph becomes essential. For arbitrary Coxeter groups, the correct Hom formula for Soergel modules is

$(R,R^s)$3

and the graded dimension of these Hom spaces agrees with the Hecke pairing. In infinite type, this gives a negative answer to Soergel’s question whether ordinary right $(R,R^s)$4-module maps between Soergel modules are always induced from bimodule maps. The same framework produces a distinguished submodule $(R,R^s)$5 that mimics ordinary cohomology inside intersection cohomology and yields consequences such as top-heaviness for Bruhat intervals (Patimo, 8 Apr 2025).

6. Topological, $(R,R^s)$6-theoretic, and braided extensions

Soergel bimodules also admit topological realizations. For a compact connected Lie group $(R,R^s)$7 of adjoint type with maximal torus $(R,R^s)$8, the basic polynomial ring is

$(R,R^s)$9

and for each simple reflection $s\in S$0 one constructs a space $s\in S$1 with

$s\in S$2

In this model, Soergel bimodules are identified with the cohomology of well-defined objects in the category of spectra, and Thom spectra of formal virtual bundles produce Rouquier-type braid complexes over $s\in S$3. After reduction modulo an odd prime, the reduced Steenrod algebra $s\in S$4 acts on these bimodules, on their Hochschild bicomplexes, and hence on triply graded link homology. For $s\in S$5, the resulting triply graded link homology is well-defined over any $s\in S$6-algebra and supports an $s\in S$7-action over $s\in S$8 for odd $s\in S$9 (Kitchloo, 2013).

A multiplicative analogue is provided by $\mathfrak h$00-theory Soergel bimodules. Here classical Soergel bimodules are viewed as a completed and infinitesimal version of a $\mathfrak h$01-theoretic theory built from equivariant $\mathfrak h$02-theory of Bott–Samelson varieties and their intersections. Morphisms are described geometrically in terms of equivariant $\mathfrak h$03-theoretic correspondences, and Bruhat-stratified torus-equivariant $\mathfrak h$04-motives on flag varieties are modeled by bounded chain complexes of $\mathfrak h$05-theory Soergel bimodules. The classical theory is recovered by completion at the augmentation ideal, in the spirit of Atiyah–Segal completion (Eberhardt, 2022).

Braided and higher-categorical structures now play a central role. For type $\mathfrak h$06, bounded chain complexes of Soergel bimodules form a semistrict monoidal 2-category equipped with a braiding whose 1-morphisms are Rouquier complexes of shuffle braids; explicit slide maps and higher homotopies give the naturality data on generating morphisms, and cohomology-vanishing arguments extend the homotopy-coherent naturality to all chain complexes (Stroppel et al., 2024). A related spectral enhancement replaces polynomial rings by $\mathfrak h$07 for a connective complex oriented $\mathfrak h$08-ring spectrum $\mathfrak h$09. The resulting $\mathfrak h$10-valued stable Soergel $\mathfrak h$11-category admits a braiding, equivalently an $\mathfrak h$12-algebra structure, whose two-strand crossing is the Rouquier complex; standard splittings such as $\mathfrak h$13 and $\mathfrak h$14 persist in this spectral setting (Liu, 2024).

These extensions show that Soergel bimodules are not confined to a single algebraic incarnation. They appear as the Hecke category in planar diagrammatics, as a Hodge-theoretic model of intersection cohomology, as a singular and 2-categorical theory controlled by parabolic invariants and idempotent completions, and as topological, $\mathfrak h$15-theoretic, and spectral objects that support braid actions, Rouquier complexes, Steenrod operations, and link-homological constructions.