Soergel bimodules turn the Hecke algebra of a Coxeter group into a category. Basis elements become graded bimodules, multiplication becomes tensor product, and Kazhdan–Lusztig positivity becomes a statement about graded dimensions rather than a mysterious cancellation inside polynomials.
This chapter continues BTT’s Kazhdan–Lusztig Theory, Hecke Algebras, Categorification and Koszul Duality.
Coxeter data
Let (W,S) be a Coxeter system with a reflection representation V. Put R=Sym(V*) with its natural grading. For a simple reflection s, the invariant subring R^s sits inside R.
The basic Soergel bimodule is, up to grading convention, B_s=R⊗_{R^s}R(1). It carries commuting left and right R-actions.
Bott–Samelson bimodules
For a word s_1…s_m in simple reflections, form B_{s_1}⊗_R···⊗_R B_{s_m}. These are Bott–Samelson bimodules.
Taking finite direct sums, grading shifts and direct summands produces the Soergel category. The category remembers more than the Hecke algebra because it retains actual morphisms and decomposition data.
Categorifying the Hecke algebra
The split Grothendieck group of the Soergel category is identified with the Hecke algebra. Tensor product of bimodules descends to multiplication, while grading shift corresponds to multiplication by the deformation parameter.
Thus a Hecke-algebra identity is the shadow of a decomposition or isomorphism in a category.
Rank-one calculation
For W={1,s}, the polynomial ring decomposes over its invariants into even and odd parts. The bimodule B_s behaves as the categorical lift of the simple Hecke generator.
Tensoring B_s with itself decomposes into grading shifts of B_s. On the Grothendieck group this becomes the quadratic Hecke relation in the chosen normalization.
Indecomposable objects
For each w∈W there is an indecomposable Soergel bimodule B_w, characterized as a distinguished summand of a Bott–Samelson object attached to a reduced expression for w.
The isomorphism class B_w does not depend on the chosen reduced expression, although the ambient Bott–Samelson presentation does.
Kazhdan–Lusztig basis
Soergel’s conjecture, proved in broad generality through Hodge-theoretic methods, identifies the character of B_w with the Kazhdan–Lusztig canonical basis element.
This gives categorical meaning to KL coefficients: graded multiplicities and morphism spaces force positivity that is difficult to see from the recursive polynomial definition alone.
Light leaves and diagrammatics
Diagrammatic presentations encode morphisms between Bott–Samelson objects using colored strands, dots and vertices subject to local relations. Light-leaves constructions provide explicit bases for important Hom spaces.
The diagrammatic language turns abstract bimodule maps into combinatorial topology while preserving the Coxeter data.
Category O
Projective and translation functors in category O are controlled by Hecke combinatorics. Soergel’s functor translates suitable projective objects into modules or bimodule structures where this combinatorics becomes explicit.
This makes Soergel theory a bridge between category O, KL theory and categorified Hecke actions.
Hodge theory
Elias and Williamson developed a Hodge theory for Soergel bimodules, including analogues of hard Lefschetz and Hodge–Riemann bilinear relations. These structural inequalities prove positivity properties of KL polynomials.
The striking lesson is that Hodge-type geometry can survive in a purely algebraic categorical model even when no classical variety is being used directly.
Verification workflow
- Fix the Coxeter realization and grading convention.
- Distinguish R from R^s.
- Track grading shifts in B_s.
- Separate Bott–Samelson objects from their indecomposable summands.
- Specify the Hecke normalization before comparing quadratic relations.
- Do not identify a reduced expression with the element w itself.
- State hypotheses when invoking Soergel’s conjecture or Hodge theory.
Practice
1. Define B_s. 2. What is a Bott–Samelson bimodule? 3. What operation categorifies Hecke multiplication? 4. What labels indecomposable Soergel bimodules? 5. Why do KL coefficients become positive? 6. What do diagrammatics encode?
Answers. R⊗_{R^s}R with a grading shift; a tensor product of simple bimodules; tensor product over R; elements of W; they become graded categorical multiplicities; morphisms and relations in the Soergel category.
Representation Mathematics — Batch 14
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