Kazhdan–Lusztig Theory | Bruhat Order, Hecke Algebras and Category O

Kazhdan–Lusztig theory begins with a combinatorial ordering of symmetries, deforms their group algebra into a Hecke algebra, selects a remarkable self-dual basis, and extracts polynomials whose coefficients reach into geometry and representation theory.

The surprise is not that one can attach polynomials to pairs of elements in a Coxeter group. The surprise is how much those polynomials know. In Weyl-group settings they encode local intersection-cohomological information about Schubert varieties. In a regular integral block of BGG category O, their values at 1 give composition multiplicities of simple modules inside Verma modules. Through Soergel bimodules, the same basis becomes the decategorified shadow of a graded monoidal category.

This article builds those bridges rather than presenting the subject as a sequence of names. We will calculate length and Bruhat order in symmetric groups, deform a simple reflection inside the Hecke algebra, verify the first Kazhdan–Lusztig basis element directly, see why every type A₂ Kazhdan–Lusztig polynomial is 1, and then meet a type A₃ interval with polynomial 1+q. The final route returns that polynomial to category O and geometry.

Scope: advanced enrichment in representation theory. The main algebraic examples use finite Coxeter and Weyl groups over characteristic zero. Indexing and q-normalisations vary across references, so this guide fixes one convention before calculating. For prerequisites, use Category O, Verma Modules and BGG Reciprocity, Hecke Algebras and Arithmetic Eigenvalues, and Symmetric Group Representations.

Coxeter groups · Bruhat order · Hecke algebra · Kazhdan–Lusztig basis · First nontrivial example · Schubert geometry · Category O · Categorification · Practice

The organising problem: a standard basis can be too literal

A group algebra has an obvious basis indexed by group elements. A Hecke algebra has an analogous standard basis. That basis is excellent for multiplication by generators, but it does not automatically expose the geometric and representation-theoretic structure one wants.

Kazhdan and Lusztig’s 1979 insight was to construct a second basis with a strong symmetry under an involution that reverses the deformation parameter. The transition coefficients between the standard basis and this new basis are the Kazhdan–Lusztig polynomials.

The new basis is therefore not an arbitrary change of coordinates. It is singled out by triangularity with respect to Bruhat order, a degree bound, and self-duality under the Hecke involution. Those constraints are strong enough to determine it uniquely.

Coxeter combinatorics → Bruhat order → Hecke deformation → self-dual basis → Kazhdan–Lusztig polynomials → geometry and multiplicities.

Coxeter groups provide generators, relations and length

A Coxeter system (W,S) consists of a group W generated by a set S={sᵢ} of involutions. The defining relations have the form

sᵢ²=1,   (sᵢsⱼ)^mᵢⱼ=1

for specified integers mᵢⱼ≥2 when i≠j. Equivalently, the generators satisfy braid relations of length mᵢⱼ.

The symmetric group Sₙ is the most concrete example. Let sᵢ swap positions i and i+1. Then sᵢ²=1, distant generators commute, and adjacent generators satisfy sᵢsᵢ₊₁sᵢ=sᵢ₊₁sᵢsᵢ₊₁.

The length ℓ(w) is the minimum number of simple reflections needed to express w. In Sₙ, this equals the number of inversions: pairs i<j with w(i)>w(j). A reduced word is an expression for w using exactly ℓ(w) generators.

Worked length calculations in S4

Take y=2143. Its inversions are 2>1 and 4>3, so ℓ(y)=2. One reduced expression is s₁s₃; because s₁ and s₃ commute, s₃s₁ is another.

Now take w=4231. Its inversions are 4>2, 4>3, 4>1, 2>1 and 3>1, so ℓ(w)=5. One reduced expression is

w=s₁s₂s₃s₂s₁.

The length difference ℓ(w)−ℓ(y)=3 will later force any Kazhdan–Lusztig polynomial P_y,w(q) to have degree at most 1. A simple inversion count is already imposing a polynomial constraint.

Reduced words are not unique, but length is

In S₃, the longest permutation 321 can be written s₁s₂s₁ or s₂s₁s₂. The braid relation identifies the two reduced expressions. A theory built from reduced words must therefore be invariant under allowed braid moves.

This is exactly why the standard Hecke basis element T_w can be defined by choosing any reduced word for w: the Hecke generators satisfy the same braid relations, so the result does not depend on that reduced choice.

Bruhat order says which symmetries sit underneath which others

Bruhat order is a partial order on W. One especially useful criterion is the subword criterion: y≤w when a reduced expression for y can be obtained by deleting letters from a reduced expression for w. More precisely, the retained letters occur in their original order and multiply to y.

The order is not the same as comparing lengths. Length is necessary but insufficient. If y≤w, then ℓ(y)≤ℓ(w); the converse need not hold.

For our S₄ example, y=s₁s₃ appears as a subword of w=s₁s₂s₃s₂s₁ by retaining the first and third generators. Therefore 2143≤4231.

This verification matters because a Kazhdan–Lusztig polynomial is zero unless its first index lies below its second in Bruhat order.

The complete type A2 Bruhat skeleton

Let W=S₃ with simple reflections s₁ and s₂. The six elements split by length:

• length 0: e;
• length 1: s₁, s₂;
• length 2: s₁s₂, s₂s₁;
• length 3: w₀=s₁s₂s₁=s₂s₁s₂.

The covering relations are e below both simple reflections, each simple reflection below both length-two elements when the subword condition permits it, and both length-two elements below w₀. The precise type A₂ order is stated explicitly in the MIT lecture notes listed in the sources below.

A Hasse diagram records only covers; the full partial order is obtained by transitivity. If e<s₁<s₁s₂<w₀, then e≤w₀ even though no edge is drawn directly between them.

Bruhat intervals localise the calculation

For y≤w, the interval [y,w] contains exactly the elements x satisfying y≤x≤w. Many recursive formulas for R-polynomials and Kazhdan–Lusztig polynomials only inspect elements in such intervals.

This is computationally useful. A large Coxeter group may contain many elements irrelevant to one requested pair. Restricting to an interval is the representation-theoretic analogue of cutting a large graph down to the nodes that can actually influence the target.

It also connects to geometry. In a Weyl group, Bruhat order describes closure relations among Schubert cells. If y≤w, the Schubert cell indexed by y lies inside the Schubert variety indexed by w.

The Hecke algebra deforms the group algebra

We use the convention of Pavel Etingof’s MIT 18.757 lecture. The Hecke algebra H_q(W) is defined over Z[q^1/2,q^−1/2] by generators Tᵢ satisfying the Coxeter braid relations and the quadratic relation

(Tᵢ+1)(Tᵢ−q)=0.

Expanding gives

Tᵢ²=(q−1)Tᵢ+q.

At q=1 this becomes Tᵢ²=1, recovering the quadratic relation of the group algebra. The Hecke algebra is therefore a deformation: the familiar group structure sits at a special parameter value.

Standard basis multiplication follows length

For w∈W choose a reduced expression and define T_w as the corresponding product of Hecke generators. The braid relations make this independent of the chosen reduced expression.

Multiplying by a simple reflection has two cases. If ℓ(sw)=ℓ(w)+1, then T_sT_w=T_sw. If ℓ(sw)=ℓ(w)−1, then

T_sT_w=(q−1)T_w+qT_sw.

The second formula is the deformed correction. At q=1 it reduces to T_sT_w=T_sw, as ordinary group multiplication requires.

Worked multiplication with one simple reflection

Let w=s. Then ℓ(ss)=0=ℓ(s)−1. The decreasing-length formula gives

T_s²=(q−1)T_s+qT_e,

which is exactly the expanded quadratic relation. This small return check confirms that our multiplication convention and basis convention agree.

Invertibility produces the bar involution

The quadratic relation shows Tᵢ is invertible:

Tᵢ⁻¹=q⁻¹(Tᵢ+1−q).

Define an involution D by sending q^1/2 to q^−1/2 and Tᵢ to Tᵢ⁻¹. More generally, D(T_w)=T_w⁻¹⁻¹.

This involution is the symmetry that the Kazhdan–Lusztig basis is designed to respect. The standard basis is natural for multiplication, while the new basis is natural for this duality.

The Kazhdan–Lusztig basis is triangular and self-dual

For each w∈W there are unique polynomials P_y,w(q)∈Z[q] satisfying three key conditions:

• P_y,w=0 unless y≤w, and P_w,w=1;
• if y<w, then deg P_y,w≤[ℓ(w)−ℓ(y)−1]/2;
• the element C_w=q^−ℓ(w)/2Σ_yP_y,w(q)T_y is fixed by D.

The elements C_w form the Kazhdan–Lusztig basis in this normalisation. Other sources may write C′_w, use v in place of q^1/2, or change signs and powers. Those are convention changes, not different mathematics, provided all formulas are translated together.

Verify the simple-reflection basis element directly

For w=s, the only y≤s are e and s. The length difference from e to s is 1, so the degree bound forces P_e,s to be constant. Its constant term is 1, hence P_e,s=1.

Therefore

C_s=q^−1/2(T_e+T_s).

Apply D:

D(C_s)=q^1/2[1+q⁻¹(T_s+1−q)]=q^−1/2(1+T_s)=C_s.

The simplest nontrivial basis vector already shows how parameter inversion and operator inversion compensate each other.

The degree bound eliminates many possibilities immediately

If y<w and ℓ(w)−ℓ(y)≤2, then the degree bound forces P_y,w to have degree zero. Since its constant term is 1, P_y,w=1.

This explains why low-rank examples can look deceptively simple. A short Bruhat interval has no degree room for a q-term.

In type A₂, even the longest interval has an especially simple answer: all Kazhdan–Lusztig polynomials are 1. The resulting category O multiplicities are therefore all either zero or one in the corresponding regular block.

Type A2: recover the category O pattern

Fix a regular dominant integral λ and abbreviate w·λ by the reduced-word label of w. In the convention used by the MIT lecture, the decomposition classes are

M_121 = L_121
M_12  = L_12 + L_121
M_21  = L_21 + L_121
M_1   = L_1 + L_12 + L_21 + L_121
M_2   = L_2 + L_12 + L_21 + L_121
M_e   = L_e + L_1 + L_2 + L_12 + L_21 + L_121.

Every coefficient shown is 1. Bruhat comparability determines where a simple can appear, and the trivial Kazhdan–Lusztig polynomials give multiplicity one on those allowed positions.

This is a useful control case before moving to S₄. The moment a polynomial such as 1+q appears, evaluation at q=1 can produce multiplicity 2, which cannot occur in the type A₂ table above.

A first nontrivial S4 calculation: P2143,4231(q)=1+q

Return to y=2143 and w=4231. We already checked ℓ(y)=2, ℓ(w)=5 and y≤w. The degree bound gives

deg P_2143,4231≤(5−2−1)/2=1.

A documented Kazhdan–Lusztig calculation gives

P_2143,4231(q)=1+q.

The polynomial passes every immediate check. Its constant term is 1. Its coefficients are nonnegative. Its degree is exactly 1, within the bound. Evaluating at q=1 gives 2.

This example is small enough to see the transition from order data to genuinely nontrivial polynomial data. Length and comparability constrain the answer, but they do not determine whether the coefficient of q is 0 or 1. Additional Hecke or geometric structure is doing real work.

Why 1+q contains more information than the number 2

The evaluation P(1)=2 is the number needed for an ungraded composition multiplicity in category O. But the polynomial retains a degree distinction that the integer 2 forgets.

Geometrically, q records cohomological degree in an intersection-cohomology interpretation after the chosen normalisation. Categorically, q can be realised through grading shifts in a Hecke or Soergel category. Setting q=1 collapses those degrees.

This is the same decategorification pattern developed in the BTT guide to Grothendieck Groups and Categorification: a richer graded object is compressed to an integer only after useful structure has been stored upstairs.

R-polynomials provide a recursive bridge to the self-dual basis

One standard computational route introduces R-polynomials through the bar involution of the standard basis. They are supported on Bruhat intervals and satisfy recursive formulas obtained by multiplying by simple reflections that decrease length.

The self-duality requirement D(C_w)=C_w, together with the degree bound, then determines P_y,w recursively. This is an exact algorithm, though not necessarily the most efficient one for large groups.

A practical implementation memoises shorter intervals and reuses descent information. The mathematical point is more important than the programming detail: the algorithm reduces a large target to smaller Bruhat problems while preserving the involutive symmetry that defines the canonical basis.

Kazhdan–Lusztig cells come from basis multiplication

Multiplying Kazhdan–Lusztig basis elements by simple generators reveals which basis elements can reach which others. This generates left and right preorders; combining them produces two-sided cells.

Cell modules provide representations of Hecke algebras and, after suitable specialisation, representations related to the underlying Coxeter group. In symmetric groups, cell combinatorics is tightly linked to Robinson–Schensted theory and the representation theory indexed by partitions.

The word cell therefore has a precise algebraic origin here. It is not simply a subset of the group chosen for convenience. The subset is generated by which canonical-basis coefficients can communicate under multiplication.

Schubert geometry explains why the polynomials are not arbitrary

For a complex reductive group G with Borel subgroup B, the flag variety G/B decomposes into Schubert cells indexed by the Weyl group W. The closure relation is controlled by Bruhat order:

X_y lies in X_w exactly when y≤w

after choosing a consistent Schubert-cell convention. Thus the same partial order supporting the Hecke recursion is already present in the geometry of the flag variety.

Schubert varieties can be singular. Ordinary cohomology does not retain Poincaré duality across singularities in the same way it does for smooth compact manifolds. Intersection cohomology repairs this failure while preserving enough local information to distinguish singular behaviour.

Kazhdan and Lusztig connected their polynomials to local intersection homology of Schubert varieties. This geometric interpretation explains the appearance of nonnegative coefficients in Weyl-group cases and shows why a nontrivial polynomial is measuring more than a combinatorial curiosity.

Patterns 3412 and 4231 mark the first type-A singular behaviour

For permutations, the classical Lakshmibai–Sandhya smoothness criterion says that a Schubert variety is smooth exactly when its permutation avoids the patterns 3412 and 4231.

Our nontrivial example uses w=4231 itself. This is not accidental. It is one of the minimal type-A patterns where singular geometry appears, and Kazhdan–Lusztig polynomials detect that local failure of smooth behaviour.

A useful caution follows. Pattern avoidance is a type-A permutation criterion. General Coxeter groups do not come with permutations of {1,…,n} in this form, so the appropriate geometry or combinatorics must be expressed differently.

Intersection cohomology is a graded explanation of positivity

If a polynomial coefficient is realised as the dimension of a cohomology group, negativity would be impossible. This is the conceptual source of positivity in geometric Weyl-group settings.

The original algebraic definition does not make coefficient positivity obvious. A recursive expression can involve subtraction even though the final answer has nonnegative coefficients. Geometry supplies an object whose graded dimensions explain the final signs.

This distinction—an invariant being positive without every computational formula being manifestly positive—appears throughout mathematics. A poor intermediate coordinate system can conceal a structural theorem that is obvious in a better model.

The Kazhdan–Lusztig theorem calculates Verma-module multiplicities in category O

Fix a complex semisimple Lie algebra and a regular dominant integral weight λ. In the convention of the MIT source used here, Verma modules and simples in the corresponding regular block can be indexed by Weyl-group elements through the dot action.

The Kazhdan–Lusztig character theorem gives

[M_y·λ:L_w·λ]=P_y,w(1).

The left side is a composition multiplicity: how many times the simple module L_w·λ occurs in a Jordan–Hölder series of the Verma module M_y·λ. The right side starts as a polynomial constructed from a Coxeter group and Hecke algebra.

This is the central representation-theoretic bridge:

Hecke polynomial → evaluate at 1 → composition multiplicity in a highest-weight category.

Indexing conventions can reverse or shift labels in other books. Before importing a formula, record whether weights are dominant or antidominant, which dot action is used, and whether the standard module is written M(w·λ) or with an inverse/longest-element reindexing.

The S4 example predicts multiplicity two

Because P_2143,4231(q)=1+q, the evaluation at 1 is 2. Under the regular integral category O indexing of the theorem above, the corresponding simple therefore occurs twice in the corresponding Verma module.

The polynomial retains the two graded contributions separately; the ungraded composition series sees only their total. This is why evaluating too early can destroy information useful for categorification and geometry.

The preceding Category O guide computed an sl₂ block where all relevant multiplicities were 0 or 1. Kazhdan–Lusztig theory explains how larger Weyl groups produce more intricate multiplicities systematically.

BGG reciprocity turns the same numbers into projective-filtration data

Category O also has projective covers with Verma filtrations. BGG reciprocity says that the multiplicity of a Verma module M(μ) in a standard filtration of P(λ) equals the composition multiplicity of L(λ) in M(μ).

Combining this with Kazhdan–Lusztig theory means one family of polynomials controls two different multiplicity questions:

• simple factors inside Verma modules;
• Verma factors inside projective modules.

The equality does not make a projective module a direct sum of Vermas. The second multiplicity belongs to a filtration. Extension data remains present even after the multiplicity matrix has been calculated.

Character formulas are obtained by matrix inversion

Verma characters are explicit, while simple characters are the difficult objects. If D is the matrix with entries D_y,w=P_y,w(1) in a compatible Bruhat ordering, then

[M]=D[L]

schematically in the Grothendieck group. Triangularity makes D invertible over the integers. Therefore simple characters can be expressed as alternating integer combinations of Verma characters using D⁻¹.

This is why solving the multiplicity problem solves the character problem. The hard representation theory is concentrated into a transition matrix whose entries come from the Hecke algebra.

Soergel bimodules categorify the Hecke algebra

Soergel introduced a monoidal category of graded bimodules associated with a Coxeter system. After taking an appropriate split Grothendieck group, its tensor product becomes Hecke multiplication.

Indecomposable Soergel bimodules correspond to Kazhdan–Lusztig basis elements. Grading shifts account for powers of the deformation parameter. Thus a polynomial coefficient can be interpreted through graded multiplicities rather than remaining a formal coefficient in a Laurent-polynomial ring.

Ben Elias and Geordie Williamson proved Soergel’s conjecture using an algebraic Hodge theory. Their result implies positivity of Kazhdan–Lusztig polynomials for arbitrary Coxeter systems and yields an algebraic route to the Kazhdan–Lusztig character theorem.

This is a strong example of categorification doing real mathematical work. The category is not merely a verbose restatement of the Hecke algebra. Its internal Hodge-type structure proves positivity that is hidden in the decategorified recursion.

What decategorification forgets

When a graded object is sent to the Hecke algebra, morphisms and extension data disappear while graded multiplicities become polynomial coefficients. Evaluating q=1 forgets even the grading and retains only total multiplicity.

There are therefore at least three levels in the same calculation:

• categorical level: graded bimodules and morphisms;
• Hecke level: Kazhdan–Lusztig basis and polynomials;
• ungraded multiplicity level: integer values P_y,w(1).

A correct argument should state which level it inhabits. An integer equality cannot automatically recover a graded object, just as a Grothendieck class cannot recover every non-split extension.

Positive characteristic creates p-Kazhdan–Lusztig theory

Classical Kazhdan–Lusztig theory is tied to characteristic-zero intersection cohomology and Soergel theory. When coefficients are taken in characteristic p, decomposition behaviour can change.

The resulting p-canonical or p-Kazhdan–Lusztig phenomena are central to modern modular representation theory. They help explain why characteristic-p decomposition numbers can differ dramatically from characteristic-zero expectations.

This article does not identify classical and p-Kazhdan–Lusztig polynomials. The latter depend on the prime and belong to a richer modular story. The earlier BTT guide to Modular Representation Theory provides the conceptual warning: changing characteristic can change decomposition and extension structure.

A computational workflow

• 1. Fix the Coxeter convention. State the generators, length function and multiplication direction.
• 2. Compute reduced words. Verify lengths independently, for example by inversion counts in Sₙ.
• 3. Test Bruhat comparability. Use the subword criterion or an equivalent rank criterion.
• 4. Restrict to [y,w]. Do not carry irrelevant group elements into the recursion.
• 5. Fix the Hecke normalisation. Record the quadratic relation before using any published formula.
• 6. Apply degree bounds early. They can force many polynomials to equal 1.
• 7. Compute recursively or categorically. Cache shorter intervals when using an algorithm.
• 8. Check positivity and constant term. A negative coefficient in the classical theory is a strong sign of a convention or recurrence error.
• 9. Evaluate at 1 only for the ungraded multiplicity question. Keep q when grading or geometry matters.
• 10. Translate representation indices explicitly. Do not mix dominant, antidominant or longest-element conventions.

Common mistakes

• Comparing only lengths: ℓ(y)≤ℓ(w) does not prove y≤w.
• Using a nonreduced word in the subword criterion without checking: the criterion is formulated through reduced expressions.
• Mixing Hecke normalisations: (T+1)(T−q)=0 and other popular conventions shift signs and powers.
• Assuming all short examples show the general behaviour: type A₂ has only trivial KL polynomials.
• Evaluating q=1 too soon: this loses graded and geometric information.
• Treating a filtration multiplicity as a direct-sum decomposition: BGG reciprocity counts standard subquotients.
• Calling every nontrivial KL polynomial “a singular point” without indexing: the pair (y,w) records local information along a Schubert stratum.
• Using characteristic-zero KL polynomials as universal modular decomposition numbers: p-canonical phenomena can intervene.

Practice questions

1. Find the inversion length of 3142. 2. Give one reduced expression for 321 in S₃. 3. Use a reduced word to show s₂≤s₁s₂s₁. 4. If y<w and ℓ(w)−ℓ(y)=2, what is P_y,w(q)?

5. Expand (T_s+1)(T_s−q)=0 to find T_s². 6. Find T_s⁻¹. 7. Write C_s in the convention used here. 8. Explain why type A₂ regular category O has no composition multiplicity greater than 1 in the displayed Verma decompositions.

9. Verify that ℓ(4231)−ℓ(2143)=3. 10. What ungraded multiplicity follows from P_2143,4231(q)=1+q? 11. Why can a positive polynomial arise from a recursion containing subtraction? 12. Name one piece of information lost when a graded Soergel object is replaced first by its Hecke class and then by evaluation at q=1.

Worked answers

1–4. Coxeter and Bruhat checks

1. In 3142, the inversions are 3>1, 3>2 and 4>2, giving length 3. The pair 1<4 is not an inversion, nor is 1<2.

2. One reduced expression is s₁s₂s₁. The braid relation gives the equally reduced expression s₂s₁s₂.

3. In the reduced word s₁s₂s₁, retain only the middle generator. The resulting subword is s₂, so s₂≤s₁s₂s₁.

4. The degree bound is at most (2−1)/2=1/2, so the polynomial has degree zero. Its constant term is 1, hence P_y,w=1.

5–8. Hecke and canonical-basis checks

5. Expanding gives T_s²+(1−q)T_s−q=0, hence T_s²=(q−1)T_s+q.

6. Rearranging the quadratic relation after multiplying by the inverse gives T_s⁻¹=q⁻¹(T_s+1−q).

7. C_s=q^−1/2(1+T_s). Direct application of D returns the same element.

8. In type A₂ all relevant Kazhdan–Lusztig polynomials are 1. The theorem identifies the corresponding composition multiplicity with P(1)=1 whenever Bruhat comparability allows the simple to occur.

9–12. Nontrivial polynomial and categorification

9. The permutation 2143 has two inversions, while 4231 has five. Their length difference is 5−2=3.

10. Evaluation at 1 gives 1+1=2, so the corresponding simple composition factor occurs with multiplicity two in the regular integral category O convention used by the theorem.

11. Positivity can be a property of the final invariant even when one computational presentation is not manifestly positive. In geometric or Soergel models, coefficients are realised through graded dimensions, making positivity structural.

12. Passing to the Hecke class forgets morphisms and much categorical extension information. Evaluating at q=1 then also forgets grading, retaining only the total multiplicity.

How to teach the topic without turning it into symbol memorisation

Begin with permutations, inversion counts and subwords. Do not introduce P_y,w before the learner can say why one pair is comparable and another is not. Next deform one reflection and verify the Hecke quadratic relation at q=1.

Then build C_s and check its self-duality. This makes the new basis feel necessary rather than decorative. Use type A₂ as the control case where the degree bound forces simplicity. Only after that introduce 2143≤4231 and the polynomial 1+q.

Finally ask the learner to translate the same object three ways: as a Hecke polynomial, as local Schubert information, and as a category O multiplicity after q=1. The subject becomes coherent when the learner can explain what each translation preserves and what it forgets.

Sources and further study

[1] David Kazhdan and George Lusztig, Representations of Coxeter Groups and Hecke Algebras, Inventiones Mathematicae 53 (1979), 165–184, the original source of the canonical basis, polynomials and character conjecture. [2] Pavel Etingof, MIT 18.757, Lecture 21: Multiplicities in Category O, for the Hecke convention, Bruhat order, uniqueness theorem and the formula [M_y·λ:L_w·λ]=P_y,w(1). [3] Alexander Woo, Permutations with Kazhdan–Lusztig Polynomial P_id,w(q)=1+q^h, for type-A singularity calculations and explicit nontrivial examples. [4] Ben Elias and Geordie Williamson, The Hodge Theory of Soergel Bimodules, Annals of Mathematics 180 (2014), for positivity in arbitrary Coxeter systems and Soergel’s conjecture. [5] George Lusztig’s publication archive records the original 1979 article and the subsequent Schubert-variety and Poincaré-duality work.

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