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Coxeter systems R-polynomials KL polynomials Coxeter groups, KL and R-polynomials Neil J.Y. Fan Department of Mathematic, Sichuan University July 16, 2013 Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 1 / 50

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Page 1: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

Coxeter systemsR-polynomials

KL polynomials

Coxeter groups, KL and R-polynomials

Neil J.Y. Fan

Department of Mathematic, Sichuan University

July 16, 2013

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 1 / 50

Page 2: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

Coxeter systemsR-polynomials

KL polynomials

Coxeter systemsThe Bruhat order

Coxeter systems

Definition

A Coxeter system is a pair (W,S) consisting of a group W and aset of generators S, subject only to

(ss′)m(s,s′) = 1,

where m(s, s) = 1,m(s, s′) = m(s′, s) ≥ 2 for s 6= s′ in S.

Ex1. The symmetric group Sn, with

S = {s1, . . . , sn−1},

where si = (i, i+ 1) and

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 2 / 50

Page 3: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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s2i = 1, for all i ∈ [n− 1];

(sisj)2 = 1(or sisj = sjsi), for i, j ∈ [n− 1] and |i− j| > 1;

(sisi+1)3 = 1(or sisi+1si = si+1sisi+1), for i ∈ [n− 2].

Ex2. Finite reflection group, such as the dihedral group

Dn = 〈σ, τ |σn = τ2 = 1, τστ = σ−1〉,

also are (An−1, n ≥ 2), (Bn, n ≥ 2), (Dn, n ≥ 4).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 3 / 50

Page 4: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Coxeter systemsThe Bruhat order

Geometric representation of W

Let V = 〈αs|s ∈ S〉. Define a symmetric bilinear form B on V by

B(αs, αs′) = − cosπ

m(s, s′).

For each s ∈ S, define a reflcetion σs : V → V by

σsλ = λ− 2B(αs, λ)αs.

Proposition

The homomorphism σ : W → GL(V ) sending s to σs is a faithfulrepresentation of W .

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 4 / 50

Page 5: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Coxeter systemsThe Bruhat order

Coxeter graphs

Definition

The Coxeter graph Γ of (W,S) is a graph with vertex set S, thereis an edge between α, β ∈ S if m(α, β) ≥ 3. Label this edge bym(α, β).

For example, the Coxeter graphs of An, Bn are

An· · ·

s1 s2 s3 sn

Bn· · ·

s0 s1 s2 sn−1

4

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 5 / 50

Page 6: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Figure: Coxeter graphs of finite Coxeter groups

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 6 / 50

Page 7: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Figure: The simple Lie algebras

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 7 / 50

Page 8: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Coxeter systemsThe Bruhat order

Definition

Any element w ∈W can be written as

w = si1si2 · · · sir ,

for some sik ∈ S. If r is minimal, then r is called the length ofw, denoted `(w). Any expression of w which is a product of `(w)elements of S is called a reduced expression of w.

For example, let π = 4213 ∈ S4. Then

π = (1, 4, 3) = (1, 3) · (1, 4)

= (1, 2)(2, 3)(1, 2) · (1, 2)(2, 3)(3, 4)(2, 3)(1, 2)

= s1s2s1 · s1s2s3s2s1

= s1s3s2s1 = s3s1s2s1 = s3s2s1s2.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 8 / 50

Page 9: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Definition

For w ∈W , denote

DL(w) = {s ∈ S : `(sw) < `(w)},DR(w) = {s ∈ S : `(ws) < `(w)}.

DL(w) and DR(w) are called the left descent set and right descentset of w, respectively.

Definition

LetT (W ) = {wsw−1 : s ∈ S,w ∈W}.

The elements of T (W ) are called reflections of W . The elementsof S are also called simple reflections.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 9 / 50

Page 10: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Root systems

Definition

The root system Φ of (W,S) is the set of unit vectors in V permutedby W , that is,

Φ = w(αs), w ∈W, s ∈ S.

Note that Φ = −Φ since s(αs) = −αs. If α ∈ Φ is a root, then

α =∑s∈S

csαs, cs ∈ R.

Call α positive, α > 0, (resp. negative, α < 0) if all cs ≥ 0 (resp.all cs ≤ 0).

Φ = Φ+ ] Φ−.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 10 / 50

Page 11: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Theorem

Let w ∈ W and s ∈ S. If `(ws) > `(w), then w(αs) > 0. If`(ws) < `(w), then w(αs) < 0.

Proposition

(a) If s ∈ S, then s sends αs to its negative, but permutes theremaining positive roots.(b) For any w ∈W , `(w) is the number of positive roots sent by wto negative.

Theorem

Let w ∈W and α ∈ Φ+.

`(wsα) > `(w)⇔ w(α) > 0.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 11 / 50

Page 12: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Theorem (Strong Exchange Property)

Suppose w = s1s2 · · · sk(si ∈ S) and t ∈ T . If `(tw) < `(w), thentw = s1 · · · si · · · sk for some i ∈ [k].

Theorem (Exchange Property)

Let w = s1 · · · sk be a reduced expression and s ∈ S. If `(sw) <`(w), then sw = s1 · · · si · · · sk for some i ∈ [k].

Theorem (Deletion Property)

If w = s1s2 · · · sk, si ∈ S and `(w) < k,then w =s1 · · · si · · · sj · · · sk, for some 1 ≤ i < j ≤ k.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 12 / 50

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The Bruhat order

Definition

The Bruhat order on W is a partial order defined by u ≤ v if and onlyif v = ut1t2 · · · tm for some ti ∈ T (W ), such that for all i ∈ [m−1],

`(ut1 · · · titi+1) > `(ut1 · · · ti).

For example,

123

321

132

312

213

231

Figure: The Bruhat order of S3.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 13 / 50

Page 14: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Figure: The Bruhat order of S4.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 14 / 50

Page 15: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Definition

The Bruhat graph of (W,S) is a directed graph with vertex set Wand there is a directed edge from u to v if u < v and v = ut forsome t ∈ T .

For example,

123

132 213

231312

321

Figure: The Bruhat graph of S3

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 15 / 50

Page 16: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Theorem (Dyer, Invent. Math., 1993)

(Deodhar’s inequality) Let u, v ∈W , for any x ∈ [u, v], we have

#{t ∈ T |u ≤ xt ≤ v} ≥ `(v)− `(u).

History:

• First conjectured: Deodhar, Commun. Algebra, 1986.

• For Sn: Lakshmibai, Seshadri, Bull. AMS,1984.

• For crystallographic Coxeter groups: Carrell, Proc. Sym-pos. Pure Math., 1994. (torus action on intersectionsof Schubert varieties for Kac-Moody groups with dualSchubert varieties.)

• For finite Weyl groups: Polo, Indag. Mathem., N.S.,1994.(Zariski tangent spaces of Schubert varieties.)

• For general Coxeter groups: Dyer, nil-Hecke ring.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 16 / 50

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Let W J denote the set of minimal coset representatives modulo theparabolic subgroup WJ .

Theorem (Bjorner-Ekedahl, Ann. Math., 2009)

Let (W,S) be a crystallographic Coxeter group, J ⊆ S, and w ∈W J . Then

0 ≤ i < j ≤ `(w)− i =⇒ fw,Ji ≤ fw,Jj ,

where fw,Ji denote the number of elements of length i below w inthe Bruhat order on W J .

When W J = W , simplify fw,Ji to fwi . The equality fwi = fw`(w)−i ischaracterized in terms of vanishing of coefficients in the KL polyno-mial Pe,w(q). (cohomology of Kac-Moody Schubert varieties.)

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 17 / 50

Page 18: Coxeter groups, KL and R-polynomialsmath.sjtu.edu.cn/conference/Bannai/2013/data/20130716A/slides.pdf · Coxeter systems R-polynomials KL polynomials Coxeter systems The Bruhat order

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Figure: The Bruhat order of W J ,W = S5, J = {s2, s3}.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 18 / 50

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Bruhat order and Schubert varieties

A (complete) flag F• is a sequence of subspaces

{0} ⊆ F1 ⊂ F2 ⊂ · · · ⊂ Fn−1 ⊂ Fn = Cn,

with dimFi = i. The flag variety Fn is the set of flags of Cn.

Fn ∼= GL(n)/B,

where B is the group of invertible upper triangular matrices.(Given a matrix g ∈ GL(n), let F• with Fi being the span of thefirst i columns of g. Two matrices g and g′ represent the same flagiff g′ = gb for some b ∈ B.)

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 19 / 50

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Given w = w(1) · · ·w(n) ∈ Sn, define

rw(p, q) = #{i|1 ≤ i ≤ p, 1 ≤ w(i) ≤ q},∀p, q ∈ [n].

u ≤ w ∈ Sn ⇔ ru(p, q) ≥ rw(p, q),∀p, q ∈ [n].

Fix an ordered basis e1, . . . , en for Cn, and let E• be the flag whereEi = 〈e1, . . . , ei〉.

Definition

Given w ∈ Sn, the Schubert cell X◦w associated to w is the subsetof Fn consisting of the flags

{F•|dim(Fp ∩ Eq) = rw(p, q),∀p, q ∈ [n]}.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 20 / 50

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Definition

The Schubert variety Xw is the closure of X◦w, whose points corre-spond to the flags

{F•|dim(Fp ∩ Eq) ≥ rw(p, q),∀p, q}.

The Schubert variety Xw is a disjoint union of Schubert cells,

Xw =∐v≤w

X◦v ,

andu ≤ w ⇔ Xu ⊆ Xw.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 21 / 50

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In each Schubert cell X◦w there is a Schubert point

ew = 〈ew(1)〉 ⊂ 〈ew(1), ew(2)〉 ⊂ · · · ⊂ 〈ew(1), . . . , ew(n)〉.

Theorem (Lakshmibai-Seshadri, Bull. Amer. Math. Soc.,1984)

The Schubert variety Xw is smooth at ex iff

#{t ∈ T |x < xt ≤ w} = `(w)− `(x).

Theorem (Lakshmibai-Sandhya, Proc. Indian Acad. Sci., 1990)

The Schubert variety Xw is smooth at every point iff the permuta-tion w avoids the patterns 3412 and 4231.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 22 / 50

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DefinitionCombinatorial interpretationComputation

The Hecke algebra H

The Hecke algebra H of W over the ring of Laurent polynomialsF = Z[q

12 , q−

12 ] is a free F -module with basis {Tw : w ∈W}. That

is,H =

⊕w∈W

FTw

The multiplication on the basis is defined by

TsTw =

{Tsw, if s /∈ DL(w),

(q − 1)Tw + qTsw, if s ∈ DL(w).(1)

for all s ∈ S and w ∈W .

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 23 / 50

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DefinitionCombinatorial interpretationComputation

There exists inverse for every basis element Ts in H. Indeed, wehave

T−1s = q−1Ts − (1− q−1)Te.

If w = s1 · · · sr (reduced expression), then

Tw = Ts1 · · ·Tsr .

Therefore every Tw is invertible in H. The R-polynomials appear ascoefficients of the inverse of the elements in the standard basis ofthe Hecke algebra H.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 24 / 50

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DefinitionCombinatorial interpretationComputation

Theorem (Kazhdan-Lusztig, Invent. Math., 1979)

For all w ∈W , there exists a unique family {Ru, v(q)}u,v∈W ⊆ Z[q]of polynomials satisfying

(Tw−1)−1 = (−1)`(w)q−`(w)∑u≤w

(−1)`(u)Ru,w(q)Tu,

where Rw,w(q) = 1 and Ru, v(q) = 0 if u � v.

The inversion formula of R-polynomials:

Theorem (Kazhdan-Lusztig, Invent. Math., 1979)∑u≤w≤v

(−1)`(w)−`(u)Ru,w(q)Rw,v(q) = δu,v.

Combinatorial proof?

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 25 / 50

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DefinitionCombinatorial interpretationComputation

Theorem (Kazhdan-Lusztig, Invent. Math., 1979)

Let u, v ∈W and s ∈ DR(v). Then

Ru, v(q) =

{Rus, vs(q), if s ∈ DR(u),

qRus, vs(q) + (q − 1)Ru, vs(q), if s /∈ DR(u),

where Ru, u(q) = 1 and Ru, v(q) = 0 if u � v.

Theorem (Deodhar, Invent. Math., 1985)

Let F be a finite field of order p and u, v ∈ Sn. Then

Ru,v(p) = |X◦v ∩ (X◦v )∗|,

where (X◦v )∗ is the Schubert cell opposite to X◦v .

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 26 / 50

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DefinitionCombinatorial interpretationComputation

Combinatorial interpretations of the R-polynomials

Because of the presence of the factor q−1, the coefficients of the R-polynomials are not necessarily positive. To study the R-polynomialscombinatorially, the R-polynomials are introduced.

Theorem (Dyer, Compos. Math., 1993)

Let u, v ∈ W . Then there exists a unique polynomial Ru, v(q) ∈N(q) such that

Ru, v(q) = q`(v)−`(u)

2 Ru, v(q12 − q−

12 ).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 27 / 50

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DefinitionCombinatorial interpretationComputation

Theorem (Dyer, Compos. Math., 1993)

Let u, v ∈W . Then we have

(i) Ru, v(q) = 0, if u � v;

(ii) Ru, v(q) = 1, if u = v;

(iii) If s ∈ DR(v) then

Ru,v(q) =

{Rus, vs(q), if s ∈ DR(u),

Rus, vs(q) + qRu, vs(q), if s /∈ DR(u).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 28 / 50

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DefinitionCombinatorial interpretationComputation

Theorem (Dyer, Compos. Math., 1993)

Let ≺ be a reflection ordering, and u, v ∈W,u ≤ v. Then,

Ru, v(q) =∑∆

q`(∆),

where ∆ ranges over all the increasing directed paths from u to vin the Bruhat graph of W and `(∆) is the length of the path ∆.

Combinatorial proof of the inversion formula of R-polynomials.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 29 / 50

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DefinitionCombinatorial interpretationComputation

Explicit formulas of the R-polynomials

Theorem (Brenti, Invent. Math., 1994)

Let u, v ∈W,u < v. Then the following are equivalent:

(i) Ru, v(q) = (q − 1)`(v)−`(u);

(ii) [u, v] does not contain any interval isomorphic to S3.

Theorem (Marietti, European J. Combin., 2002)

Given a Coxeter system (W,S), let w = s1 · · · sn−1snsn−1 · · · s1 bea reduced expression such that si 6= sj if i 6= j for all i, j ∈ [n]. Ifu, v ∈W and u ≤ v ≤ w, then there exists a ∈ N such that

Ru, v(q) = (q − 1)a(q2 − q + 1)`(v)−`(u)−a

2 .

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 30 / 50

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DefinitionCombinatorial interpretationComputation

For Sn, R-polynomials do not factor in general. For example,

R12345, 54321(q) = q2(1 + 5q2 + 10q4 + 6q6 + q8),

R123456, 654321(q) = q3(1 + 9q2 + 39q4 + 57q6 + 36q8 + 10q10 + q12).

Let σ = σ1 · · ·σn ∈ Sn and i, j ∈ [n]. Denote by

Ci,j(σ) = {(σi1 , . . . , σik) : k ∈ [n], i = i1 < · · · < ik = j

and σi1 < · · · < σik},

C (σ) =⋃

1≤i<j≤nCi,j(σ).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 31 / 50

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DefinitionCombinatorial interpretationComputation

Theorem (Brenti, Adv. Math., 1997)

Let σ ∈ Sn and w ∈ C (σ). Then we have

Rσ,wσ(q) = qk(w)−1(q2 + 1)d−k(w)+1

2 ,

where d = inv(wσ)− inv(σ) and k(w) is the length of the cycle w.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 32 / 50

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Theorem (Pagliacci, J. Combin. Theory, Series A, 2001)

Let u = u1 · · ·un ∈ Sn, 1 ≤ i < k < l < j ≤ n and

ui < ul < uk < uj .

Suppose that v = u(i, j)(k, l). Then

Ru, v(q) = q4(q2 + 1)inv(v)−inv(u)−4

2 .

For example, letu = 1 5 2 4 3 6,

v = 6 4 2 5 3 1.

Then inv(u) = 4, inv(v) = 12.

Ru, v(q) = q4(q2 + 1)2.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 33 / 50

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Definition of KL polynomials

Define ι : H → H by

ι(q12 ) = q−

12 and ι(Tw) = (Tw−1)−1.

ι is a ring automorphism, which is clearly an involution. Let

C ′s = q−12 (Ts + Te).

Since T−1s = q−1Ts − (1− q−1)Te, we have ι(C ′s) = C ′s.

Theorem (Kazhdan-Lusztig, Invent. Math., 1979)

There exists a unique basis C = {C ′w : w ∈W} of H such that

(i) ι(C ′w) = C ′w;

(ii) C ′w = q−`(w)

2∑

u≤w Pu,w(q)Tu.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 34 / 50

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Theorem (Kazhdan-Lusztig, Invent. Math., 1979)

Let (W,S) be a Coxeter system. Then there is a unique family ofpolynomials {Pu, v(q)}u,v∈W ⊆ Z[q] satisfying the following condi-tions:

(i) Pu, v(q) = 0 if u � v;

(ii) Pu, u(q) = 1;

(iii) deg(Pu, v(q)) ≤ 12(`(v)− `(u)− 1), if u < v;

(iv) If u ≤ v, then

q`(v)−`(u)Pu, v(q−1) =

∑u≤z≤v

Ru, z(q)Pz, v(q).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 35 / 50

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Theorem (Kazhdan-Lusztig, Invent. Math., 1979)

Let (W,S) be a Coxeter system, u ≤ v, and s ∈ DR(v). Then

Pu, v(q) = q1−cPus, vs(q)+qcPu, vs(q)−

∑z:s∈DR(z)

q`(u,v)

2 µ(z, vs)Pu, z(q),

where µ(u, v) is the coefficient of q12

(`(u,v)−1) in Pu, v(q) and

c =

{1, if s ∈ DR(u);

0, if s /∈ DR(u).

Corollary (Kazhdan-Lusztig, Invent. Math., 1979)

Let u, v ∈W,u ≤ v. If s ∈ DR(v), then Pu,v(q) = Pus,v(q).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 36 / 50

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Connection with Schubert varieties

Let IH∗(Xv,C)X◦u be the local intersection cohomology of Xv at apointX◦u. This is a graded vector space, and denote by IH i(Xv,C)X◦u(i ∈ N) its graded pieces.

Theorem (Kazhdan-Lusztig, Proc. Sympos. Pure Math., 1980)

Let W be a Weyl group and u, v ∈W,u ≤ v. Then

Pu,v(q) =∑i≥0

qidimC(IH2i(Xv)X◦u).

Theorem (Elias-Williamson, arXiv:1212.0791)

Let W be an arbitrary Coxeter group. The coefficients of Pu,v(q)are nonnegative for all u, v ∈W .

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 37 / 50

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Theorem (Carrell, Proc. Sympos. Pure Math., 1994)

Let u, v ∈W . Then the following statements are equivalent:

(i) Pu,v(q) = 1;

(ii) Px,v(q) = 1 for all x ∈ [u, v];

(iii) |{t ∈ T |x < xt ≤ v}| = `(v)− `(x) for all x ∈ [u, v].

Theorem (Lakshmibai-Sandhya, Proc. Indian Acad. Sci., 1990)

Let v ∈ Sn. Then

Pe,v(q) = 1⇔ Xv is smooth⇔ v avoids 3412 and 4231.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 38 / 50

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DefinitionCombinatorial interpretationsComputation

Combinatorial interpretations of KL polynomials

For j ∈ Q, let

Uj

(∑i∈Z

aiqi

)=∑i≥j

aiqi.

For any composition β = β1, . . . , βs, βi ∈ N, define γβ1,...,βs as−1, β1 = 1, s = 1;

(q − 1)U |β|2

(q|β|−1γβ2,...,βs(

1q )), β1 = 1, s ≥ 1;

(q − 1)γβ′(q) + U |β|+12

(q|β|−1(1− q)γβ′(1

q )), β1 > 1.

where β′ = β1 − 1, . . . , βs and |β| = β1 + · · ·+ βs.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 39 / 50

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Denote by B(u, v) the set of direct Bruhat graphs from u to v. Ify = x(i, j) and x < y, label this direct edge (x, y) by λ(x, y) =(i, j). For a direct path Γ = (x0, x1, . . . , xn), say i is a descent ifλ(xi−1, xi) > λ(xi, xi+1). Let

D(Γ) = (β1, . . . , βs),

if Γ has s− 1 descents and βi is the length of path from xi−1 to xi.

Theorem (Brenti, J. London Math. Soc., 1997)

Let u < v ∈ Sn. Then

Pu,v(q) = (−1)`(u,v)∑

Γ

q`(u,v)−|D(Γ)|

2 γD(Γ)(q),

where the sum is over all directed paths in B(u, v).

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 40 / 50

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There are 22 paths, for Γ = 15243→ 15342→ 35142→ 35241→45231, D(Γ) = (1, 2, 1). Pu,v(q) = qγ(2) + qγ(1,1) + γ(4) + 2γ(1,3) +4γ(2,2) + 3γ(3,1) + 3γ(1,1,2) + 4γ(1,2,1) + 2γ(2,1,1) + γ(1,1,1,1).Since γ(1,1) = −q2+q, γ(2) = q2−q+1, . . . . We have Pu,v(q) = 1+q.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 41 / 50

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Let a, b ∈ Z, a ≤ b. By a lattice path on [a, b] we mean a functionΓ : [a, b]→ Z such that Γ(a) = 0 and

|Γ(i+ 1)− Γ(i)| = 1,

for all i ∈ [a, b− 1]. Let

N(Γ) = {i ∈ [a+ 1, b− 1] : Γ(i) < 0},d+(Γ) = |{i ∈ [a, b− 1] : Γ(i+ 1)− Γ(i) = 1}|,`(Γ) = b− a, and Γ≥0 = `(Γ)− 1− |N(Γ)|.

We call N(Γ) the negative set of Γ and `(Γ) the length of Γ, and

d+(Γ) =Γ(b) + b− a

2.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 42 / 50

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DefinitionCombinatorial interpretationsComputation

For example,

N(Γ) = {3, 4, 5}, d+(Γ) = 2, `(Γ) = 6,Γ≥0 = 2.

Given a path ∆ = (a0, a1, . . . , ar) in the Bruhat graph of W , definethe length of ∆ to be `(∆) = r, its descent set w.r.t. a givenreflection order ≺ to be

D(∆) = {i ∈ [r − 1] : ai(ai−1)−1 � ai+1(ai)−1}.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 43 / 50

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Theorem (Brenti, J. Amer. Math. Soc., 1998)

Let u, v ∈W,u < v. Then

Pu,v(q) =∑

(Γ,∆)

(−1)Γ≥0+d+(Γ)q`(u,v)+Γ(`(Γ))

2 ,

where the sum is over all pairs (Γ,∆) such that Γ is a lattice path,∆ is a Bruhat path from u to v, `(Γ) = `(∆), N(Γ) = `(∆) −D(∆), and Γ(`(Γ)) < 0.

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Explicit formulas

Theorem (Brenti-Simion, J. Algebraic Combin., 2000)

For n ≥ 5, we have

Pe,34···n12(q) = Fn−2(q) =

b(n−2)/2c∑i=0

(n− i− 2

i

)qi,

Pe,34···n−2nn−1 12(q) = Fn−3(q).

Theorem (Marietti, J. Algebra, 2006)

Let u ≤ v ≤ (1, n+ 1) ∈ Sn+1. Then

Pu,v(q) = (1 + q)b,

where b = |{k ∈ [n] : vk = 2, vk+1 = 2, uk+1 = 0}|.Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 45 / 50

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Theorem (Woo, Electronic J. Combin., 2009)

Suppose the singular locus of Xw has exactly one irreducible compo-nent, and w avoids the patterns 653421, 632541, 463152, 526413,546213, and 465132. Then

Pe,w(q) = 1 + qh,

where h is the minimum height of a 3412 embedding, with h = 1 ifno such embedding exists.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 46 / 50

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Combinatorial invariance conjecture

Conjecture (Lusztig-Dyer)

Let u, v, τ, σ ∈W be such that [u, v] ∼= [τ, σ] (as posets). Then

Pu,v(q) = Pτ,σ(q).

Importance:

• equality of certain intersection cohomology vector spaces

• inclusion relations of Schubert varieties Xu and Xv

• It is thought that intersection homology is a deeper propertythan adjacency of Schubert cells.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 47 / 50

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Theorem (Cloux, Adv. Math., 2003)

Let W,W ′ be two Coxeter groups, w ∈ W and w′ ∈ W ′ such that[e, w] ∼= [e, w′]. Then any poset isomorphism φ : [e, w] → [e, w′]preserves KL polynomials, in the following cases:

(i) the Coxeter graph of one of the two groups W,W ′ is a tree;

(ii) one of the two groups W,W ′ is affine of type An;

(iii) each irreducible component of one of the two groups W,W ′ isof one of types (i),(ii) above.

In particular, the result holds in all cases where one of the two groupsW,W ′ is finite or affine.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 48 / 50

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Theorem (Brenti-Caselli-Marietti, Adv. Math., 2006)

Let v ∈W , and M be a special matching of [e, v]. Then

Ru,v(q) =

{RM(u),M(v)(q), if M(u) C u;qRM(u),M(v)(q) + (q − 1)Ru,M(v)(q), if M(u) B u,

for all u ∈ [e, v]. So the polynomials Rx,y(q) (x, y ∈ [e, v]) dependonly on [e, v] as abstract poset.

Theorem (Incitti, J. Combin. Theory, Ser. A, 2007)

The KL polynomials are combinatorial invariants for intervals up tolength 6 in Coxeter groups of type B and D and for intervals up tolength 8 in Coxeter groups of type A.

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 49 / 50

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Thank You !

Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 50 / 50