coxeter groups, kl and...
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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
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
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Coxeter systemsThe Bruhat order
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).
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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 .
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Coxeter systemsR-polynomials
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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
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Coxeter systemsThe Bruhat order
Figure: Coxeter graphs of finite Coxeter groups
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Figure: The simple Lie algebras
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Coxeter systemsR-polynomials
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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.
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Coxeter systemsR-polynomials
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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.
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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−.
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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.
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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.
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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.
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Coxeter systemsThe Bruhat order
Figure: The Bruhat order of S4.
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Coxeter systemsR-polynomials
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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
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Coxeter systemsThe Bruhat order
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
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Figure: The Bruhat order of W J ,W = S5, J = {s2, s3}.
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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.)
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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]}.
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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.
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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 .
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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.
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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?
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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 .
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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).
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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 .
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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).
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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.
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DefinitionCombinatorial interpretationComputation
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.
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DefinitionCombinatorial interpretationsComputation
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.
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DefinitionCombinatorial interpretationsComputation
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).
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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).
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DefinitionCombinatorial interpretationsComputation
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 .
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DefinitionCombinatorial interpretationsComputation
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.
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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.
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DefinitionCombinatorial interpretationsComputation
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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DefinitionCombinatorial interpretationsComputation
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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DefinitionCombinatorial interpretationsComputation
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.
Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 44 / 50
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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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DefinitionCombinatorial interpretationsComputation
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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DefinitionCombinatorial interpretationsComputation
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
Coxeter systemsR-polynomials
KL polynomials
DefinitionCombinatorial interpretationsComputation
Thank You !
Neil J.Y. Fan (Department of Math, SCU) Coxeter groups, KL and R-polynomials July 16, 2013 50 / 50