Flag Enumeration in Polytopes Eulerian Partially Ordered Sets and

Flag Enumeration in Polytopes Eulerian PartiallyOrdered Sets and Coxeter Groups

Louis BilleraCornell University, Ithaca, NY 14853 USA

ICM - Hyderabad, August 2010

Introduction: Face Enumeration in Convex PolytopesSimplicial polytopesCounting flags in polytopes

Eulerian Posets and the cd-indexFlags in graded and Eulerian posetsInequalities for flags in polytopes and spheres

Algebraic Approches to Counting FlagsConvolution productQuasisymmetric function of a posetPeak functions and Eulerian posets

Bruhat Intervals in Coxeter GroupsR-polynomials and Kazhdan-Lusztig polynomialsComplete quasisymmetric function; complete cd-indexKazhdan-Lusztig polynomial from the complete cd-index

Epilog: Combinatorial Hopf Algebras

f -vectors of polytopes

For a d-dimensional convex polytope Q, letfi = fi (Q) = the number of i-dimensional faces of Q

f0 = the number of verticesf1 = the number of edges...fd−1 = the number of facets (defining inequalities)

The f -vector of Q: f (Q) = (f0, f1, . . . , fd−1)

Problem: Determine when a vector f = (f0, f1, . . . , fd−1)is f (Q) for some d-polytope Q.

d = 2: Exercised = 3: Steinitz (1906)d ≥ 4: open
























d = 2: Exercise

d = 3: Steinitz (1906)d ≥ 4: open






d = 2: Exercised = 3: Steinitz (1906)

d ≥ 4: open







Simplicial polytopes

A polytope is simplicial if all faces are simplices (equiv: vertices arein general position)

The h-vector (h0, . . . , hd) of a simplicial d-polytope is defined bythe polynomial relation

d∑i=0

hixd−i =

d∑i=0

fi−1(x − 1)d−i

The corresponding g -vector (g0, . . . , gbd/2c) is defined by g0 = 1and gi = hi − hi−1, for i ≥ 1.




d∑i=0

hixd−i =

d∑i=0

fi−1(x − 1)d−i





d∑i=0

hixd−i =

d∑i=0

fi−1(x − 1)d−i


The g -Theorem

Theorem(BL/S,1980): (h0, h1, . . . , hd) is the h-vector of asimplicial convex d-polytope if and only if

for all i ,

hi = hd−i (Dehn-Sommerville equations)

and gi = hi − hi−1 satisfy, for 0 ≤ i ≤⌊

d2

⌋,

gi ≥ 0 (Generalized Lower Bound Thm)

and for i ≥ 1

gi+1 ≤ g〈i〉i (Macaulay-McMullen conditions)

The g -Theorem

Theorem(BL/S,1980): (h0, h1, . . . , hd) is the h-vector of asimplicial convex d-polytope if and only if for all i ,



d2

⌋,


and for i ≥ 1


The g -Theorem




d2

⌋,


and for i ≥ 1


The g -Theorem




d2

⌋,


and for i ≥ 1


General polytopes

For general convex polytopes, the situation for f -vectors is muchless satisfactory.

I The only equation they all satisfy is the Euler relation

f0 − f1 + f2 − · · · ± fd−1 = 1− (−1)d

I Already in d = 4, we do not know all linear inequalities onf -vectors.

I There is little hope at this point of giving an analog to theMacaulay-McMullen conditions.

A possible approach is to try to solve a harder problem: count notfaces, but chains of faces.

General polytopes



f0 − f1 + f2 − · · · ± fd−1 = 1− (−1)d




General polytopes



f0 − f1 + f2 − · · · ± fd−1 = 1− (−1)d




General polytopes



f0 − f1 + f2 − · · · ± fd−1 = 1− (−1)d




General polytopes



f0 − f1 + f2 − · · · ± fd−1 = 1− (−1)d




Flag f -vectors of polytopes

For a d-dimensional polytope Q and a set S of possibledimensions, define fS(Q) to be the number of chains of faces of Qhaving dimensions prescribed by the set S .

The functionS 7→ fS(Q)

is called the flag f -vector of Q.

I It includes the f -vector, by counting chains of one element:(fS : |S | = 1).











Face lattices of polytopes

The best setting in which to study the flag f -vector of ad-polytope Q is that of its lattice of faces P = F(Q), a gradedposet of rank d + 1

Q

e

��EEEEEEE

@@@@@

a b

cd

��

��

P = F(Q)

ab

��

...........................................................................

ccc

��

AAA

eee

��

��

SSS

llll

��

��

AAA

@@@

"""

""

BBB

��

##

##

JJJ

SS

bbbb@@

bbbb .............

.............................................................................

,,,bbbb��

��Z

ZZ

@@%%%

HHHH

HHHHH

lll�

��BBB@

@@��

��

HHHH"

"""

abcd abe bce cde ade

bc cd ad ae be ce de

a b c d

∅

e

Q��

��



Q

e

��EEEEEEE

@@@@@

a b

cd

��

��

P = F(Q)

ab

��

...........................................................................

ccc

��

AAA

eee

��

��

SSS

llll

��

��

AAA

@@@

"""

""

BBB

��

##

##

JJJ

SS

bbbb@@

bbbb .............

.............................................................................

,,,bbbb��

��Z

ZZ

@@%%%

HHHH

HHHHH

lll�

��BBB@

@@��

��

HHHH"

"""



a b c d

∅

e

Q��

��



Q

e

��EEEEEEE

@@@@@

a b

cd

��

��

P = F(Q)

ab

��

...........................................................................

ccc

��

AAA

eee

��

��

SSS

llll

��

��

AAA

@@@

"""

""

BBB

��

##

##

JJJ

SS

bbbb@@

bbbb .............

.............................................................................

,,,bbbb��

��Z

ZZ

@@%%%

HHHH

HHHHH

lll�

��BBB@

@@��

��

HHHH"

"""



a b c d

∅

e

Q��

��

Flag f -vectors of graded posets

P a graded poset (with 0̂ and 1̂) of rank n + 1, with rank function ρ

Flag f -vector is the function S 7→ fS = fS(P), where forS = {i1, . . . , ik} ⊂ [n] := {1, . . . , n},

fS = #{y1 < y2 < · · · < yk | yj ∈ P, ρ(yj) = ij}

P is Eulerian if it is graded and for all x < y ∈ P, the number ofelements of even rank in [x , y ] = number of elements of odd rank.

• Face posets of polytopes and spheres are Eulerian.




fS = #{y1 < y2 < · · · < yk | yj ∈ P, ρ(yj) = ij}






fS = #{y1 < y2 < · · · < yk | yj ∈ P, ρ(yj) = ij}






fS = #{y1 < y2 < · · · < yk | yj ∈ P, ρ(yj) = ij}



The cd-index for Eulerian posets

For S ⊂ [n] let the flag h-vector be defined by

hS =∑T⊂S

(−1)|S |−|T |fT

and for noncommuting indeterminates a and b letuS = u1u2 · · · un, where

ui =

{b if i ∈ S

a if i /∈ S

Then for Eulerian posets, the generating function

ΨP =∑S

hS(P)uS

is always a polynomial in c and d, where c = a + b andd = ab + ba. This polynomial ΦP(c,d) is called the cd-index of P.



hS =∑T⊂S

(−1)|S |−|T |fT


ui =

{b if i ∈ S

a if i /∈ S


ΨP =∑S

hS(P)uS




hS =∑T⊂S

(−1)|S |−|T |fT


ui =

{b if i ∈ S

a if i /∈ S


ΨP =∑S

hS(P)uS

is always a polynomial in c and d, where c = a + b andd = ab + ba.

This polynomial ΦP(c,d) is called the cd-index of P.



hS =∑T⊂S

(−1)|S |−|T |fT


ui =

{b if i ∈ S

a if i /∈ S


ΨP =∑S

hS(P)uS


An example: The Boolean algebra B3

Ex. For P = B3 = 2[3],

cu u uuuu

u��

��

HHHH

HHH

@@@@

��

@@@@�

��

�� S

SSSSSS

{a, b, c}

{a, c} {a, b} {b, c}

{a} {c}

∅

{b}= faces of

a b

u

f∅ = 1, f{1} = 3, f{2} = 3, f{1,2} = 6 soh∅ = 1, h{1} = 2, h{2} = 2, h{1,2} = 1 and so

ΨP = aa + 2ba + 2ab + bb= (a + b)2 + (ab + ba)= c2 + d = ΦP


Ex. For P = B3 = 2[3],

cu u uuuu

u��

��

HHHH

HHH

@@@@

��

@@@@�

��

�� S

SSSSSS

{a, b, c}

{a, c} {a, b} {b, c}

{a} {c}

∅

{b}= faces of

a b

u

f∅ = 1, f{1} = 3, f{2} = 3, f{1,2} = 6 so

h∅ = 1, h{1} = 2, h{2} = 2, h{1,2} = 1 and so



Ex. For P = B3 = 2[3],

cu u uuuu

u��

��

HHHH

HHH

@@@@

��

@@@@�

��

�� S

SSSSSS

{a, b, c}

{a, c} {a, b} {b, c}

{a} {c}

∅

{b}= faces of

a b

u




Ex. For P = B3 = 2[3],

cu u uuuu

u��

��

HHHH

HHH

@@@@

��

@@@@�

��

�� S

SSSSSS

{a, b, c}

{a, c} {a, b} {b, c}

{a} {c}

∅

{b}= faces of

a b

u



Inequalities on cd coefficients

If P has rank n + 1 then the degree of ΦP(c,d) is n (deg c = 1,deg d = 2).

There are Fibonacci many cd words of degree n. WriteΦP =

∑w [w ]P w over cd-words w .

I Stanley: [w ]P ≥ 0 for polytopes (S-shellable CW spheres)

I Karu: [w ]P ≥ 0 for all Gorenstein∗ posets (includes all faceposets of regular CW -spheres)

I B , Ehrenborg & Readdy: Among all n-dimensional zonotopes(resp. hyperplane arrangements), the cd-index is termwiseminimized on the n-cube Cn (resp. coordinate arrangement).

I B & Ehrenborg: Among all n-dimensional polytopes, thecd-index is termwise minimized on the n-simplex ∆n.

I Karu & Ehrenborg: The simplex (Boolean lattice) minimizestermwise among all Gorenstein* lattices of the same rank.


If P has rank n + 1 then the degree of ΦP(c,d) is n (deg c = 1,deg d = 2). There are Fibonacci many cd words of degree n. WriteΦP =















































Convolutions of flag f -vectors

Notation: Write f(n)S , S ⊂ [n − 1], when counting chains in a poset

of rank n.

Kalai: Given f(n)S and f

(m)T , S ⊂ [n − 1], T ⊂ [m − 1] and P a

poset of rank n + m, define f(n)S ∗ f

(m)T by

f(n)S ∗ f

(m)T (P) =

∑x∈P : r(x)=n

f(n)S ([0̂, x ]) · f (m)

T ([x , 1̂])

Note: Flag numbers are closed under convolution

f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

where T + n := {x + n | x ∈ T}



of rank n.


(m)T , S ⊂ [n − 1], T ⊂ [m − 1] and P a


(m)T

by

f(n)S ∗ f

(m)T (P) =

∑x∈P : r(x)=n

f(n)S ([0̂, x ]) · f (m)

T ([x , 1̂])


f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

where T + n := {x + n | x ∈ T}



of rank n.


(m)T , S ⊂ [n − 1], T ⊂ [m − 1] and P a


(m)T by

f(n)S ∗ f

(m)T (P) =

∑x∈P : r(x)=n

f(n)S ([0̂, x ]) · f (m)

T ([x , 1̂])


f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

where T + n := {x + n | x ∈ T}



of rank n.


(m)T , S ⊂ [n − 1], T ⊂ [m − 1] and P a


(m)T by

f(n)S ∗ f

(m)T (P) =

∑x∈P : r(x)=n

f(n)S ([0̂, x ]) · f (m)

T ([x , 1̂])


f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

where T + n := {x + n | x ∈ T}

Convolution preserves equations

If F =∑αS f

(n)S and F (P) = 0 for all rank n polytopes (graded

posets, Eulerian posets) P, then f(k)S ∗ F ∗ f

(m)T also vanishes, i.e.

f(k)S ∗ F ∗ f

(m)T (P ′) = 0

for rank k + n + m polytopes (graded posets, Eulerian posets) P ′

Polytopes of dimension d − 1 (Eulerian posets of rank d) satisfythe Euler relations:

f(d)∅ − f

(d){1} + f

(d){2} − · · ·

· · ·+ (−1)d−1f(d){d−1} + (−1)d f

(d)∅ = 0

Thm (Bayer & B ): All linear relations on the f(d)S for polytopes,

and so for Eulerian posets, come from these via convolution.


If F =∑αS f




f(k)S ∗ F ∗ f

(m)T (P ′) = 0



f(d)∅ − f

(d){1} + f

(d){2} − · · ·

· · ·+ (−1)d−1f(d){d−1} + (−1)d f

(d)∅ = 0




If F =∑αS f




f(k)S ∗ F ∗ f

(m)T (P ′) = 0



f(d)∅ − f

(d){1} + f

(d){2} − · · ·

· · ·+ (−1)d−1f(d){d−1} + (−1)d f

(d)∅ = 0



Subsets ←→ Compositions

Let [n] := {1, . . . , n}. β = (β1, . . . , βk) is a composition of n + 1(written β |= n + 1) if each βi > 0, and β1 + · · ·+ βk = n + 1.

β = (β1, . . . , βk) |= n + 1 e.g. (1, 2, 1, 3) |= 7

l

S(β) := {β1, β1 + β2, . . . , β1 + · · ·+ βk−1} ⊂ [n] {1, 3, 4} ⊂ [6]

Subsets ←→ Compositions

Let [n] := {1, . . . , n}. β = (β1, . . . , βk) is a composition of n + 1(written β |= n + 1) if each βi > 0, and β1 + · · ·+ βk = n + 1.

β = (β1, . . . , βk) |= n + 1 e.g. (1, 2, 1, 3) |= 7

l

S(β) := {β1, β1 + β2, . . . , β1 + · · ·+ βk−1} ⊂ [n] {1, 3, 4} ⊂ [6]

Enumeration algebras

Let A = Q〈y1, y2, . . . 〉 = A0 ⊕ A1 ⊕ A2 · · · be the free associativealgebra on noncommuting yi , deg(yi ) = i .

Via the association yk ←→ f k∅ and so

yβ := yβ1 · · · yβkβ = (β1, . . . , βk) |= n

l

fS(β) = fS S ⊂ [n − 1]

multiplication in A is the analogue of Kalai’s convolution of flagf -vectors, in which

f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)



Via the association yk ←→ f k∅

and so

yβ := yβ1 · · · yβkβ = (β1, . . . , βk) |= n

l

fS(β) = fS S ⊂ [n − 1]


f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)




yβ := yβ1 · · · yβkβ = (β1, . . . , βk) |= n

l

fS(β) = fS S ⊂ [n − 1]


f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)




yβ := yβ1 · · · yβkβ = (β1, . . . , βk) |= n

l

fS(β) = fS S ⊂ [n − 1]


f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

Euler elements of An

F ∈ An ←→ functionals on graded posets of rank n,

i.e., expressions of the form∑

S⊂[n−1] αS f(n)S .

Ex. As an element of A4

2y4 − y1y3 + y2y2 − y3y1 = 2f(4)∅ − f

(4){1} + f

(4){2} − f

(4){3}

the Euler relation for posets of rank 4.

For k ≥ 1 define in Ak

χk :=∑

i+j=k

(−1)iyiyj =k∑

i=0

(−1)i f(k)i ,

the kth Euler relation




S⊂[n−1] αS f(n)S .


2y4 − y1y3 + y2y2 − y3y1 = 2f(4)∅ − f

(4){1} + f

(4){2} − f

(4){3}



χk :=∑

i+j=k

(−1)iyiyj =k∑

i=0

(−1)i f(k)i ,





S⊂[n−1] αS f(n)S .


2y4 − y1y3 + y2y2 − y3y1 = 2f(4)∅ − f

(4){1} + f

(4){2} − f

(4){3}



χk :=∑

i+j=k

(−1)iyiyj =k∑

i=0

(−1)i f(k)i ,


Eulerian Enumeration Algebra

LetIE = 〈χk : k ≥ 1〉 ⊂ A

be the 2-sided ideal of all relations on Eulerian posets and

AE = A/IE

be the algebra of functionals on Eulerian posets.

Theorem (B. & Liu): As graded algebras,

AE ∼= Q〈y1, y3, y5, . . . 〉


LetIE = 〈χk : k ≥ 1〉 ⊂ A


AE = A/IE



AE ∼= Q〈y1, y3, y5, . . . 〉


LetIE = 〈χk : k ≥ 1〉 ⊂ A


AE = A/IE



AE ∼= Q〈y1, y3, y5, . . . 〉


LetIE = 〈χk : k ≥ 1〉 ⊂ A


AE = A/IE



AE ∼= Q〈y1, y3, y5, . . . 〉

Quasisymmetric functions

Let QSym ⊂ Q[[x1, x2, . . . ]] the algebra of quasisymmetricfunctions

QSym := QSym0 ⊕QSym1 ⊕ · · ·

where QSymn := span{Mβ | β = (β1, . . . , βk) |= n} and

Mβ :=∑

i1<i2<···<ik

xβ1

i1xβ2

i2· · · xβk

ik

Ex. (1, 2, 1) |= 4←→ {1, 3} ⊂ [3] and so

M(1,2,1) = M(4){1,3} =

∑i1<i2<i3

x1i1x

2i2x

1i3





Mβ :=∑

i1<i2<···<ik

xβ1

i1xβ2

i2· · · xβk

ik

Ex. (1, 2, 1) |= 4←→ {1, 3} ⊂ [3] and so

M(1,2,1) = M(4){1,3} =

∑i1<i2<i3

x1i1x

2i2x

1i3





Mβ :=∑

i1<i2<···<ik

xβ1

i1xβ2

i2· · · xβk

ik

Ex. (1, 2, 1) |= 4←→ {1, 3} ⊂ [3] and so

M(1,2,1) = M(4){1,3} =

∑i1<i2<i3

x1i1x

2i2x

1i3

Quasisymmetric function of a graded poset

For a finite graded poset P, with rank function ρ(·), define theformal power series (Ehrenborg)

F (P) :=∑

0̂=u0≤···≤uk−1<uk=1̂

xρ(u0,u1)1 x

ρ(u1,u2)2 · · · xρ(uk−1,uk )

k

where the sum is over all finite multichains in P whose last twoelements are distinct and ρ(x , y) = ρ(y)− ρ(x).

I F (P) ∈ QSym, in fact F (P) ∈ QSymn where n = ρ(P)

I F (P) =∑

α fαMα where fα is the flag f -vector of P



F (P) :=∑

0̂=u0≤···≤uk−1<uk=1̂

xρ(u0,u1)1 x

ρ(u1,u2)2 · · · xρ(uk−1,uk )

k



I F (P) =∑




F (P) :=∑

0̂=u0≤···≤uk−1<uk=1̂

xρ(u0,u1)1 x

ρ(u1,u2)2 · · · xρ(uk−1,uk )

k



I F (P) =∑


Peak subalgebra Π

For a cd-word w of degree n, w = cn1dcn2d · · · cnkdcm, let

Iw = {{i1 − 1, i1}, {i2 − 1, i2}, . . . , {ik − 1, ik}}

where ij = deg(cn1dcn2d · · · cnj d) (deg c = 1, deg d = 2) and

b[Iw ] ={S ⊆ [n]

∣∣ S ∩ I 6= ∅,∀I ∈ Iw}

The peak algebra Π is defined to be the subalgebra of QSymgenerated by the peak quasisymmetric functions

Θw =∑

T∈b[Iw ]

2|T |+1M(n+1)T

where w is any cd-word.

Peak subalgebra Π

For a cd-word w of degree n, w = cn1dcn2d · · · cnkdcm, let

Iw = {{i1 − 1, i1}, {i2 − 1, i2}, . . . , {ik − 1, ik}}

where ij = deg(cn1dcn2d · · · cnj d) (deg c = 1, deg d = 2) and

b[Iw ] ={S ⊆ [n]

∣∣ S ∩ I 6= ∅,∀I ∈ Iw}

The peak algebra Π is defined to be the subalgebra of QSymgenerated by the peak quasisymmetric functions

Θw =∑

T∈b[Iw ]

2|T |+1M(n+1)T

where w is any cd-word.

Eulerian posets and peak functions

Theorem(BMSvW): If P is an Eulerian poset then F (P) ∈ Π.

Proof involves showing Π and AE are dual as Hopf algebras, withcoproducts ∆(Mβ) =

∑β=β1·β2

Mβ1 ⊗Mβ2 and∆(yk) =

∑i+j=k yi ⊗ yj

Theorem (B HvW): If P is any Eulerian poset, then

F (P) =∑w

1

2|w |d+1[w ]P Θw

where the [w ]P are the coefficients of the cd-index of P and |w |dis the number of d’s in w .

Note: This could serve as a definition for the cd-index of P.




∑β=β1·β2


∑i+j=k yi ⊗ yj


F (P) =∑w

1

2|w |d+1[w ]P Θw






∑β=β1·β2


∑i+j=k yi ⊗ yj


F (P) =∑w

1

2|w |d+1[w ]P Θw






∑β=β1·β2


∑i+j=k yi ⊗ yj


F (P) =∑w

1

2|w |d+1[w ]P Θw



Coxeter Groups

A Coxeter group is a group W generated by a set S with therelations

I s2 = e for all s ∈ S (e = identity)

I and otherwise only relations of the form

(ss ′)m(s,s′) = e

for s 6= s ′ ∈ S with m(s, s ′) = m(s ′, s) ≥ 2

Each v ∈W can be written v = s1s2 · · · sk with si ∈ S

This is a reduced expression if k is minimal; k = l(v) length of v.

Coxeter Groups




(ss ′)m(s,s′) = e




Coxeter Groups




(ss ′)m(s,s′) = e




Coxeter Groups




(ss ′)m(s,s′) = e




Coxeter Groups




(ss ′)m(s,s′) = e




Bruhat order on (W , S)

v = s1s2 · · · sk reduced expression for v ; u ≤ v for u ∈W if someexpression for u is a subword u = si1si2 · · · si` of v .

Theorem(Verma): With this order, for u ≤ v ∈W , the Bruhatinterval [u, v ] is an Eulerian poset of rank l(v)− l(u).

Note: As a consequence of this, there is a cd-index Φu,v for eachBruhat interval [u, v ], homogeneous of degree l(v)− l(u)− 1.

Goal: To extend to a nonhomogeneous cd-polynomial Φ̃u,v
















R-polynomials

H(W ) the Hecke algebra associated to W : the freeZ[q, q−1]-module having the set {Tv : v ∈W } as a basis andmultiplication such that for all v ∈W and s ∈ S :

TvTs =

{Tvs , if l(vs) > l(v)qTvs + (q − 1)Tv , if l(vs) < l(v)

H(W ) is an associative algebra having Te as unity. Each Tv isinvertible in H(W ): for v ∈W ,

(Tv−1)−1 = q−l(v)∑u≤v

(−1)l(v)−l(u) Ru,v (q) Tu

where Ru,v (q) ∈ Z[q], the R-polynomial of [u, v ].

R-polynomials

H(W ) the Hecke algebra associated to W : the freeZ[q, q−1]-module having the set {Tv : v ∈W } as a basis andmultiplication such that for all v ∈W and s ∈ S :

TvTs =

{Tvs , if l(vs) > l(v)qTvs + (q − 1)Tv , if l(vs) < l(v)

H(W ) is an associative algebra having Te as unity. Each Tv isinvertible in H(W ): for v ∈W ,

(Tv−1)−1 = q−l(v)∑u≤v

(−1)l(v)−l(u) Ru,v (q) Tu

where Ru,v (q) ∈ Z[q], the R-polynomial of [u, v ].

Kazhdan-Lusztig polynomials

There is a unique family of polynomials {Pu,v (q)}u,v∈W ⊆ Z[q],such that, for all u, v ∈W ,

1. Pu,v (q) = 0 if u 6≤ v

2. Pu,u(q) = 1

3. deg(Pu,v (q)) ≤ b12 (l(v)− l(u)− 1)c, if u < v

4.

ql(v)−l(u) Pu,v

(1

q

)=∑

u≤z≤v

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

if u ≤ v .



1. Pu,v (q) = 0 if u 6≤ v

2. Pu,u(q) = 1


4.

ql(v)−l(u) Pu,v

(1

q

)=∑

u≤z≤v

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

if u ≤ v .



1. Pu,v (q) = 0 if u 6≤ v

2. Pu,u(q) = 1


4.

ql(v)−l(u) Pu,v

(1

q

)=∑

u≤z≤v

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

if u ≤ v .



1. Pu,v (q) = 0 if u 6≤ v

2. Pu,u(q) = 1


4.

ql(v)−l(u) Pu,v

(1

q

)=∑

u≤z≤v

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

if u ≤ v .



1. Pu,v (q) = 0 if u 6≤ v

2. Pu,u(q) = 1


4.

ql(v)−l(u) Pu,v

(1

q

)=∑

u≤z≤v

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

if u ≤ v .

Complete quasisymmetric function of a Bruhat interval

There exists a unique polynomial R̃u,v (q) ∈ N[q] such that

Ru,v (q) = q12(l(v)−l(u)) R̃u,v

(q

12 − q−

12

)

For Bruhat interval [u, v ], the complete quasisymmetric function

F̃ (u, v) :=∑

u=t0≤t1≤···≤tk−1<tk=v

R̃t0t1(x1)R̃t1t2(x2) · · · R̃tk−1tk (xk)

F̃ (u, v) =∑

α cα(u, v) Mα cα count paths in the Bruhat graph



Ru,v (q) = q12(l(v)−l(u)) R̃u,v

(q

12 − q−

12

)


F̃ (u, v) :=∑

u=t0≤t1≤···≤tk−1<tk=v


F̃ (u, v) =∑




Ru,v (q) = q12(l(v)−l(u)) R̃u,v

(q

12 − q−

12

)


F̃ (u, v) :=∑

u=t0≤t1≤···≤tk−1<tk=v


F̃ (u, v) =∑


The complete cd-index

Theorem: For any Bruhat interval [u, v ], F̃ (u, v) ∈ Π, in fact

F̃ (u, v) ∈ Πl(u,v) ⊕ Πl(u,v)−2 ⊕ Πl(u,v)−4 ⊕ · · ·

Since F̃ (u, v) ∈ Π, we can express it in terms of the peak basisΘw . We define the complete cd-index of the Bruhat interval [u, v ]

Φ̃u,v :=∑w

[w ]u,v w

by the unique expression

F̃ (u, v) =∑w

[w ]u,v

[1

2|w |d+1Θw

]sum over all cd-words w , deg(w) = l(u, v)− 1, l(u, v)− 3, . . .





Φ̃u,v :=∑w

[w ]u,v w


F̃ (u, v) =∑w

[w ]u,v

[1

2|w |d+1Θw






Φ̃u,v :=∑w

[w ]u,v w


F̃ (u, v) =∑w

[w ]u,v

[1

2|w |d+1Θw


Ballot polynomials and Kazhdan-Lusztig polynomials

Theorem:

Pu,v (q) =

bn/2c∑i=0

ai qi Bn−2i (−q)

where n = l(v)− l(u)− 1 and

ai = ai (u, v) = [cn−2i ]u,v +∑w

(−1)|w|2

+|w |d Cwd [wdcn−2i ]u,v

Bk(q) :=∑bk/2c

i=0k+1−2i

k+1

(k+1i

)qi are the ballot polynomials

and Cw is a product of Catalan numbers (or 0 if w is not even).


Theorem:

Pu,v (q) =

bn/2c∑i=0



ai = ai (u, v) = [cn−2i ]u,v +∑w

(−1)|w|2


Bk(q) :=∑bk/2c

i=0k+1−2i

k+1

(k+1i




Theorem:

Pu,v (q) =

bn/2c∑i=0



ai = ai (u, v) = [cn−2i ]u,v +∑w

(−1)|w|2


Bk(q) :=∑bk/2c

i=0k+1−2i

k+1

(k+1i




Theorem:

Pu,v (q) =

bn/2c∑i=0



ai = ai (u, v) = [cn−2i ]u,v +∑w

(−1)|w|2


Bk(q) :=∑bk/2c

i=0k+1−2i

k+1

(k+1i



Conjectures

Main conjectures: For all Coxeter systems (W , S), and all Bruhatintervals [u, v ] in W , the Kazhdan-Lusztig polynomial Pu,v

is nonngegative (Kazhdan-Lusztig) and

depends only on the poset [u, v ] and not on theunderlying group (Lusztig, Dyer).

The first holds for all finite Coxeter groups; the second for all lowerintervals, i.e. u = e = idW . Both hold when [u, v ] is a lattice.

Conjecture: For all [u, v ], Φ̃u,v ≥ 0. (Φu,v ≥ 0 is known)

Conjecture: For all lower Bruhat intervals [e, v ],Φ̃e,v (1, 1) ≤ ΦBl(v)

(1, 1). (Φe,v ≤ ΦBl(v)is known (Reading))

Conjecture: For each Bruhat interval [u, v ] of rank l(u, v) = n + 1,ai (u, v) ≥ 0 for i = 0, 1, . . . , bn/2c. (Implied by Pu,v ≥ 0)

Conjectures


is nonngegative (Kazhdan-Lusztig)

and







Conjectures









Conjectures









Conjectures









Conjectures









Conjectures









Combinatorial Hopf Algebra (H , ζ)

where H = H0⊕H1⊕H2⊕ · · · is a graded connected Hopf algebra(over Q) and ζ : H → Q is a character (algebra morphism).

A morphism f : (H ′, ζ ′)→ (H, ζ) is one for which ζ ′ = ζ ◦ f .

Ex. 1: (QSym, ζQ) where ζQ(Mα) = 1 if α = (n), n ≥ 0,ζQ(Mα) = 0 otherwise.

Ex. 2: (P, ζ), where P is formal Q-vector space of graded posets,P1 · P2 := P1 × P2, ∆(P) =

∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.

Theorem(Aguiar...): For any (H, ζH), there is a unique morphismF : (H, ζH)→ (QSym, ζQ) (i.e., (QSym, ζQ) is a terminal object).

For (H, ζH) = (P, ζ) from Ex. 2, F (P) =∑

α fαMα is this F

B & Aguiar: Extends to nonhomogeneous polynomial charactersζ : H → Q[x ]. Bruhat interval, ζ([u, v ]) = R̃u,v , F = F̃ (u, v)






∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F







∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F







∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F







∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F







∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F







∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F

B & Aguiar: Extends to nonhomogeneous polynomial charactersζ : H → Q[x ].

Bruhat interval, ζ([u, v ]) = R̃u,v , F = F̃ (u, v)






∑x∈P [ 0̂, x ]⊗ [ x , 1̂ ], ζ(P) = 1

for all posets P.



α fαMα is this F


Documents

Flag Enumeration in Polytopes Eulerian Partially Ordered Sets and