Enumeration of Flags in Eulerian Posets

Louis Billera

Cornell University, Ithaca, NY 14853

* * * * *Conference on

Algebraic and Geometric CombinatoricsAnogia, Crete

20-26 August, 2005

Lecture I: Faces in polytopes and chains in posets

Lecture II: Enumeration algebra, quasisymmetric functions and

the peak algebra

Lecture III: Applications: arrangements, convex closures and

Coxeter groups

Lecture I: Faces in polytopes and chains in posets

• f-vectors of convex polytopes and the g-theorem

• flag f-vectors of graded posets

• Eulerian posets and the cd-index

• Convolutions of flag f-vectors

1

Preamble on f-vectors of polytopes

For a d-dimensional convex polytope Q, let

fi = fi(Q) = the number of i-dimensional faces of Q

f0 = the number of vertices,

f1 = the number of edges,...

fd−1 = the number of facets (or 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: Exercise

d = 3: Steinitz (1906)

d ≥ 4: open

2

Simplicial polytopes

A polytope is simplicial if all faces are simplices (equiv: verticesare in 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.

The h-vector and the f-vector of a polytope mutually determineeach other via the formulas (for 0 ≤ i ≤ d):

hi =i∑

j=0

(−1)i−j(d− j

i− j

)fj−1 ,

fi−1 =i∑

j=0

(d− j

i− j

)hj ,

3

The g-Theorem

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

hi = hd−i (Dehn-Sommerville equations)

for all i, and gi = hi − hi−1, 0 ≤ i ≤⌊d2

⌋, satisfy

gi ≥ 0 (Generalized Lower Bound Thm)

and

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

for i ≥ 1.

Note: The last conditions derive from (but are not quite thesame as) the conditions of Kruskal and Katona for f-vectors ofgeneral simplicial complexes, but with gi in place of fi.

Equivalently, the gi’s are a Hilbert function. Necessity proof(Stanley) depends on producing a commutative ring with thisHilbert function.

4

General polytopes

For general convex polytopes, the situation for f-vectors is much

less satisfactory.

1) The only equation they all satisfy is the Euler relation

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

2) Already in d = 4, we do not know all linear inequalities on

f-vectors.

3) There is little hope at this point of giving an analog to the

Macaulay-McMullen conditions.

A possible solution is to try to solve a harder problem: count

not faces, but chains of faces.

5

Flag f-vectors of Polytopes (first pass)

For a d-dimensional polytope Q and a set S of possible dimen-sions, define fS(Q) to be the number of chains of faces of Q

having dimensions prescribed by the set S.

The function

S 7→ fS(Q)

is called the flag f-vector of Q.

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

• It has a straightforwardly defined flag h-vector that turns outto be a (finely graded) Hilbert function.

• It satisfies an analog of the Dehn-Sommerville equations, whichcut their dimension down to the Fibonacci numbers (comparedto

⌊n2

⌋).

• And more ...6

Face Lattices of Polytopes

The best setting in which to study the flag f-vector or a d-

polytope Q is that of its lattice of faces P = F(Q), a graded

poset of rank d + 1

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7

Flag f-vectors of Graded Posets

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

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

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

To begin to understand flag f-vectors of convex polytopes, itmight be helpful to first be able to answer:

Question 1: Determine all flag f-vectors of graded posets,

Question 1a: Determine all linear inequalities satisfied by flagf-vectors of graded posets.

The former is a Kruskal-Katona analog and remains open. Thelatter are DS and GLB analogs for graded posets. They havecomplete solutions (B & Hetyei, 1998).

8

Eulerian Posets

P is Eulerian if for all x < y ∈ P ,

µ(x, y) = (−1)ρ(y)−ρ(x)

where µ is the Mobius function of P .

(Equivalently, number of elements of even rank in [x, y] = numberof elements of odd rank.)

• Face posets of polytopes and spheres are Eulerian.

Again, two natural questions arise:

Question 2: Determine all flag f-vectors of Eulerian posets,

Question 2a: Determine all linear inequalities satisfied by flagf-vectors of Eulerian posets.

• The linear equations are known: For Eulerian posets, onlyFibonacci many fS are linearly independent.

9

Generalized Dehn-Sommerville Equations

There are 2n flag numbers fS, S ⊂ [n] for graded posets of rank

n + 1. For Eulerian posets, these are not independent:

n= 0: f∅

n= 1: f∅, f{1} but f{1} = 2f∅ (Euler relation)

n= 2: f∅, f{1}, f{2}, f{1,2} but f{1} = f{2} (Euler relation) and

f{1,2} = 2f{2}

n= 3: f∅, f{1}, f{2}, f{3}, f{1,2}, . . . , f{1,2,3} but

f{1} − f{2}+ f{3} = 2f∅ (Euler relation), f{1,2} = 2f{2}, etc.

n=4: f∅, f{1}, f{2}, f{3}, f{1,3}

The relevant relations for P are all derived from Euler relations

in P and in intervals [x, y] of P . Details later ...

10

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 let uS = u1u2 · · ·un,

where

ui =

b if i ∈ S

a if i /∈ S

Let c = a + b and d = ab + ba. Then for Eulerian posets, the

generating function

ΨP =∑S

hS(P )uS

is always a polynomial in c and d; this polynomial ΦP (c,d) is

called the cd-index of P .

11

cd monomials and cd coefficients

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

• There are Fibonacci many cd monomials of degree n.

• Write ΦP =∑

w [w]P w over cd-words w.

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

Karu: [w]P ≥ 0 for all Gorenstein∗ posets.

• B , Ehrenborg & Readdy: Among all n-dimensional zonotopes,

the cd-index is termwise minimized on the n-cube Cn.

• B & Ehrenborg: Among all n-dimensional polytopes, the cd-

index is termwise minimized on the n-simplex ∆n.

12

An example: The Boolean algebra B3

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

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

{a} {c}

∅

{b}= faces of

a b

w

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

ΨP = aa + 2ba + 2ab + bb

= (a + b)2 + (ab + ba)

= c2 + d = ΦP

13

Convolutions of flag f-vectors

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

of rank n.

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 (P ) =

∑x∈P : r(x)=n

f(n)S ([0, x]) · f(m)

T ([x, 1])

Claim: f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

where T + n := {x + n | x ∈ T}

14

If rank P is n + m

f(n)S ∗ f

(m)T (P ) =

∑x∈P : r(x)=n f

(n)S ([0, x]) · f(m)

T ([x, 1])

[0,x]

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P

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f(n)S ∗ f

(m)T = f

(n+m)S∪{n}∪(T+n)

Ex. f(2){1} ∗ f

(3){2} = f

(5){1,2,4}

f(2)∅ ∗ f

(3)∅ = f

(5){2}

15

Convolved Inequalities

Kalai: If F =∑

αS f(n)S , G =

∑βS f

(m)S where

F (P1) ≥ 0 and G(P2) ≥ 0

for all polytopes (graded posets, Eulerian posets) P1 and P2 of

ranks n and m, respectively, then

F ∗G(P ) ≥ 0

for all polytopes (graded posets, Eulerian posets) P of rank n+m

Ex. Polygons have at least 3 vertices, so

f(3){1} − 3f

(3)∅ ≥ 0

for all polygons (rank = dimension + 1). Thus(f(3){1} − 3f

(3)∅

)∗ f

(1)∅ = f

(4){1,3} − 3f

(4){3} ≥ 0

for all 3-polytopes (number of vertices in 2-faces ≥ 3 times the

number of 2-faces).

16

Derived Inequalities for Polytopes

Most of the inequalities described earlier are of the form

F (P ) =∑

αS f(n)S (P ) ≥ 0

and so can be convolved to give further inequalities. For exam-

ple:

Let w = cn1dcn2dcn3 · · · cnpdcnp+1 be a cd-word, and define m0, . . . , mp

by m0 = 1 and mi = mi−1 + ni +2. Then the coefficient of w in

the cd-index is given by∑i1,...,ip

(−1)(m1−i1)+(m2−i2)+···+(mp−ip) ki1i2···ip

where the sum is over all p-tuples (i1, i2, . . . , ip) such that mj−1 ≤ij ≤ mj − 2 and

kS =∑

T⊆S

(−2)|S|−|T | fT

17

cd coefficients as forms

The cd-indices for posets of ranks 1–5:

f(1)∅

f(2)∅ c

f(3)∅ c2+

(f(3){1} − 2f

(3)∅

)d

f(4)∅ c3+

(f(4){1} − 2f

(4)∅

)dc+

(f(4){2} − f

(4){1}

)cd

f(5)∅ c4+

(f(5){1} − 2f

(5)∅

)dc2

(f(5){2} − f

(5){1}

)cdc

+(f(5){3} − f

(5){2} + f

(5){1} − 2f

(5)∅

)c2d

+(f(5){1,3} − 2f

(5){3} − 2f

(5){1} + 4f

(5)∅

)d2

18

Relations on the fS

Polytopes of dimension d− 1 (Eulerian posets of rank d) satisfy

the Euler relations:

f(d)∅ − f

(d){1}+ f

(d){2} − · · ·

· · ·+ (−1)d−1f(d){d−1}+ (−1)df

(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.

Proof consists of producing Fibonacci many polytopes whose

flag f-vectors span. These can be made by considering repeated

pyramids (P ) and prisms (B) starting with an edge, never taking

two B’s in a row. (Count the number of words of length d − 1

in P and B with no repeated B, get Fibonacci number.)

19

Lecture I: Faces in polytopes and chains in posets

Lecture II: Enumeration algebra, quasisymmetric functions and

the peak algebra

Lecture III: Applications: arrangements, convex closures and

Coxeter groups

Lecture II: Enumeration algebra, quasisymmetric functions and

the peak algebra

• The enumeration algebra over Eulerian posets

• Quasisymmetric functions and P -partitions

• The quasisymmetric function of a graded poset

• Peak quasisymetric functions and enriched P -partitions

• Connection to the enumeration algebra via Hopf algebra du-

ality

20

Subsets ←→ Compositions

Let [n] := {1, . . . , n}. Then β = (β1, . . . , βk) |= n + 1 means eachβi > 0, and β1 + · · ·+ βk = n + 1.

β = (β1, . . . , βk) |= n + 1

l

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

and

S = {i1, i2, . . . , ik−1} ⊂ [n]

l

β(S) := (i1, i2 − i1, i3 − i2, . . . , n + 1− ik−1) |= n + 1

21

Enumeration algebras

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

Via the association

yβ := yβ1· · · yβk

β = (β1, . . . , βk) |= n + 1

l

fS(β) = fS S ⊂ [n]

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)

This corresponds to summing over faces or links of a fixed rank.

Ex. f(3){1} = y1 y2 so

f(3){1} ∗ f

(3){1} = y1 y2 y1 y2 = f

(6){1,3,4}

22

Euler elements of An+1

F ∈ An ←→ functionals on graded posets

of rank n,

i.e., expressions of the form∑

S⊂[n−1] αSf(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)if(k)i ,

the kth Euler relation, where y0 = 1 and f(k)0 = f

(k)k = f

(k)∅ .

23

Eulerian Enumeration Algebra

Ex. In A4

χ4 = y0y4 − y1y3 + y2y2 − y3y1 + y4y0

Let

IE = 〈χk : k ≥ 1〉 ⊂ A

2-sided ideal of all relations on Eulerian posets

AE = A/IE

algebra of functionals on Eulerian posets.

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

AE∼= Q〈y1, y3, y5, . . . 〉

(“odd jump” algebra), and so dimQ(AE)n is the nth Fibonacci

number.

24

Quasisymmetric functions

Let Q ⊂ Q[[x1, x2, . . . ]] the algebra of quasisymmetric functions

Q := Q0 ⊕Q1 ⊕ · · ·

where

Qn := span{Mβ | β = (β1, . . . , βk) |= n}

Mβ :=∑

i1<i2<···<ik

xβ1i1

xβ2i2· · ·xβk

ik.

Here M0 = 1 so Q0 = Q; otherwise k > 0, each βi > 0, and

β1 + · · ·+ βk = n.

Ex. (1,2,1) |= 4 and

M(1,2,1) =∑

i1<i2<i3

x1i1

x2i2

x1i3

25

Relabel Mβ: For S ⊂ [n], define

MS = M(n+1)S := Mβ(S)

Ex. If S = {1,3} ⊂ [3] then β = (1,2,1) |= 4 and so

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

Note: Quasisymmetric functions arise naturally as weight enu-

merators of P -partitions of labeled posets (Gessel).

In this context, a more natural basis arises as weight enumerators

of labeled chains:

LS =∑

T⊃S

MT

Here S ⊂ T ⊂ [n] and S is the descent set of the labeling.

26

P -partitions

P = {1 < 2 < 3 < · · · } positive integers,

P an arbitrary poset and γ : P −→ [n] a 1-1 labeling of P , wheren = |P |.

A P -partition is an order preserving function

f : P −→ Pthat is nearly strict, i.e.,

p < q ⇒

f(p) < f(q) or

f(p) = f(q) and γ(p) < γ(q)

Ex. 1) P = n-element chain (naturally labeled): P -partitions arepartitions f(1) ≤ f(2) ≤ · · · ≤ f(n) of f(1) + f(2) + · · ·+ f(n).

Ex. 2) P = n-element antichain: P -partitions are compositions(f(1), f(2), . . . , f(n)) of f(1) + f(2) + · · ·+ f(n).

27

Weight Enumerators

Weight enumerator of all P -partitions

Γ(P, γ) =∑f

P−partition

∏p∈P

xf(p)

Proposition(Gessel):

1. If |P | = n, then

Γ(P, γ) ∈ Qn.

2. For P a chain labeled γ(1), γ(2), . . . , γ(n), Γ(P, γ) dependsonly on the descent set of γ, in fact,

Γ(P, γ) = LS,

where

S = {i | γ(i) > γ(i + 1)}.

28

Shuffle Product on Q

Poset sum P + Q: x ≤P+Q y ⇐⇒

x, y ∈ P, x ≤P y or x, y ∈ Q, x ≤Q y

Proposition:

Γ((P, γ) + (Q, δ)

)= Γ(P, γ) · Γ(Q, δ)

and so LS · LT =∑

R LR, where the sum is over all descent sets

of shuffles of a sequence with descent set S with one having

descent set T .

Ex.

L(2){1} · L

(2){1} = L

(4){1,2,3}+ 2L

(4){1,3}

+ L(4){1,2}+ L

(4){2}+ L

(4){2,3}

29

Shuffle Example: L(2){1} · L

(2){1}

Both 21 and 43 have descent set {1}, so consider the six shuffles

of these two sequences and their descent sets:

2143 {1,3}2413 {2}2431 {2,3}4231 {1,3}4213 {1,2}4321 {1,2,3}

Thus

L(2){1} · L

(2){1} = L

(4){1,2,3}+ 2L

(4){1,3}

+ L(4){1,2}+ L

(4){2}+ L

(4){2,3}

30

Connection to flag f-vectors

Given graded poset P with rank function ρ(·), we associate theformal quasisymmetric function

F (P ) =∑

0=t0≤t1≤···≤tk=1

xρ(t0,t1)1 x

ρ(t1,t2)2 · · ·xρ(tk−1,tk)

k

with the sum over all multichains in P and ρ(x, y) = ρ(y)− ρ(x).

F (P ) ∈ Qρ(P )

Ehrenborg: This association is multiplicative, in the sense that

F (P1 × P2) = F (P1) · F (P2).

In this context, changing invariants for P corresponds to chang-ing basis in Q:

F (P ) =∑S

fS(P )MS

=∑S

hS(P )LS

31

The algebra of Peak Functions

For a cd-word w of degree n,

w = cn1dcn2d · · · cnkdcm

(deg c = 1, degd = 2), let

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

where ij = deg(cn1dcn2d · · · cnjd).

b[Iw] = {S ⊂ [n] | S ∩ I 6= ∅, ∀I ∈ Iw}

The peak algebra Π is defined to be the subalgebra of Q gener-

ated by the peak quasisymmetric functions

Θw =∑

T∈b[Iw]

2|T |+1MT ,

where w is any cd-word (including empty cd-word 1, for which

I1 = ∅). Fibonacci many!

32

Why Π?

Stembridge: Peak quasisymmetric functions arise naturally as

weight enumerators of enriched P -partitions of labeled posets,

where we associate

w = cn1dcn2d · · · cnkdcm

a cd-word of degree n (deg c = 1, degd = 2)

l

Sw = {i1, i2, . . . , ik} ⊂ [n]

where ij = deg(cn1dcn2d · · · cnjd).

Stembridge considers the basis for Π to be indexed by sets S

of the form Sw. In this context, his basis ΘS arises as weight

enumerators of labeled chains, where S is the peak set of the

labeling. (A peak is a descent preceded by an ascent.)

33

Enriched P -partitions

Z∗ = {−1 < 1 < −2 < 2 < −3 < 3 < · · · } nonzero integers,

P an arbitrary poset and γ : P −→ [n] a 1-1 labeling of P , wheren = |P |.

An enriched P -partition is an order preserving function

f : P −→ Z∗

that is nearly strict, i.e.,

p < q ⇒

f(p) < f(q) or

f(p) = f(q) > 0 and γ(p) < γ(q)

f(p) = f(q) < 0 and γ(p) > γ(q)

Weight enumerator of all enriched P -partitions

∆(P, γ) =∑

f enrichedP−partition

∏p∈P

x|f(p)|

34

Peak Sets

Proposition(Stembridge):

• If |P | = n, then

∆(P, γ) ∈ Πn.

• For P a chain labeled γ(1), γ(2), . . . , γ(n), ∆(P, γ) depends onlyon the peak set of γ, in fact,

∆(P, γ) = ΘS,

where

S = {i | γ(i− 1) < γ(i) > γ(i + 1)}.

• If Πn = Π∩Qn, then dimQ(Πn) = an, the nth Fibonacci number(indexed so a1 = a2 = 1).

• Multiplication of the ΘS has a shuffle interpretation, but thistime in terms of peaks.

35

Brief interlude on dual Hopf algebras

The product on an algebra A can be viewed as a linear map

A⊗A −→ A, a⊗ b 7→ a · b

A coalgebra C has instead a coproduct

C −→ C ⊗ C

A Hopf algebra H has both (and more).

In the dual vector space H∗ to a Hopf algebra H, the adjoint of

the product on H

H∗ ⊗H∗ ←− H∗

gives a coproduct on H∗, and the adjoint of the coproduct on H

H∗ ←− H∗ ⊗H∗

gives a product on H∗, making H∗ a Hopf algebra as well.

36

Coproducts on A and Q

Π and AE both have Fibonacci Hilbert series; not isomorphic (Π

commutative, AE not).

Bergeron, Mykytiuk, Sottile, van Willigenburg: coproducts on Qand A

∆(Mβ) =∑

β=β1·β2

Mβ1⊗Mβ2

∆(yk) =∑

i+j=k

yi ⊗ yj

extend to coproducts on Π and AE, resp.

Ex.

∆(M(2,1,1)

)= 1⊗M(2,1,1) + M(2) ⊗M(1,1)+ M(2,1) ⊗M(1) + M(2,1,1) ⊗ 1

∆(y2) = 1⊗ y2 + y1 ⊗ y1 + y2 ⊗ 1

37

Hopf Duality

Theorem (BMSV): These coproducts make Π and AE into a dual

pair of Hopf algebras.

For graded poset P , recall the formal quasisymmetric function

F (P ) =∑

0=t0≤t1≤···≤tk=1

xρ(t0,t1)1 x

ρ(t1,t2)2 · · ·xρ(tk−1,tk)

k =∑S

fS(P )MS

Corollary: If P is Eulerian, then F (P ) ∈ Π.

Question: How to represent F (P ) in terms of the basis of peak

functions {Θw} for Π?

Equivalently, what is the dual basis in AE to the basis {Θw}?

38

Theorem(B.,Hsiao & van Willigenburg): 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.

Corollary: The elements

1

2|w|d+1[w] ∈ AE

form a dual basis to the basis Θw in Π.

As a consequence of a result of B., Ehrenborg and Readdy, we

get a slick way to see the relationship between enumerative in-

variants of hyperplane arrangements and zonotopes and those of

the associated geometric lattices.

39

Lecture I: Faces in polytopes and chains in posets

Lecture II: Enumeration algebra, quasisymmetric functions and

the peak algebra

Lecture III: Applications: arrangements, convex closures and

Coxeter groups

Lecture III: Applications: arrangements, convex closures and

Coxeter groups

• The Stembridge map Q −→ Π

• Geometric lattices and arrangements

• Meet distributive lattices and convex closures

• Kazhdan-Lusztig polynomials of Coxeter groups

40

From Descents to Peaks

Peaks

Descents

Ascents

41

The Stembridge map ϑ

Consider the (algebra) map

ϑ : Q −→ Π

defined by associating the weight enumerator of P -partitions to

that of enriched P -partitions for the same labeled poset (P, γ);

considering chains, we get

ϑ(LS) = ΘΛ(S),

where for S ⊆ [n],

Λ(S) = {i ∈ S | i 6= 1, i− 1 /∈ S}.

Note that if S is a descent set then Λ(S) is the associated peak

set.

Proposition: If poset P has a nonnegative flag h-vector (say,

if P is Cohen-Macaulay), then ϑ(F (P )) has a nonnegative “cd-

index”.42

Arrangements of hyperplanes

Six planes in general position in R3

A B C D

The number of i-gons in each:

i A B C D3 20 14 12 124 0 12 16 185 12 6 4 06 0 0 0 2

All different, yet each has 32 regions (in fact, each has same flag

f-vector).

43

The braid arrangement for S4

��� ��

� ��2

3

1

2413 2431

2341

23142134

2143

1243

1234

13243124

3214

3412

3421

314213421432

1423

3241

1-2

2-4

2-3

3-4

1-4

1-3

Planes:(42

)= 6 of the form xi = xj (i < j)

Regions: sortings of the coordinates and so correspond to oneof the 4! = 24 permutations

44

The cd-index of zonotopes and arrangements

Let z1, z2, . . . , zm ∈ Rn and let L be the geometric lattice of sub-spaces spanned by subsets of the z′is, ordered by inclusion.

L is graded, and so is L ∪ 0 (add a new 0 to L, increasing therank by one). Thus

F (L ∪ 0) ∈ Q.

Now consider the arrangement H of m hyperplanes {H1, H2, . . . , Hm}in Rn having normals

z1, z2, . . . , zm

Note that L can be seen as the lattice of all intersections of thehyperplanes Hi, ordered by reverse inclusion (the “intersectionlattice”).

H carves the (n − 1)-sphere in Rn into regions, that can beordered by inclusion (and so the resulting graded poset H has aflag f-vector).

45

The dual zonotope

Dual to the arrangement H is a zonotope

Z = [−z1, z1] + [−z2, z2] + · · ·+ [−zm, zm]

z2

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc

cc#

##

##

##

##

##

##

##

#cc

cc

cc

cc

cc

cc

cc

cc

##

##

##

##

##

##

##

##cc

cc

cc

cc

cc

cc

cc

cc #

##

##

##

##

##

##

##

#z3

−z2

−z1

−z3

z1

##

##

##

##

##

##

##

##

whose lattice of faces F(Z) is Eulerian. Thus

F (Z) := F (F(Z)) ∈ Π.

46

Geometric Lattice → Lattice of Regions

In the 70’s, Tom Zaslavsky showed how interesting enumerative

invariants of arrangements can be obtained from the simpler

underlying geometric lattice. For example, the Mobius function

of L determines the numbers of regions of H (the f-vector).

Later, Bayer-Sturmfels showed that L determines the flag f-

vector of H.

Recently B , Ehrenborg, Readdy + B , Hsiao, vanWilligenburg

made this determination explicit via the descents-to-peaks map

ϑ as

Theorem: ϑ(F (L ∪ 0)

)= 2 F (Z).

47

Labeling Chains in Geometric Lattices

L geometric lattice, of rank n, x <· y in L cover relation (∃ no

z ∈ L, x < z < y)

Then y = x ∨ a where a is an atom in L (a ·> 0)

Totally order the atoms in L:

z1 < z2 < · · · < zm or H1 < H2 < · · · < Hm

and label the cover relation x <· y by the least i such that the

atom ai satisfies y = x ∨ ai.

In L ∪ 0, label the additional cover relation 0 ·> 0 by a smallest

label 0.

As a result, every maximal chain C in L ∪ 0 has received a se-

quence `(C) = (a0, a1, . . . , an) of labels from the set {0, 1, 2, . . . ,m},with a0 = 0.

48

Descents and Peaks in `(C)

For each chain C in L∪ 0, the label sequence `(C) has a descentset and a peak set.

Bjorner-Stanley (R-labeling): If hS denotes the number of max-imal chains C in L ∪ 0 with descent set S, then

F (L ∪ 0) =∑S

hS LS.

BER + BHvW: If tS denotes the number of maximal chains C

in L ∪ 0 with peak set S, then

F (Z) =1

2

∑S

tS ΘS.

This is, perhaps, the most explicit description of the relation

ϑ(F (L ∪ 0)

)= 2 F (Z).

However, a simple understanding of why this works is yet to be.

49

Example: Boolean lattice → Cube

If zi = ei, where e1, e2, e3 are the coordinate vectors in R3, then

L = B3, the Boolean algebra, and

Z = 3-cube (H = coordinate arrangement)

The labeling scheme assigns all π ∈ S3 to B3 and so to B3 ∪ 0the labels

0123, 0132, 0213, 0231, 0312, 0321.So

F (L ∪ 0) = L∅+ 2L{2}+ 2L{3}+ L{2,3}.

and

ϑ(F (L ∪ 0)

)= Θ∅+ 3Θ{2}+ 2Θ{3}

= 2(1

2Θc3 +

6

4Θdc +

4

4Θcd

)⇒ cd-index of the 3-cube is c3 + 6dc + 4cd and that of the

coordinate arrangement is (via w 7→ w∗, reversal of cd words)

c3 + 4dc + 6cd.

50

Other examples where ϑ (F (Q)) = 2 F (P )?

Hsiao: If L is a distributive lattice then

ϑ(F (L ∪ 0)

)= 2 F (L)

for some shellable Eulerian poset L.

This L is the face poset of a regular CW-sphere.

More generally (B , Hsiao & Provan), if L is meet-distributive,

(closed subsets of an anti-exchange closure [Edelman, et al.]),

e.g., the lattice of convex subsets of a finite subset of Euclidean

space:

ϑ(F (L∗ ∪ 0)

)= 2 F (L)

51

Order complex of a distributive lattice

Let L be a distributive lattice: L = J(P ), where J(P ) is the

lattice of order ideals in a poset P , ordered by inclusion.

The order complex ∆(L) is the simplicial complex on the ele-

ments of L with simplices being the chains of L.

Provan: ∆(L \ 0) can be obtained from a (|P | − 1)-simplex by a

sequence of stellar subdivisions:

∆<c>

<b> <a,c>

<b,c>

<a>Pca

b

<b>

<c>

<a>

<b,c><a,c>

L=J(P)< >

52

To construct the sphere with face poset L:• Reflect the order complex into the boundary of the crosspoly-tope;• equivalently, do Provan construction over the boundary of the|P |-dimensional crosspolytope.

Resulting simplicial polytope is the barycentric subdivision ∆(L)of the regular CW-sphere with face poset L.

<b,−c>

<a>

<c>

<b>

<a,c>

<b,c>

<−a,c>

<−a,−c>

<−a>

<−c>

<a,−c>

F (L ∪ 0) = L∅+ L{2}+ L{3}ϑ(F (L ∪ 0)

)= Θ∅+ Θ{2}+ Θ{3}= 2F (L)

53

Example: convex subsets of three collinear points a < b < c.

{b,c}

{a,b,c}

{a,b}

{a} {c}

0

{b}

L L

F (L∗ ∪ 0) = L∅+ L{2}+ 2L{3}

ϑ(F (L∗ ∪ 0)

)= Θ∅+ Θ{2}+ 2Θ{3}= 2F (L∗)

54

Comments and possible extensions

• For any oriented matroid with geometric lattice L, ϑ(F (L ∪ 0)

)gives the cd-index of the associated pseudoplane arrange-

ment.

• Swartz: For nonorientable matroids L, there is an arrange-

ment of homotopy spheres having L as intersection lattice.

– Conjecture: F (L ∪ 0) gives a lower bound for the flag

f-vector in this case.

(Equality ⇔ L orientable??)

• Does ϑ(F (L ∪ 0)

)= 2 F (P ) hold for any semimodular lat-

tice?

55

Coxeter Groups (work with F. Brenti)

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

relations

• s2 = e for all s ∈ S (e = identity),

• 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.

Examples include the symmetry groups of regular polytopes (and

so the symmetric groups) and much more (see Bjorner-Brenti,

Combinatorics of Coxeter Groups, Springer, 2005).

56

Bruhat order on (W, S)

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

If k is minimal among all such expressions for v, then

s1s2 · · · sk is called a reduced expression for v and

k = l(v) is called the length of v.

Bruhat order on (W, S): if v = s1s2 · · · sk is a reduced expression

for v, then u ≤ v for u ∈W if some reduced expression for u is a

subword u = si1si2 · · · si` of v.

Fact: for each u ≤ v ∈W the Bruhat interval [u, v] is an Eulerian

poset of rank l(v)− l(u)

57

R-polynomials

H(W ) the Hecke algebra associated to W : the free Z[q, q−1]-

module having the set {Tv : v ∈W} as a basis and multiplication

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 is

invertible 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 polynomials Ru,v are called the R-polynomials of W . For

u, v ∈W , u ≤ v, deg(Ru,v) = l(v)− l(u) and Ru,u(q) = 1.

58

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.

59

Extended quasisymmetric function of a Bruhat interval

For u ≤ v ∈ W , there exists a (necessarily unique) polynomialRu,v(q) ∈ N[q] such that

Ru,v(q) = q12(l(v)−l(u)) Ru,v

(q12 − q−

12

)

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

F ([u, v]) :=∑

u=t0≤t1≤···≤tk=v

Rt0t1(x1)Rt1t2(x2) · · · Rtk−1tk(xk),

where, again, the sum is over all multichains in [u, v].

Properties:

• F ([u1, v1]× [u2, v2]) = F ([u1, v1]) F ([u2, v2]),

• F ([u, v]) =∑

α cα(u, v)Mα =∑

α bα(u, v)Lα, where cα and bα

count paths in the Bruhat graph (related to Bruhat order [u, v])

60

F ([u, v]) is a peak function

Brenti: The cα satisfy the generalized Dehn-Sommerville equa-

tions, so we may conclude:

• F ([u, v]) ∈ Π, in fact

F ([u, v]) ∈ Πl(u,v) ⊕Πl(u,v)−2 ⊕Πl(u,v)−4 ⊕ · · ·

Note: Bruhat order [u, v] is always Eulerian, so F ([u, v]) ∈ Π, but

usually F ([u, v]) 6= F ([u, v]). In fact

F ([u, v]) = F ([u, v]) + lower terms.

Since F ([u, v]) ∈ Π, we define the extended cd-index of [u, v] by

F ([u, v]) =∑w

[w]u,v

[1

2|w|d+1Θw

]=∑w

[w]u,v Θw

where the sum is over all cd-words w

(with deg(w) = `(u, v)− 1, l(u, v)− 3, . . . ).

61

Ballot polynomials and Kazhdan-Lusztig polynomials

Brenti gave an expresion for Pu,v, u < v, in terms of the cα(u, v),

q−l(v)−l(u)

2 Pu,v(q)− ql(v)−l(u)

2 Pu,v

(1

q

)=

∑β∈C

bβ(u, v)[q−|β|2 Υβ(q)

]where Υβ(q) enumerates certain implicitly defined lattice paths.

By expressing this in terms of the extended cd-index of [u, v], the

resulting paths are now explicit, and we can get

Pu,v(q) =bn/2c∑i=0

ai qi Bn−2i(−q)

where

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+1

i

)qi are the ballot polynomials and

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

62

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