enumeration of flags in eulerian...
TRANSCRIPT
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, b, c}
{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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1
P
[x,1]
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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