flag enumeration in polytopes eulerian partially ordered sets and
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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
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
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
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
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: Exercise
d = 3: Steinitz (1906)d ≥ 4: open
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
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
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.
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.
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.
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 ,
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 ,
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 ,
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)
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
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
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
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
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.
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).
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).
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
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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.
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.
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.
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.
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.
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.
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.
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.
An example: The Boolean algebra B3
Ex. For P = B3 = 2[3],
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{a} {c}
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{b}= faces of
a b
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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
An example: The Boolean algebra B3
Ex. For P = B3 = 2[3],
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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
An example: The Boolean algebra B3
Ex. For P = B3 = 2[3],
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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
An example: The Boolean algebra B3
Ex. For P = B3 = 2[3],
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{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
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.
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.
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.
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.
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.
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.
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.
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}
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}
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}
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}
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.
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.
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.
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)
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)
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)
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)
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
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
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
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, . . . 〉
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, . . . 〉
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, . . . 〉
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, . . . 〉
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
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
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
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
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
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
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.
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.
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.
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.
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
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
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
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
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.
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
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
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
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 .
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 .
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 .
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 .
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 .
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
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
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
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, . . .
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, . . .
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, . . .
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).
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).
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).
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).
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
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
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
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
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
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
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)
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)
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)
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)
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)
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)
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)
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)
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)
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