Flag Enumeration in Polytopes Eulerian Partially Ordered Sets and
Transcript of Flag Enumeration in Polytopes Eulerian Partially Ordered Sets and
Flag Enumeration in Polytopes Eulerian PartiallyOrdered Sets and Coxeter Groups
Louis BilleraCornell University, Ithaca, NY 14853 USA
ICM - Hyderabad, August 2010
Introduction: Face Enumeration in Convex PolytopesSimplicial polytopesCounting flags in polytopes
Eulerian Posets and the cd-indexFlags in graded and Eulerian posetsInequalities for flags in polytopes and spheres
Algebraic Approches to Counting FlagsConvolution productQuasisymmetric function of a posetPeak functions and Eulerian posets
Bruhat Intervals in Coxeter GroupsR-polynomials and Kazhdan-Lusztig polynomialsComplete quasisymmetric function; complete cd-indexKazhdan-Lusztig polynomial from the complete cd-index
Epilog: Combinatorial Hopf Algebras
f -vectors of polytopes
For a d-dimensional convex polytope Q, letfi = fi (Q) = the number of i-dimensional faces of Q
f0 = the number of verticesf1 = the number of edges...fd−1 = the number of facets (defining inequalities)
The f -vector of Q: f (Q) = (f0, f1, . . . , fd−1)
Problem: Determine when a vector f = (f0, f1, . . . , fd−1)is f (Q) for some d-polytope Q.
d = 2: Exercised = 3: Steinitz (1906)d ≥ 4: open
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
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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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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
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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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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)