The Sperner Property - MIT Mathematicsrstan/transparencies/sperner.pdf · A poset (partially...
Transcript of The Sperner Property - MIT Mathematicsrstan/transparencies/sperner.pdf · A poset (partially...
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The Sperner Property
Richard P. StanleyM.I.T. and U. Miami
January 20, 2019
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Sperner’s theorem
Theorem (E. Sperner, 1927). Let S1,S2, . . . ,Sm be subsets of ann-element set X such that Si 6⊆ Sj for i 6= j , Then m ≤
(n
⌊n/2⌋
),
achieved by taking all ⌊n/2⌋-element subsets of X .
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Sperner’s theorem
Theorem (E. Sperner, 1927). Let S1,S2, . . . ,Sm be subsets of ann-element set X such that Si 6⊆ Sj for i 6= j , Then m ≤
(n
⌊n/2⌋
),
achieved by taking all ⌊n/2⌋-element subsets of X .
Emanuel Sperner9 December 1905 – 31 January 1980
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Posets
A poset (partially ordered set) is a set P with a binary relation ≤
satisfying:
Reflexivity: t ≤ t
Antisymmetry: s ≤ t, t ≤ s ⇒ s = t
Transitivity: s ≤ t, t ≤ u ⇒ s ≤ u
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Graded posets
chain: u1 < u2 < · · · < uk
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Graded posets
chain: u1 < u2 < · · · < uk
Assume P is finite. P is graded of rank n if
P = P0 ∪ P1 ∪ · · · ∪ Pn,
such that every maximal chain has the form
t0 < t1 < · · · < tn, ti ∈ Pi .
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Diagram of a graded poset
..
.
P
P
P
P
P
0
1
2
n −1
n
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Rank-symmetry and unimodality
Let pi = #Pi .
Rank-generating function: FP(q) =
n∑
i=0
piqi
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Rank-symmetry and unimodality
Let pi = #Pi .
Rank-generating function: FP(q) =
n∑
i=0
piqi
Rank-symmetric: pi = pn−i ∀i
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Rank-symmetry and unimodality
Let pi = #Pi .
Rank-generating function: FP(q) =
n∑
i=0
piqi
Rank-symmetric: pi = pn−i ∀i
Rank-unimodal: p0 ≤ p1 ≤ · · · ≤ pj ≥ pj+1 ≥ · · · ≥ pn for some j
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Rank-symmetry and unimodality
Let pi = #Pi .
Rank-generating function: FP(q) =
n∑
i=0
piqi
Rank-symmetric: pi = pn−i ∀i
Rank-unimodal: p0 ≤ p1 ≤ · · · ≤ pj ≥ pj+1 ≥ · · · ≥ pn for some j
rank-unimodal and rank-symmetric ⇒ j = ⌊n/2⌋
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The Sperner property
antichain A ⊆ P :
s, t ∈ A, s ≤ t ⇒ s = t
• • • •
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The Sperner property
antichain A ⊆ P :
s, t ∈ A, s ≤ t ⇒ s = t
• • • •
Note. Pi is an antichain
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The Sperner property
antichain A ⊆ P :
s, t ∈ A, s ≤ t ⇒ s = t
• • • •
Note. Pi is an antichain
P is Sperner (or has the Sperner property) if
maxA
#A = maxi
pi
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An example
rank-symmetric, rank-unimodal, FP(q) = 3 + 3q
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An example
rank-symmetric, rank-unimodal, FP(q) = 3 + 3q not Sperner
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The boolean algebra
Bn: subsets of {1, 2, . . . , n}, ordered by inclusion
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The boolean algebra
Bn: subsets of {1, 2, . . . , n}, ordered by inclusion
pi =(ni
), FBn
(q) = (1 + q)n
rank-symmetric, rank-unimodal
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Diagram of B3
2
231312
1 3
φ
123
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Sperner’s theorem, restated
Theorem. The boolean algebra Bn is Sperner.
Proof (D. Lubell, 1966).
Bn has n! maximal chains.
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Sperner’s theorem, restated
Theorem. The boolean algebra Bn is Sperner.
Proof (D. Lubell, 1966).
Bn has n! maximal chains.
If S ∈ Bn and #S = i , then i !(n − i)! maximal chains of Bn
contain S .
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Sperner’s theorem, restated
Theorem. The boolean algebra Bn is Sperner.
Proof (D. Lubell, 1966).
Bn has n! maximal chains.
If S ∈ Bn and #S = i , then i !(n − i)! maximal chains of Bn
contain S .
Let A be an antichain. Since a maximal chain can intersect atmost one element of A, we have
∑
S∈A
|S |! (n − |S |)! ≤ n!.
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Sperner’s theorem, restated
Theorem. The boolean algebra Bn is Sperner.
Proof (D. Lubell, 1966).
Bn has n! maximal chains.
If S ∈ Bn and #S = i , then i !(n − i)! maximal chains of Bn
contain S .
Let A be an antichain. Since a maximal chain can intersect atmost one element of A, we have
∑
S∈A
|S |! (n − |S |)! ≤ n!.
Divide by n!:∑
S∈A
1(n|S|
) ≤ 1.
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Lubell’s proof (cont.)
Divide by n!:∑
S∈A
1(n|S|
) ≤ 1.
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Lubell’s proof (cont.)
Divide by n!:∑
S∈A
1(n|S|
) ≤ 1.
Now(n|S|
)≤(
n⌊n/2⌋
), so
∑
A∈S
1(
n⌊n/2⌋
) ≤ 1.
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Lubell’s proof (cont.)
Divide by n!:∑
S∈A
1(n|S|
) ≤ 1.
Now(n|S|
)≤(
n⌊n/2⌋
), so
∑
A∈S
1(
n⌊n/2⌋
) ≤ 1.
⇒ |A| ≤(
n⌊n/2⌋
)�
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A q-analogue
Lubell’s proof carries over to some other posets.
Theorem. Let Bn(q) denote the poset of all subspaces of Fnq.
Then Bn(q) is Sperner.
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Linear algebra
P = P0 ∪ · · · ∪ Pm : graded poset
QPi : vector space with basis Q
U : QPi → QPi+1 is order-raising if for all s ∈ Pi ,
U(s) ∈ spanQ{t ∈ Pi+1 : s < t}
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Order-matchings
Order matching: µ : Pi → Pi+1: injective and µ(t) < t
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Order-matchings
Order matching: µ : Pi → Pi+1: injective and µ(t) < t
P
Pi +1
i
µ
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Order-raising and order-matchings
Key Lemma. If U : QPi → QPi+1 is injective and order-raising,then there exists an order-matching µ : Pi → Pi+1.
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Order-raising and order-matchings
Key Lemma. If U : QPi → QPi+1 is injective and order-raising,then there exists an order-matching µ : Pi → Pi+1.
Proof. Consider the matrix of U with respect to the bases Pi andPi+1.
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Key lemma proof
Pi+1︷ ︸︸ ︷
t1 · · · tm · · · tn
Pi
s1...sm
6= 0 | ∗. . . | ∗
6= 0| ∗
det 6= 0
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Key lemma proof
Pi+1︷ ︸︸ ︷
t1 · · · tm · · · tn
Pi
s1...sm
6= 0 | ∗. . . | ∗
6= 0| ∗
det 6= 0
⇒ s1 < t1, . . . , sm < tm �
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Minor variant
Similarly if there exists surjective order-raising U : QPi → QPi+1,then there exists an order-matching µ : Pi+1 → Pi .
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A criterion for Spernicity
P = P0 ∪ · · · ∪ Pn : finite graded poset
Proposition. If for some j there exist order-raising operators
QP0inj.→ QP1
inj.→ · · ·
inj.→ QPj
surj.→ QPj+1
surj.→ · · ·
surj.→ QPn,
then P is rank-unimodal and Sperner.
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A criterion for Spernicity
P = P0 ∪ · · · ∪ Pn : finite graded poset
Proposition. If for some j there exist order-raising operators
QP0inj.→ QP1
inj.→ · · ·
inj.→ QPj
surj.→ QPj+1
surj.→ · · ·
surj.→ QPn,
then P is rank-unimodal and Sperner.
Proof. Rank-unimodal clear: p0 ≤ p1 ≤ · · · ≤ pj ≥ pj+1 · · · ≥ pn.
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A criterion for Spernicity
P = P0 ∪ · · · ∪ Pn : finite graded poset
Proposition. If for some j there exist order-raising operators
QP0inj.→ QP1
inj.→ · · ·
inj.→ QPj
surj.→ QPj+1
surj.→ · · ·
surj.→ QPn,
then P is rank-unimodal and Sperner.
Proof. Rank-unimodal clear: p0 ≤ p1 ≤ · · · ≤ pj ≥ pj+1 · · · ≥ pn.
“Glue together” the order-matchings.
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Gluing example
j
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Gluing example
j
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Gluing example
j
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Gluing example
j
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Gluing example
j
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Gluing example
j
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A chain decomposition
P = C1 ∪ · · · ∪ Cpj (chains)
A = antichain,C = chain ⇒ #(A ∩ C ) ≤ 1
⇒ #A ≤ pj . �
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
131 2 3 4 5 6 7 8 9 10 11 12
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
)1 2 3 4 5 6 7 8 9 10 11 12 13
) ) ) )
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
(1 2 3 4 5 6 7 8 9 10 11 12 13
) ) ) ) )( ( ( ( ( ( (
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
(1 2 3 4 5 6 7 8 9 10 11 12 13
) ) ) ) )( ( ( ( ( ( (
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
(1 2 3 4 5 6 7 8 9 10 11 12 13
) ) ) ) )( ( ( ( ( ( (
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Back to Bn
Explicit order matching (Bn)i → (Bn)i+1 for i < n/2:
Example. S = {1, 4, 6, 7, 11} ∈ (B13)5
(1 2 3 4 5 6 7 8 9 10 11 12 13
) ) ) ) )( ( ( ( ( ( (
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Order-raising for Bn
DefineU : Q(Bn)i → Q(Bn)i+1
by
U(S) =∑
#T=i+1S⊂T
T , S ∈ (Bn)i .
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Order-raising for Bn
DefineU : Q(Bn)i → Q(Bn)i+1
by
U(S) =∑
#T=i+1S⊂T
T , S ∈ (Bn)i .
Similarly define D : Q(Bn)i+1 → Q(Bn)i by
D(T ) =∑
#S=iS⊂T
S , T ∈ (Bn)i+1.
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Order-raising for Bn
DefineU : Q(Bn)i → Q(Bn)i+1
by
U(S) =∑
#T=i+1S⊂T
T , S ∈ (Bn)i .
Similarly define D : Q(Bn)i+1 → Q(Bn)i by
D(T ) =∑
#S=iS⊂T
S , T ∈ (Bn)i+1.
Note. UD is positive semidefinite, and hence has nonnegative realeigenvalues, since the matrices of U and D with respect to thebases (Bn)i and (Bn)i+1 are transposes.
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A commutation relation
Lemma. On Q(Bn)i we have
DU − UD = (n − 2i)I ,
where I is the identity operator.
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A commutation relation
Lemma. On Q(Bn)i we have
DU − UD = (n − 2i)I ,
where I is the identity operator.
Corollary. If i < n/2 then U is injective.
Proof. UD has eigenvalues θ ≥ 0, and eigenvalues of DU areθ + n− 2i > 0. Hence DU is invertible, so U is injective. �
Similarly U is surjective for i ≥ n/2.
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A commutation relation
Lemma. On Q(Bn)i we have
DU − UD = (n − 2i)I ,
where I is the identity operator.
Corollary. If i < n/2 then U is injective.
Proof. UD has eigenvalues θ ≥ 0, and eigenvalues of DU areθ + n− 2i > 0. Hence DU is invertible, so U is injective. �
Similarly U is surjective for i ≥ n/2.
Corollary. Bn is Sperner.
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What’s the point?
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What’s the point?
The symmetric group Sn acts on Bn by
w · {a1, . . . , ak} = {w · a1, . . . ,w · ak}.
If G is a subgroup of Sn, define the quotient poset Bn/G to bethe poset on the orbits of G (acting on Bn), with
o ≤ o′ ⇔ ∃S ∈ o,T ∈ o
′, S ⊆ T .
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An example
n = 3, G = {(1)(2)(3), (1, 2)(3)}
3
φ
1,2
12
123
13,23
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Spernicity of Bn/G
Easy: Bn/G is graded of rank n and rank-symmetric.
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Spernicity of Bn/G
Easy: Bn/G is graded of rank n and rank-symmetric.
Theorem. Bn/G is rank-unimodal and Sperner.
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Spernicity of Bn/G
Easy: Bn/G is graded of rank n and rank-symmetric.
Theorem. Bn/G is rank-unimodal and Sperner.
Crux of proof. The action of w ∈ G on Bn commutes with U, sowe can “transfer” U to Bn/G , preserving injectivity on the bottomhalf.
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An interesting example
R: set of squares of an m × n rectangle of squares.
Gmn ⊂ SR : can permute elements in each row, and permute rowsamong themselves, so #Gmn = n!mm!.
Gmn∼= Sn ≀Sm (wreath product)
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Young diagrams
A subset S ⊆ R is a Young diagram if it is left-justified, withweakly decreasing row lengths from top to bottom.
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Young diagrams
A subset S ⊆ R is a Young diagram if it is left-justified, withweakly decreasing row lengths from top to bottom.
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BR/Gmn
Lemma. Each orbit o ∈ BR/Gmn contains exactly one Youngdiagram.
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BR/Gmn
Lemma. Each orbit o ∈ BR/Gmn contains exactly one Youngdiagram.
Proof. Let S ∈ BR . Let αi be the number of elements of S in rowi . Let λ1 ≥ · · · ≥ λm be the decreasing rearrangement ofα1, . . . , αm. Then the unique Young diagram in the orbitcontaining S has λi elements in row i . �
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Poset structure of Br/Gmn
Yo: Young diagram in the orbit o
Easy : o ≤ o′ in BR/Gmn if and only if Yo ⊆ Yo′ (containment of
Young diagram).
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Poset structure of Br/Gmn
Yo: Young diagram in the orbit o
Easy : o ≤ o′ in BR/Gmn if and only if Yo ⊆ Yo′ (containment of
Young diagram).
L(m, n): poset of Young diagrams in an m × n rectangle
Corollary. BR/Gmn∼= L(m, n)
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Examples of L(m, n)
φ
φ
1
2
3
4
φ
1
2
21
22
11
1
2
3
31
32
33
22
21
11
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L(3, 3)
11
φ
1
2
3
31
32
33
21
22
311
321
221
211
111
322 331
333
222
332
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q-binomial coefficients
For 0 ≤ k ≤ n, define the q-binomial coefficient
[n
k
]
=(1− qn)(1− qn−1) · · · (1− qn−k+1)
(1− qk)(1− qk−1) · · · (1− q).
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q-binomial coefficients
For 0 ≤ k ≤ n, define the q-binomial coefficient
[n
k
]
=(1− qn)(1− qn−1) · · · (1− qn−k+1)
(1− qk)(1− qk−1) · · · (1− q).
Example.[42
]
= 1 + q + 2q2 + q3 + q4
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Properties
[nk
]∈ N[q]
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Properties
[nk
]∈ N[q]
[nk
]
q=1=(nk
)
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Properties
[nk
]∈ N[q]
[nk
]
q=1=(nk
)
If q is a prime power,[nk
]is the number of k-dimensional
subspaces of Fnq (irrelevant here).
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Properties
[nk
]∈ N[q]
[nk
]
q=1=(nk
)
If q is a prime power,[nk
]is the number of k-dimensional
subspaces of Fnq (irrelevant here).
F (L(m, n), q) =[m+nm
]
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Unimodality
Corollary.[m+nm
]has unimodal coefficients.
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Unimodality
Corollary.[m+nm
]has unimodal coefficients.
First proved by J. J. Sylvester (1878) using invariant theoryof binary forms.
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Unimodality
Corollary.[m+nm
]has unimodal coefficients.
First proved by J. J. Sylvester (1878) using invariant theoryof binary forms.
Combinatorial proof by K. O’Hara (1990): explicit injectionL(m, n)i → L(m, n)i+1, 0 ≤ i < 1
2mn.
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Unimodality
Corollary.[m+nm
]has unimodal coefficients.
First proved by J. J. Sylvester (1878) using invariant theoryof binary forms.
Combinatorial proof by K. O’Hara (1990): explicit injectionL(m, n)i → L(m, n)i+1, 0 ≤ i < 1
2mn.
Not an order-matching. Still open to find an explicitorder-matching L(m, n)i → L(m, n)i+1.
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Algebraic geometry
X : smooth complex projective variety of dimension n
H∗(X ;C) = H0(X ;C)⊕ H1(X ;C)⊕ · · · ⊕ H2n(X ;C):cohomology ring, so H i ∼= H2n−i .
Hard Lefschetz Theorem. There exists ω ∈ H2 (the class of ageneric hyperplane section) such that for 0 ≤ i ≤ n, the map
ωn−2i : H i → H2n−i
is a bijection. Thus ω : H i → H i+1 is injective for i ≤ n andsurjective for i ≥ n.
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Cellular decompositions
X has a cellular decomposition if X = ⊔Ci , each Ci∼= Cdi (as
affine varieties), and each Ci is a union of Cj ’s.
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Cellular decompositions
X has a cellular decomposition if X = ⊔Ci , each Ci∼= Cdi (as
affine varieties), and each Ci is a union of Cj ’s.
Fact. If X has a cellular decomposition and [Ci ] ∈ H2(n−i) denotesthe corresponding cohomology classes, then the [Ci ]’s form aC-basis for H∗.
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The cellular decomposition poset
Let X = ⊔Ci be a cellular decomposition. Define a posetPX = {Ci}, by
Ci ≤ Cj if Ci ⊆ Cj
(closure in Zariski or classical topology).
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The cellular decomposition poset
Let X = ⊔Ci be a cellular decomposition. Define a posetPX = {Ci}, by
Ci ≤ Cj if Ci ⊆ Cj
(closure in Zariski or classical topology).
Easy:
PX is graded of rank n.
#(PX )i = dimC H2(n−i)(X ;C)
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The cellular decomposition poset
Let X = ⊔Ci be a cellular decomposition. Define a posetPX = {Ci}, by
Ci ≤ Cj if Ci ⊆ Cj
(closure in Zariski or classical topology).
Easy:
PX is graded of rank n.
#(PX )i = dimC H2(n−i)(X ;C)
PX is rank-symmetric (Poincare duality)
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The cellular decomposition poset
Let X = ⊔Ci be a cellular decomposition. Define a posetPX = {Ci}, by
Ci ≤ Cj if Ci ⊆ Cj
(closure in Zariski or classical topology).
Easy:
PX is graded of rank n.
#(PX )i = dimC H2(n−i)(X ;C)
PX is rank-symmetric (Poincare duality)
PX is rank-unimodal (hard Lefschetz)
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Spernicity of PX
Identify CP with H∗(X ;C) via Ci ↔ [Ci ].
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Spernicity of PX
Identify CP with H∗(X ;C) via Ci ↔ [Ci ].
Recall: ω ∈ H2(X ;C) (class of hyperplane section)
Interpretation of cup product on H∗(X ;C) as intersection impliesthat ω is order-raising.
Hard Lefschetz ⇒ ω : H2i → H2(i+1) is injective for i < n/2 andsurjective for i ≥ n/2.
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Spernicity of PX
Identify CP with H∗(X ;C) via Ci ↔ [Ci ].
Recall: ω ∈ H2(X ;C) (class of hyperplane section)
Interpretation of cup product on H∗(X ;C) as intersection impliesthat ω is order-raising.
Hard Lefschetz ⇒ ω : H2i → H2(i+1) is injective for i < n/2 andsurjective for i ≥ n/2.
⇒ Theorem. PX has the Sperner property.
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Main example
What smooth projective varieties have cellular decompositions?
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Main example
What smooth projective varieties have cellular decompositions?
Generalized flag variety: G/Q, where G is a semisimplealgebraic group over C, and Q is a parabolic subgroup
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Main example
What smooth projective varieties have cellular decompositions?
Generalized flag variety: G/Q, where G is a semisimplealgebraic group over C, and Q is a parabolic subgroup
Example. Gr(n, k) = SL(n,C)/Q for a certain Q, theGrassmann variety of all k-dimensional subspaces of Cn.
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Main example
What smooth projective varieties have cellular decompositions?
Generalized flag variety: G/Q, where G is a semisimplealgebraic group over C, and Q is a parabolic subgroup
Example. Gr(n, k) = SL(n,C)/Q for a certain Q, theGrassmann variety of all k-dimensional subspaces of Cn.
rational canonical form ⇒ PGr(m+n,m)∼= L(m, n)
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“Best” special case
G = SO(2n + 1,C), Q = “spin” maximal parabolic subgroup
M(n) := PG/Q∼= Bn/Sn, where Bn is the hyperoctahedral group
(symmetries of n-cube) of order 2nn!, so #M(n) = 2n
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“Best” special case
G = SO(2n + 1,C), Q = “spin” maximal parabolic subgroup
M(n) := PG/Q∼= Bn/Sn, where Bn is the hyperoctahedral group
(symmetries of n-cube) of order 2nn!, so #M(n) = 2n
M(n) is isomorphic to the set of all subsets of {1, 2, . . . , n} withthe ordering
{a1 > a2 > · · · > ar} ≤ {b1 > b2 > · · · > bs},
if r ≤ s and ai ≤ bi for 1 ≤ i ≤ r .
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Examples of M(n)
3
21
21
31
321
421
φ
φ
φ
φ
(1)M
(2)M
(3)M
M(4)
2
2
1
21
2
1
321
32
31
1
4321
432
431
43
32
42
41
4
3
1
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Rank-generating function of M(n)
rank of {a1, . . . , ar} in M(n) is∑
ai
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Rank-generating function of M(n)
rank of {a1, . . . , ar} in M(n) is∑
ai
⇒ F (M(n), q) :=
(n2)∑
i=0
|M(n)i | · qi = (1 + q)(1 + q2) · · · (1 + qn)
Corollary. The polynomial (1 + q)(1 + q2) · · · (1 + qn) hasunimodal coefficients.
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Rank-generating function of M(n)
rank of {a1, . . . , ar} in M(n) is∑
ai
⇒ F (M(n), q) :=
(n2)∑
i=0
|M(n)i | · qi = (1 + q)(1 + q2) · · · (1 + qn)
Corollary. The polynomial (1 + q)(1 + q2) · · · (1 + qn) hasunimodal coefficients.
No combinatorial proof known, though can be done with justelementary linear algebra (Proctor).
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The function f (S, α)
Let S ⊂ R, #S < ∞, α ∈ R.
f (S, α) = #{T ⊆ S :∑
i∈T
i = α}
Note.∑
i∈∅ i = 0
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The function f (S, α)
Let S ⊂ R, #S < ∞, α ∈ R.
f (S, α) = #{T ⊆ S :∑
i∈T
i = α}
Note.∑
i∈∅ i = 0
Example. f ({1, 2, 4, 5, 7, 10}, 7) = 3:
7 = 2 + 5 = 1 + 2 + 4
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The Erdos-Moser conjecture for R+
Let R+ = {i ∈ R : i > 0}.
Erdos-Moser Conjecture for R+
S ⊂ R+, #S = n
⇒ f (S , α) ≤ f
(
{1, 2, . . . , n},
⌊1
2
(n+ 1
2
)⌋)
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The Erdos-Moser conjecture for R+
Let R+ = {i ∈ R : i > 0}.
Erdos-Moser Conjecture for R+
S ⊂ R+, #S = n
⇒ f (S , α) ≤ f
(
{1, 2, . . . , n},
⌊1
2
(n+ 1
2
)⌋)
Note. 12
(n+12
)= 1
2 (1 + 2 + · · ·+ n)
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The proof
Proof. Suppose S = {a1, . . . , ak}, a1 > · · · > ak . Let
ai1 + · · ·+ air = aj1 + · · ·+ ajs ,
where i1 > · · · > ir , j1 > · · · > js .
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The proof
Proof. Suppose S = {a1, . . . , ak}, a1 > · · · > ak . Let
ai1 + · · ·+ air = aj1 + · · ·+ ajs ,
where i1 > · · · > ir , j1 > · · · > js .
Now {i1, . . . , ir} ≥ {j1, . . . , js} in M(n)
⇒ r ≥ s, i1 ≥ j1, . . . , is ≥ js
⇒ ai1 ≥ bj1 , . . . , ais ≥ bjs
⇒ r = s, aik = bik ∀k .
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Conclusion of proof
Thus ai1 + · · ·+ air = bj1 + · · ·+ bjs
⇒ {i1, . . . , ir} and {j1, . . . , js} are incomparable
or equal in M(n)
⇒ #S ≤ maxA
#A = f
(
{1, . . . , n},
⌊1
2
(n + 1
2
)⌋)
�
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The weak order on Sn
si :== (i , i + 1) ∈ Sn, 1 ≤ i ≤ n− 1 (adjacent transposition)
For w ∈ Sn,
ℓ(w) := #{(i , j) : i < j ,w(i) > w(j)}
= min{p : w = si1 · · · sip}.
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The weak order on Sn
si :== (i , i + 1) ∈ Sn, 1 ≤ i ≤ n− 1 (adjacent transposition)
For w ∈ Sn,
ℓ(w) := #{(i , j) : i < j ,w(i) > w(j)}
= min{p : w = si1 · · · sip}.
weak (Bruhat) order Wn on Sn: u < v if
v = usi1 · · · sip , p = ℓ(v)− ℓ(u).
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The weak order on Sn
si :== (i , i + 1) ∈ Sn, 1 ≤ i ≤ n− 1 (adjacent transposition)
For w ∈ Sn,
ℓ(w) := #{(i , j) : i < j ,w(i) > w(j)}
= min{p : w = si1 · · · sip}.
weak (Bruhat) order Wn on Sn: u < v if
v = usi1 · · · sip , p = ℓ(v)− ℓ(u).
ℓ is the rank function of Wn, so
F (Wn, q) = (1 + q)(1 + q + q2) · · · (1 + q + · · ·+ qn−1).
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The weak order on Sn
si :== (i , i + 1) ∈ Sn, 1 ≤ i ≤ n− 1 (adjacent transposition)
For w ∈ Sn,
ℓ(w) := #{(i , j) : i < j ,w(i) > w(j)}
= min{p : w = si1 · · · sip}.
weak (Bruhat) order Wn on Sn: u < v if
v = usi1 · · · sip , p = ℓ(v)− ℓ(u).
ℓ is the rank function of Wn, so
F (Wn, q) = (1 + q)(1 + q + q2) · · · (1 + q + · · ·+ qn−1).
A. Bjorner (1984): does Wn have the Sperner property?
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Examples of weak order
4
1234
2134 1324 1243
134221433124
2314
3214
3241
3421
4321
4312
4132
4123
1423
23412413 3142 1432
34124213
2431
4231
123
213
231
321
312
132
W3
W
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An order-raising operator
theory of Schubert polynomials suggests:
U(w) :=∑
1≤i≤n−1wsi>si
i · wsi
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An order-raising operator
theory of Schubert polynomials suggests:
U(w) :=∑
1≤i≤n−1wsi>si
i · wsi
Fact (Macdonald, Fomin-S). Let u < v in Wn, ℓ(v)− ℓ(u) = p.The coefficient of v in Up(u) is
p!Svu−1(1, 1, . . . , 1),
where Sw (x1, . . . , xn−1) is a Schubert polynomial.
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A down operator
C. Gaetz and Y. Gao (2018): constructedD : Q(Wn)i → Q(Wn)i−1 such that
DU − UD =
((n
2
)
− 2i
)
I .
Suffices for Spernicity.
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A down operator
C. Gaetz and Y. Gao (2018): constructedD : Q(Wn)i → Q(Wn)i−1 such that
DU − UD =
((n
2
)
− 2i
)
I .
Suffices for Spernicity.
Note. D is order-lowering on the strong Bruhat order. Leads toduality between weak and strong order.
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Another method
Z. Hamaker, O. Pechenik, D. Speyer, and A. Weigandt(2018): for k < 1
2
(n2
), let
D(n, k) = matrix of U(n2)−2k : Q(Wn)k → Q(Wn)(n2)−k
with respect to the bases (Wn)k and (Wn)(n2)−k(in some order).
Then (conjectured by RS):
detD(n, k) = ±
((n
2
)
− 2k
)
!#(Wn)k
k−1∏
i=0
((n2
)− (k + i)
k − i
)#(Wn)i
.
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Another method
Z. Hamaker, O. Pechenik, D. Speyer, and A. Weigandt(2018): for k < 1
2
(n2
), let
D(n, k) = matrix of U(n2)−2k : Q(Wn)k → Q(Wn)(n2)−k
with respect to the bases (Wn)k and (Wn)(n2)−k(in some order).
Then (conjectured by RS):
detD(n, k) = ±
((n
2
)
− 2k
)
!#(Wn)k
k−1∏
i=0
((n2
)− (k + i)
k − i
)#(Wn)i
.
Also suffices to prove Sperner property (just need detD(n, k) 6= 0).
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An open problem
The weak order W (G ) can be defined for any (finite) Coxetergroup G . Is W (G ) Sperner?
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The final slide
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The final slide