sweep polytopes and sweep oriented matroids
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Sweep polytopes and sweep oriented matroids
Eva Philippe
École Normale Supérieure de Paris
April 9th, 2021, Polytopics
Joint work with Arnau Padrol
arXiv:2102.06134
Eva PhilippeSweep polytopes and sweep oriented matroids
1 / 26 April 9th, 2021, Polytopics 1 / 26
Sommaire
1 Sweeps of a point configuration
2 Sweep polytopes
3 Abstraction to oriented matroids
4 Generalized Baues problem
Eva PhilippeSweep polytopes and sweep oriented matroids
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A = {a1, . . . , an} a configuration of n points in Rd .
a1 a3
a2 a5
a4
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A = {a1, . . . , an} a configuration of n points in Rd .
u
a1 a3
a2 a5
a4
Sweep permutation 1, 2, 3, 4, 5.〈u , a1〉 < 〈u , a2〉 < 〈u , a3〉 < 〈u , a4〉 < 〈u , a5〉 .
Eva PhilippeSweep polytopes and sweep oriented matroids
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A = {a1, . . . , an} a configuration of n points in Rd .
u
a1 a3
a2 a5
a4
Sweep 1, 23, 4, 5.〈u , a1〉 < 〈u , a2〉 = 〈u , a3〉 < 〈u , a4〉 < 〈u , a5〉 .
Eva PhilippeSweep polytopes and sweep oriented matroids
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ua1 a3
a2 a5
a4
12345132451342531452354125341254321
In dimension 2, Perrin (1882), Goodman and Pollack (1980-1993) studiedallowable sequences.
Edelman (2000) and Stanley (2015) gave bounds on the number of sweeppermutations (or valid order arrangements of an affine hyperplane arrangement).
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A = {a1, . . . , an} a configuration of n points in Rd .
a1 a3
a2 a5
a4
1, 2, 3, 4, 5
1, 23, 4, 5
13, 4, 25
12345
a5 − a1a4 − a1
a5 − a4
a3 − a2
SH(A) = arrangement of the hyperplanes{u ∈ Rd
∣∣ 〈u , ai 〉 = 〈u , aj〉}, for all
(i , j) ∈(
[n]2
).
Its cones are in bijection with the sweeps of A.
Eva PhilippeSweep polytopes and sweep oriented matroids
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Sweep polytope
DefinitionThe sweep polytope SP(A) of the point configuration A is the zonotope :
SP(A) =∑
1≤i<j≤n
[−ai − aj
2,ai − aj
2
].
Its face poset is in bijection with the poset of sweeps of A.Eva Philippe
Sweep polytopes and sweep oriented matroids8 / 26 April 9th, 2021, Polytopics 8 / 26
Sweeps of the simplex : the permutahedron
A = {e1, . . . , en} in Rn.
Pn =∑
1≤i<j≤n
[−ei − ej
2,ei − ej
2
]= conv
({n∑
i=1
σ(i)ei , σ ∈ Sn
})−n + 1
2
n∑i=1
ei .
Eva PhilippeSweep polytopes and sweep oriented matroids
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Projection of the permutahedron
A = {a1, . . . , an} a configuration of n points in Rd .We consider the projection
MA : Rn → Rd
ei 7→ ai .
Then SP(A) = MA(Pn).
Conversely, if M : Rn → Rd is a linear map, M(Pn) is a sweep polytope.
Eva PhilippeSweep polytopes and sweep oriented matroids
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Projection of the permutahedron
A = {a1, . . . , an} a configuration of n points in Rd .We consider the projection
MA : Rn → Rd
ei 7→ ai .
Then SP(A) = MA(Pn).
Conversely, if M : Rn → Rd is a linear map, M(Pn) is a sweep polytope.
Eva PhilippeSweep polytopes and sweep oriented matroids
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Monotone path polytope
w
hmin hmax h
P
Σ(P, h) ={ 1hmax − hmin
∫ hmax
hmin
γ(y) dy | γ is a section of h}.
Theorem (Billera-Sturmfels, 1992)The vertices of Σ(P, h) are in bijection with the coherent h-monotone paths of P.
Eva PhilippeSweep polytopes and sweep oriented matroids
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Monotone path polytopes
w
hmin hmax h
P
Theorem (Billera-Sturmfels, 1992)The face poset of Σ(P, h) is isomorphic to the poset of coherent h-cellular stringsof P.
Eva PhilippeSweep polytopes and sweep oriented matroids
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The permutahedron, monotone path polytope of the cube
���n = [−1, 1]n, s : x ∈ Rn 7→∑n
i=1 xi .
e1
e2
e3
s
���n
Σ(���n, s) =2nPn.
Eva PhilippeSweep polytopes and sweep oriented matroids
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The permutahedron, monotone path polytope of the cube
���n = [−1, 1]n, s : x ∈ Rn 7→∑n
i=1 xi .
e1
e2
e3
s
���n
Σ(���n, s) =2nPn.
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The sweep polytope as a monotone path polytope
ZA =∑n
i=1[−(ai , 1), (ai , 1)] ⊂ Rd+1, h : x ∈ Rd+1 7→ xd+1
b1
b2
b−1
b−2
h
Σ( n2ZA, h) = SP(A)× {0}.
Eva PhilippeSweep polytopes and sweep oriented matroids
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Non-coherent paths and pseudo-sweeps
b1
b2
b−1
b−2
Eva PhilippeSweep polytopes and sweep oriented matroids
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Non-coherent paths and pseudo-sweeps
b2
b1b1
b2
Eva PhilippeSweep polytopes and sweep oriented matroids
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Sweeps and pseudo-sweeps
2,1,1,2
2,1,2,1
2,1,1,2
2,1,2,1
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
2,1,2,1
2,1,1,2
2,1,2,1
2,1,1,2
2,1,12
2,11,2
21,1,2
21,2,1
2,1,12
21,1,2
21,2,1
1,2,12
1,22,1
1,2,21
12,2,1
12,1,2
1,2,12 1,22,1
1,2,21 12,2,1
12,1,2
2,1,21 2,11,2
2,1,21
21,12
12,21
12,21
21,12
Eva PhilippeSweep polytopes and sweep oriented matroids
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Quick introduction to oriented matroids
An oriented matroidM on ground set E can be described as a set of elements in{+,−, 0}E that satisfies certain properties.
3
2
1
+,+,+−,−,−
−,+,+
+,−,−
−,+,−
+,−,+
0,+,+
+, 0,+
+,−, 0
0,−,−
−, 0,−
−,+, 0
0, 0, 0
Eva PhilippeSweep polytopes and sweep oriented matroids
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Sweep oriented matroids are oriented matroids whose covectors can be associatedto ordered partitions.
Eva PhilippeSweep polytopes and sweep oriented matroids
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Generalized Baues problem
2,1,1,2
2,1,2,1
2,1,1,2
2,1,2,1
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
2,1,2,1
2,1,1,2
2,1,2,1
2,1,1,2
2,1,12
2,11,2
21,1,2
21,2,1
2,1,12
21,1,2
21,2,1
1,2,12
1,22,1
1,2,21
12,2,1
12,1,2
1,2,12 1,22,1
1,2,21 12,2,1
12,1,2
2,1,21 2,11,2
2,1,21
21,12
12,21
12,21
21,12
2112
Eva PhilippeSweep polytopes and sweep oriented matroids
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Generalized Baues problem
2,1,1,2
2,1,2,1
2,1,1,2
2,1,2,1
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
2,1,2,1
2,1,1,2
2,1,2,1
2,1,1,2
2,1,12
2,11,2
21,1,2
21,2,1
2,1,12
21,1,2
21,2,1
1,2,12
1,22,1
1,2,21
12,2,1
12,1,2
1,2,12 1,22,1
1,2,21 12,2,1
12,1,2
2,1,21 2,11,2
2,1,21
21,12
12,21
12,21
21,12
Eva PhilippeSweep polytopes and sweep oriented matroids
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Topology of posets
The order complex of a poset P is the simplicial complex of its chains.
O
a b c
d e f
a b
c
d
e f
O
Eva PhilippeSweep polytopes and sweep oriented matroids
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Generalized Baues Problem for cellular strings of a zonotope
P a d-polytope, h a linear functional in Rd
ω(P, h) := non trivial cellular strings,ωcoh(P, h) := non trivial coherentcellular strings.
Weak GBP. Is ω(P, h) homotopyequivalent to a (d − 2)-sphere ?Strong GBP. Is ωcoh(P, h) adeformation retract of ω(P, h) ?
Answer : YES(Billera-Kapranov-Sturmfels, 1992)
2,1,1,2
2,1,2,1
2,1,1,2
2,1,2,1
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
2,1,2,1
2,1,1,2
2,1,2,1
2,1,1,2
2,1,12
2,11,2
21,1,2
21,2,1
2,1,12
21,1,2
21,2,1
1,2,12
1,22,1
1,2,21
12,2,1
12,1,2
1,2,12 1,22,1
1,2,21 12,2,1
12,1,2
2,1,21 2,11,2
2,1,21
21,12
12,21
12,21
21,12
Eva PhilippeSweep polytopes and sweep oriented matroids
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Generalized Baues Problem for cellular strings of a zonotope
P a d-polytope, h a linear functional in Rd
ω(P, h) := non trivial cellular strings,ωcoh(P, h) := non trivial coherentcellular strings.
Weak GBP. Is ω(P, h) homotopyequivalent to a (d − 2)-sphere ?Strong GBP. Is ωcoh(P, h) adeformation retract of ω(P, h) ?
Answer : YES(Billera-Kapranov-Sturmfels, 1992)
2,1,1,2
2,1,2,1
2,1,1,2
2,1,2,1
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
1,2,1,2
1,2,2,1
1,2,2,1
1,2,1,2
2,1,2,1
2,1,1,2
2,1,2,1
2,1,1,2
2,1,12
2,11,2
21,1,2
21,2,1
2,1,12
21,1,2
21,2,1
1,2,12
1,22,1
1,2,21
12,2,1
12,1,2
1,2,12 1,22,1
1,2,21 12,2,1
12,1,2
2,1,21 2,11,2
2,1,21
21,12
12,21
12,21
21,12
Eva PhilippeSweep polytopes and sweep oriented matroids
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Generalized Baues Problem for cellular strings of orientedmatroids
M an oriented matroid of rank r , T one of its tope.
Weak GBP. Is ω(M,T ) homotopy equivalent to a (d − 2)-sphere ?
Answer : YES (Björner, 1992)
Strong GBP. Does ω(M,T ) retract to a subcomplex homeomorphic to a(d − 2)-sphere ?
Theorem (Padrol-P., 2021)IfM is the little oriented matroid of a sweep oriented matroidMsw. Then theposet of non-trivial sweeps ofMsw is a (r − 2)-sphere and a strong deformationretract of ω(M,+).
Eva PhilippeSweep polytopes and sweep oriented matroids
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Generalized Baues Problem for cellular strings of orientedmatroids
M an oriented matroid of rank r , T one of its tope.
Weak GBP. Is ω(M,T ) homotopy equivalent to a (d − 2)-sphere ?Answer : YES (Björner, 1992)
Strong GBP. Does ω(M,T ) retract to a subcomplex homeomorphic to a(d − 2)-sphere ?
Theorem (Padrol-P., 2021)IfM is the little oriented matroid of a sweep oriented matroidMsw. Then theposet of non-trivial sweeps ofMsw is a (r − 2)-sphere and a strong deformationretract of ω(M,+).
Eva PhilippeSweep polytopes and sweep oriented matroids
25 / 26 April 9th, 2021, Polytopics 25 / 26
Generalized Baues Problem for cellular strings of orientedmatroids
M an oriented matroid of rank r , T one of its tope.
Weak GBP. Is ω(M,T ) homotopy equivalent to a (d − 2)-sphere ?Answer : YES (Björner, 1992)
Strong GBP. Does ω(M,T ) retract to a subcomplex homeomorphic to a(d − 2)-sphere ?
Theorem (Padrol-P., 2021)IfM is the little oriented matroid of a sweep oriented matroidMsw. Then theposet of non-trivial sweeps ofMsw is a (r − 2)-sphere and a strong deformationretract of ω(M,+).
Eva PhilippeSweep polytopes and sweep oriented matroids
25 / 26 April 9th, 2021, Polytopics 25 / 26
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