SLIDE 1
Introduction The Cayley trick Hom-complexes Dissection complexes
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissections, Hom-complexes and the Cayley Trick Julian Pfeifle MA - - PowerPoint PPT Presentation
Dissections, Hom-complexes and the Cayley Trick Julian Pfeifle MA - - PowerPoint PPT Presentation
Introduction The Cayley trick Hom-complexes Dissection complexes Dissections, Hom-complexes and the Cayley Trick Julian Pfeifle MA II, Universitat Politcnica de Catalunya julian.pfeifle@upc.edu Work in progress Introduction The Cayley
SLIDE 2
SLIDE 3
Introduction The Cayley trick Hom-complexes Dissection complexes
Outline
Introduction The associahedron Dissections of polygons Results in this talk The Cayley trick Joins and projections . . . and the Cayley trick Hom-complexes Hom-complexes and the Cayley trick Symmetry classes The case Hom(Kg, H) Dissection complexes
SLIDE 4
Introduction The Cayley trick Hom-complexes Dissection complexes
The associahedron
Its face lattice records the incidence structure of the dissections
- f a convex (n + 2)-gon into (j + 2)-gons, j = 1, 2, . . . , n
SLIDE 5
Introduction The Cayley trick Hom-complexes Dissection complexes
Polygon dissections
Let’s dissect into k-gons instead of triangles:
Definition
Let k ≥ 3 and m ≥ 1. (a) An allowable diagonal of a convex N-gon is one that can be completed to a dissection of the N-gon into m convex k-gons. (So N = m(k − 2) + 2.) (b) [Vic Reiner] Let T(k, m) be the simplicial complex on the allowable diagonals whose faces are the partial dissections.
SLIDE 6
Introduction The Cayley trick Hom-complexes Dissection complexes
Polygon dissections
Let’s dissect into k-gons instead of triangles:
Definition
Let k ≥ 3 and m ≥ 1. (a) An allowable diagonal of a convex N-gon is one that can be completed to a dissection of the N-gon into m convex k-gons. (So N = m(k − 2) + 2.) (b) [Vic Reiner] Let T(k, m) be the simplicial complex on the allowable diagonals whose faces are the partial dissections.
SLIDE 7
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissections of polygons
Theorem (Tzanaki 2005)
The complex T(k, m) (a) is vertex-decomposable, hence shellable; (b) has the homotopy type of a wedge of 1
m
(k−2)m
m−1
- spheres of
dimension m − 2. In particular, for k = 3 it really is a sphere: the boundary complex of the polar of the associahedron! But for k > 3, the complex T(k, m) cannot be the boundary complex of a convex polytope.
SLIDE 8
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissections of polygons
Theorem (Tzanaki 2005)
The complex T(k, m) (a) is vertex-decomposable, hence shellable; (b) has the homotopy type of a wedge of 1
m
(k−2)m
m−1
- spheres of
dimension m − 2. In particular, for k = 3 it really is a sphere: the boundary complex of the polar of the associahedron! But for k > 3, the complex T(k, m) cannot be the boundary complex of a convex polytope.
SLIDE 9
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissections of polygons
Theorem (Tzanaki 2005)
The complex T(k, m) (a) is vertex-decomposable, hence shellable; (b) has the homotopy type of a wedge of 1
m
(k−2)m
m−1
- spheres of
dimension m − 2. In particular, for k = 3 it really is a sphere: the boundary complex of the polar of the associahedron! But for k > 3, the complex T(k, m) cannot be the boundary complex of a convex polytope.
SLIDE 10
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissections of polygons
Theorem (Tzanaki 2005)
The complex T(k, m) (a) is vertex-decomposable, hence shellable; (b) has the homotopy type of a wedge of 1
m
(k−2)m
m−1
- spheres of
dimension m − 2. In particular, for k = 3 it really is a sphere: the boundary complex of the polar of the associahedron! But for k > 3, the complex T(k, m) cannot be the boundary complex of a convex polytope.
SLIDE 11
Introduction The Cayley trick Hom-complexes Dissection complexes
Flip graphs
D(k, m) is the dual graph of T(k, m): two dissections are connected if they only differ in one diagonal.
SLIDE 12
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 13
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 14
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 15
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 16
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 17
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 18
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 19
Introduction The Cayley trick Hom-complexes Dissection complexes
Preliminary results
- 1. We relate Hom-complexes to the Cayley trick, and study
Hom(G, H)/SG, where SG is the symmetry group of G
- 2. We focus on Hom(Kg, H).
(For coloring problems, people look at Hom(G, Kh).)
- 3. We obtain results on T(k, m) and D(k, m):
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
- 4. Also: T(k, m) = skm−2 ♦(k, m)
(motivated by a question of Fomin & Zelevinski)
- 5. Hom(Km−1, I(k, m)) contains copies of skd/2
C↓
d(n)
∆
- 6. D(k, m) contains copies of C(r, s), the “flip graph” of weak
compositions of r into s parts
SLIDE 20
Introduction The Cayley trick Hom-complexes Dissection complexes
Outline
Introduction The associahedron Dissections of polygons Results in this talk The Cayley trick Joins and projections . . . and the Cayley trick Hom-complexes Hom-complexes and the Cayley trick Symmetry classes The case Hom(Kg, H) Dissection complexes
SLIDE 21
Introduction The Cayley trick Hom-complexes Dissection complexes
The polyhedral Cayley trick
Let P1, . . . , Pg ⊂ ❘d be polytopes. The Cayley embedding C(P1, . . . , Pg) is C(P1, . . . , Pg) = conv
g
- i=1
Pi × ei ⊂ ❘d × ❘g
Theorem (The polyhedral Cayley trick)
- polyhedral subdivisions of C(P1, . . . , Pg)
- ∼
=
- mixed subdivisions of P1 + · · · + Pg
- as posets, by intersecting with ❘d × ( 1
g, . . . , 1 g).
SLIDE 22
Introduction The Cayley trick Hom-complexes Dissection complexes
The polyhedral Cayley trick
Let P1, . . . , Pg ⊂ ❘d be polytopes. The Cayley embedding C(P1, . . . , Pg) is C(P1, . . . , Pg) = conv
g
- i=1
Pi × ei ⊂ ❘d × ❘g
Theorem (The polyhedral Cayley trick)
- polyhedral subdivisions of C(P1, . . . , Pg)
- ∼
=
- mixed subdivisions of P1 + · · · + Pg
- as posets, by intersecting with ❘d × ( 1
g, . . . , 1 g).
SLIDE 23
Introduction The Cayley trick Hom-complexes Dissection complexes
The polyhedral Cayley trick
31 51 32 52 42 41
C(∆{3,4,5}, ∆{3,4,5}) = ∆{3,4,5} × ∆{1,2} a polyhedral subdivision . . . . . . intersected with ❘2 × (1
2) yields a
mixed subdivision of the intersection.
SLIDE 24
Introduction The Cayley trick Hom-complexes Dissection complexes
The polyhedral Cayley trick
31 51 32 52 42 41
C(∆{3,4,5}, ∆{3,4,5}) = ∆{3,4,5} × ∆{1,2} a polyhedral subdivision . . . . . . intersected with ❘2 × (1
2) yields a
mixed subdivision of the intersection.
SLIDE 25
Introduction The Cayley trick Hom-complexes Dissection complexes
The polyhedral Cayley trick
31 51 32 52 42 41
C(∆{3,4,5}, ∆{3,4,5}) = ∆{3,4,5} × ∆{1,2} a polyhedral subdivision . . . . . . intersected with ❘2 × (1
2) yields a
mixed subdivision of the intersection.
SLIDE 26
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
- A, B sets of cardinality a = |A| and b = |B|.
- ∆A := conv{ei : i ∈ A} ⊂ ❘a,
so that dim ∆A = a − 1.
- ⋆i∈A ∆B is a simplex of dimension ab − 1.
- µi : ❘b ֒
→ ❘ab = ❘b × · · · × ❘b inclusion into i-th factor
Observation
⋆
i∈A ∆B
= conv
a
- i=1
µi(∆B) × ei = C
- µ1(∆B), . . . , µa(∆B)
- ⋆
i∈A σi
= C
- µ1(σ1), . . . , µa(σa)
- for faces σ1, . . . , σa of ∆B.
SLIDE 27
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
- A, B sets of cardinality a = |A| and b = |B|.
- ∆A := conv{ei : i ∈ A} ⊂ ❘a,
so that dim ∆A = a − 1.
- ⋆i∈A ∆B is a simplex of dimension ab − 1.
- µi : ❘b ֒
→ ❘ab = ❘b × · · · × ❘b inclusion into i-th factor
Observation
⋆
i∈A ∆B
= conv
a
- i=1
µi(∆B) × ei = C
- µ1(∆B), . . . , µa(∆B)
- ⋆
i∈A σi
= C
- µ1(σ1), . . . , µa(σa)
- for faces σ1, . . . , σa of ∆B.
SLIDE 28
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
- A, B sets of cardinality a = |A| and b = |B|.
- ∆A := conv{ei : i ∈ A} ⊂ ❘a,
so that dim ∆A = a − 1.
- ⋆i∈A ∆B is a simplex of dimension ab − 1.
- µi : ❘b ֒
→ ❘ab = ❘b × · · · × ❘b inclusion into i-th factor
Observation
⋆
i∈A ∆B
= conv
a
- i=1
µi(∆B) × ei = C
- µ1(∆B), . . . , µa(∆B)
- ⋆
i∈A σi
= C
- µ1(σ1), . . . , µa(σa)
- for faces σ1, . . . , σa of ∆B.
SLIDE 29
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
- A, B sets of cardinality a = |A| and b = |B|.
- ∆A := conv{ei : i ∈ A} ⊂ ❘a,
so that dim ∆A = a − 1.
- ⋆i∈A ∆B is a simplex of dimension ab − 1.
- µi : ❘b ֒
→ ❘ab = ❘b × · · · × ❘b inclusion into i-th factor
Observation
⋆
i∈A ∆B
= conv
a
- i=1
µi(∆B) × ei = C
- µ1(∆B), . . . , µa(∆B)
- ⋆
i∈A σi
= C
- µ1(σ1), . . . , µa(σa)
- for faces σ1, . . . , σa of ∆B.
SLIDE 30
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
- A, B sets of cardinality a = |A| and b = |B|.
- ∆A := conv{ei : i ∈ A} ⊂ ❘a,
so that dim ∆A = a − 1.
- ⋆i∈A ∆B is a simplex of dimension ab − 1.
- µi : ❘b ֒
→ ❘ab = ❘b × · · · × ❘b inclusion into i-th factor
Observation
⋆
i∈A ∆B
= conv
a
- i=1
µi(∆B) × ei = C
- µ1(∆B), . . . , µa(∆B)
- ⋆
i∈A σi
= C
- µ1(σ1), . . . , µa(σa)
- for faces σ1, . . . , σa of ∆B.
SLIDE 31
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
- A, B sets of cardinality a = |A| and b = |B|.
- ∆A := conv{ei : i ∈ A} ⊂ ❘a,
so that dim ∆A = a − 1.
- ⋆i∈A ∆B is a simplex of dimension ab − 1.
- µi : ❘b ֒
→ ❘ab = ❘b × · · · × ❘b inclusion into i-th factor
Observation
⋆
i∈A ∆B
= conv
a
- i=1
µi(∆B) × ei = C
- µ1(∆B), . . . , µa(∆B)
- ⋆
i∈A σi
= C
- µ1(σ1), . . . , µa(σa)
- for faces σ1, . . . , σa of ∆B.
SLIDE 32
Introduction The Cayley trick Hom-complexes Dissection complexes
Projections
Define the projections
- π : ❘ab × ❘a → ❘b × ❘a with matrix
✶b · · · ✶b · · · ✶a
- ,
so that π
- (x1, . . . , xa, y)T
= (x1 + · · · + xa, y)T
- π∆ : ❘b × ❘a → ❘b is the projection onto the first factor.
Observation
π(σ) = π C
- µ1(σ1), . . . , µa(σa)
- = C(σ1, . . . , σa)
SLIDE 33
Introduction The Cayley trick Hom-complexes Dissection complexes
Projections
Define the projections
- π : ❘ab × ❘a → ❘b × ❘a with matrix
✶b · · · ✶b · · · ✶a
- ,
so that π
- (x1, . . . , xa, y)T
= (x1 + · · · + xa, y)T
- π∆ : ❘b × ❘a → ❘b is the projection onto the first factor.
Observation
π(σ) = π C
- µ1(σ1), . . . , µa(σa)
- = C(σ1, . . . , σa)
SLIDE 34
Introduction The Cayley trick Hom-complexes Dissection complexes
Projections
Define the projections
- π : ❘ab × ❘a → ❘b × ❘a with matrix
✶b · · · ✶b · · · ✶a
- ,
so that π
- (x1, . . . , xa, y)T
= (x1 + · · · + xa, y)T
- π∆ : ❘b × ❘a → ❘b is the projection onto the first factor.
Observation
π(σ) = π C
- µ1(σ1), . . . , µa(σa)
- = C(σ1, . . . , σa)
SLIDE 35
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
Set L = ❘ab × 1
a ⊂ ❘ab × ❘a, with 1 a = ( 1 a, . . . , 1 a).
Let σ = ⋆i∈Aσi = C
- µ1(σ1), . . . , µa(σa)
- be a face of ⋆i∈A ∆B.
Proposition
The following diagram commutes: ⋆
i∈A ∆B
⊃ σ
ιL
− − →
1 a
- µ1(σ1) + · · · + µa(σa)
- × 1
a
This is exactly the Cayley trick! ↓ π ↓ π ∆B × ∆A ⊃ C(σ1, . . . , σa
- ιπ (L)
− − − − − →
1 a
- σ1 + · · · + σa
- × 1
a
This is again the Cayley trick! ↓ π∆ ↓ π∆ ∆B ⊃ conv a
i=1 σi 1 a
- σ1 + · · · + σa
SLIDE 36
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
Set L = ❘ab × 1
a ⊂ ❘ab × ❘a, with 1 a = ( 1 a, . . . , 1 a).
Let σ = ⋆i∈Aσi = C
- µ1(σ1), . . . , µa(σa)
- be a face of ⋆i∈A ∆B.
Proposition
The following diagram commutes: ⋆
i∈A ∆B
⊃ σ
ιL
− − →
1 a
- µ1(σ1) + · · · + µa(σa)
- × 1
a
This is exactly the Cayley trick! ↓ π ↓ π ∆B × ∆A ⊃ C(σ1, . . . , σa
- ιπ (L)
− − − − − →
1 a
- σ1 + · · · + σa
- × 1
a
This is again the Cayley trick! ↓ π∆ ↓ π∆ ∆B ⊃ conv a
i=1 σi 1 a
- σ1 + · · · + σa
SLIDE 37
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
Set L = ❘ab × 1
a ⊂ ❘ab × ❘a, with 1 a = ( 1 a, . . . , 1 a).
Let σ = ⋆i∈Aσi = C
- µ1(σ1), . . . , µa(σa)
- be a face of ⋆i∈A ∆B.
Proposition
The following diagram commutes: ⋆
i∈A ∆B
⊃ σ
ιL
− − →
1 a
- µ1(σ1) + · · · + µa(σa)
- × 1
a
This is exactly the Cayley trick! ↓ π ↓ π ∆B × ∆A ⊃ C(σ1, . . . , σa
- ιπ (L)
− − − − − →
1 a
- σ1 + · · · + σa
- × 1
a
This is again the Cayley trick! ↓ π∆ ↓ π∆ ∆B ⊃ conv a
i=1 σi 1 a
- σ1 + · · · + σa
SLIDE 38
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
Set L = ❘ab × 1
a ⊂ ❘ab × ❘a, with 1 a = ( 1 a, . . . , 1 a).
Let σ = ⋆i∈Aσi = C
- µ1(σ1), . . . , µa(σa)
- be a face of ⋆i∈A ∆B.
Proposition
The following diagram commutes: ⋆
i∈A ∆B
⊃ σ
ιL
− − →
1 a
- µ1(σ1) + · · · + µa(σa)
- × 1
a
This is exactly the Cayley trick! ↓ π ↓ π ∆B × ∆A ⊃ C(σ1, . . . , σa
- ιπ (L)
− − − − − →
1 a
- σ1 + · · · + σa
- × 1
a
This is again the Cayley trick! ↓ π∆ ↓ π∆ ∆B ⊃ conv a
i=1 σi 1 a
- σ1 + · · · + σa
SLIDE 39
Introduction The Cayley trick Hom-complexes Dissection complexes
Joins and the Cayley trick
Set L = ❘ab × 1
a ⊂ ❘ab × ❘a, with 1 a = ( 1 a, . . . , 1 a).
Let σ = ⋆i∈Aσi = C
- µ1(σ1), . . . , µa(σa)
- be a face of ⋆i∈A ∆B.
Proposition
The following diagram commutes: ⋆
i∈A ∆B
⊃ σ
ιL
− − →
1 a
- µ1(σ1) + · · · + µa(σa)
- × 1
a
This is exactly the Cayley trick! ↓ π ↓ π ∆B × ∆A ⊃ C(σ1, . . . , σa
- ιπ (L)
− − − − − →
1 a
- σ1 + · · · + σa
- × 1
a
This is again the Cayley trick! ↓ π∆ ↓ π∆ ∆B ⊃ conv a
i=1 σi 1 a
- σ1 + · · · + σa
SLIDE 40
Introduction The Cayley trick Hom-complexes Dissection complexes
Outline
Introduction The associahedron Dissections of polygons Results in this talk The Cayley trick Joins and projections . . . and the Cayley trick Hom-complexes Hom-complexes and the Cayley trick Symmetry classes The case Hom(Kg, H) Dissection complexes
SLIDE 41
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes
Let G and H be graphs on g = |V(G)| and h = |V(H)| vertices.
Definition (Lovász; Babson & Kozlov)
- A homomorphism from G to H is a map ϕ : V(G) → V(H)
such that for any edge (x, y) of G, (ϕ(x), ϕ(y)) is an edge
- f H.
- Hom(G, H) is the polytopal subcomplex of ×x∈G∆V(H) of
all cells ×x∈V(G)σx such that if (x, y) ∈ E(G), then (σx, σy) is a complete bipartite subgraph of H.
- Hom+(G, H) is the simplicial subcomplex of ⋆x∈G ∆V(H) of
all simplices ⋆x∈V(G) σx such that if (x, y) ∈ E(G) and both σx and σy are nonempty, then (σx, σy) is a complete bipartite subgraph of H.
SLIDE 42
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes
Let G and H be graphs on g = |V(G)| and h = |V(H)| vertices.
Definition (Lovász; Babson & Kozlov)
- A homomorphism from G to H is a map ϕ : V(G) → V(H)
such that for any edge (x, y) of G, (ϕ(x), ϕ(y)) is an edge
- f H.
- Hom(G, H) is the polytopal subcomplex of ×x∈G∆V(H) of
all cells ×x∈V(G)σx such that if (x, y) ∈ E(G), then (σx, σy) is a complete bipartite subgraph of H.
- Hom+(G, H) is the simplicial subcomplex of ⋆x∈G ∆V(H) of
all simplices ⋆x∈V(G) σx such that if (x, y) ∈ E(G) and both σx and σy are nonempty, then (σx, σy) is a complete bipartite subgraph of H.
SLIDE 43
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes
Let G and H be graphs on g = |V(G)| and h = |V(H)| vertices.
Definition (Lovász; Babson & Kozlov)
- A homomorphism from G to H is a map ϕ : V(G) → V(H)
such that for any edge (x, y) of G, (ϕ(x), ϕ(y)) is an edge
- f H.
- Hom(G, H) is the polytopal subcomplex of ×x∈G∆V(H) of
all cells ×x∈V(G)σx such that if (x, y) ∈ E(G), then (σx, σy) is a complete bipartite subgraph of H.
- Hom+(G, H) is the simplicial subcomplex of ⋆x∈G ∆V(H) of
all simplices ⋆x∈V(G) σx such that if (x, y) ∈ E(G) and both σx and σy are nonempty, then (σx, σy) is a complete bipartite subgraph of H.
SLIDE 44
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes and the Cayley trick
Definition
The simplicial complex Homt
+(G, H) of transversal faces is the
subcomplex of Hom+(G, H) induced by the set {⋆x∈V(G) σx : |σx| > 0 for all x ∈ V(G)}.
Proposition
Set L = ❘gh × 1
- g. Then
ιL Homt
+(G, H) = ιL Hom+(G, H) = Hom(G, H).
In particular, we obtain an embedding of all these complexes into Euclidean space.
Proof.
This is precisely the Cayley trick!
SLIDE 45
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes and the Cayley trick
Definition
The simplicial complex Homt
+(G, H) of transversal faces is the
subcomplex of Hom+(G, H) induced by the set {⋆x∈V(G) σx : |σx| > 0 for all x ∈ V(G)}.
Proposition
Set L = ❘gh × 1
- g. Then
ιL Homt
+(G, H) = ιL Hom+(G, H) = Hom(G, H).
In particular, we obtain an embedding of all these complexes into Euclidean space.
Proof.
This is precisely the Cayley trick!
SLIDE 46
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes and the Cayley trick
Definition
The simplicial complex Homt
+(G, H) of transversal faces is the
subcomplex of Hom+(G, H) induced by the set {⋆x∈V(G) σx : |σx| > 0 for all x ∈ V(G)}.
Proposition
Set L = ❘gh × 1
- g. Then
ιL Homt
+(G, H) = ιL Hom+(G, H) = Hom(G, H).
In particular, we obtain an embedding of all these complexes into Euclidean space.
Proof.
This is precisely the Cayley trick!
SLIDE 47
Introduction The Cayley trick Hom-complexes Dissection complexes
Hom-complexes and the Cayley trick
Proposition
The following diagram commutes: ⋆
x∈V(G) ∆V(H)
⊃ Hom+(G, H)
ιL
− − → Hom(G, H) × 1
g
↓ π ↓ π ∆V(H) × ∆V(G) ⊃ π Hom+(G, H)
ιπ (L)
− − − − − → Hom(G, H)/SG × 1
g
↓ π∆ Hom(G, H)/SG Here Hom(G, H)/SG “:=” π∆π Hom(G, H) .
SLIDE 48
Introduction The Cayley trick Hom-complexes Dissection complexes
Symmetry classes of Hom-complexes
Let SG be the symmetry group of G.
Definition
- Hom(G, H)/SG is the union of the cells
{π(σ) : σ ∈ Hom(G, H)}, where π = π∆π This is not necessarily a polytopal complex!
- Hom(t)
+ (G, H)/SG is the simplicial complex induced by the
faces {π(σ) : σ ∈ Hom(t)
+ (G, H)}
SLIDE 49
Introduction The Cayley trick Hom-complexes Dissection complexes
Symmetry classes of Hom-complexes
Let SG be the symmetry group of G.
Definition
- Hom(G, H)/SG is the union of the cells
{π(σ) : σ ∈ Hom(G, H)}, where π = π∆π This is not necessarily a polytopal complex!
- Hom(t)
+ (G, H)/SG is the simplicial complex induced by the
faces {π(σ) : σ ∈ Hom(t)
+ (G, H)}
SLIDE 50
Introduction The Cayley trick Hom-complexes Dissection complexes
Symmetry classes of Hom-complexes
Let SG be the symmetry group of G.
Definition
- Hom(G, H)/SG is the union of the cells
{π(σ) : σ ∈ Hom(G, H)}, where π = π∆π This is not necessarily a polytopal complex!
- Hom(t)
+ (G, H)/SG is the simplicial complex induced by the
faces {π(σ) : σ ∈ Hom(t)
+ (G, H)}
SLIDE 51
Introduction The Cayley trick Hom-complexes Dissection complexes
Symmetry classes of Hom-complexes
Proposition
(a) Each cell π(σ) of Hom(G, H)/SG represents an SG-equivalence class
- f
faces
- f
the polytopal com- plex Hom(G, H).
31 51 32 52 42 41
(b) Each cell π(σ) of Hom(G, H)/SG is a generalized permutohedron in the sense of Postnikov, and all generalized permutohedra arise in this way.
SLIDE 52
Introduction The Cayley trick Hom-complexes Dissection complexes
Symmetry classes of Hom-complexes
Proposition
(a) Each cell π(σ) of Hom(G, H)/SG represents an SG-equivalence class
- f
faces
- f
the polytopal com- plex Hom(G, H).
31 51 32 52 42 41
(b) Each cell π(σ) of Hom(G, H)/SG is a generalized permutohedron in the sense of Postnikov, and all generalized permutohedra arise in this way.
SLIDE 53
Introduction The Cayley trick Hom-complexes Dissection complexes
The case Hom(Kg, H)
Theorem
(a) π Hom+(Kg, H) is a simplicial immersion of Hom+(Kg, H) into ∆V(H) × ∆[g]. Another way of expressing this is to say that Hom+(Kg, H) is a “horizontal” complex, i.e., it has no faces in ker π. (b) Each cell of π Homt
+(Kg, H) represents an Sg-equivalence
class of faces of the simplicial complex Homt
+(Kg, H).
(c) π Hom+(Kg, H) = ∆H, because Hom+(Kg, H) contains faces like (∆V(H), ∅, . . . , ∅)
SLIDE 54
Introduction The Cayley trick Hom-complexes Dissection complexes
The case Hom(Kg, H)
Theorem
(a) π Hom+(Kg, H) is a simplicial immersion of Hom+(Kg, H) into ∆V(H) × ∆[g]. Another way of expressing this is to say that Hom+(Kg, H) is a “horizontal” complex, i.e., it has no faces in ker π. (b) Each cell of π Homt
+(Kg, H) represents an Sg-equivalence
class of faces of the simplicial complex Homt
+(Kg, H).
(c) π Hom+(Kg, H) = ∆H, because Hom+(Kg, H) contains faces like (∆V(H), ∅, . . . , ∅)
SLIDE 55
Introduction The Cayley trick Hom-complexes Dissection complexes
The case Hom(Kg, H)
Theorem
(a) π Hom+(Kg, H) is a simplicial immersion of Hom+(Kg, H) into ∆V(H) × ∆[g]. Another way of expressing this is to say that Hom+(Kg, H) is a “horizontal” complex, i.e., it has no faces in ker π. (b) Each cell of π Homt
+(Kg, H) represents an Sg-equivalence
class of faces of the simplicial complex Homt
+(Kg, H).
(c) π Hom+(Kg, H) = ∆H, because Hom+(Kg, H) contains faces like (∆V(H), ∅, . . . , ∅)
SLIDE 56
Introduction The Cayley trick Hom-complexes Dissection complexes
The case Hom(Kg, H)
Theorem
(a) For any loopless graph H, the 1-skeleton of Hom(Kg, H)/Sg is part of the 1-skeleton of the hypersimplex ∆(h, g). (b) Hom(Kg, H)/Sg is a polytopal complex if and only if any complete g-partite subgraph of H is induced. This is the case if and only if ω(H) = g, i.e., the size of a largest clique in H is g.
SLIDE 57
Introduction The Cayley trick Hom-complexes Dissection complexes
The case Hom(Kg, H)
Theorem
(a) For any loopless graph H, the 1-skeleton of Hom(Kg, H)/Sg is part of the 1-skeleton of the hypersimplex ∆(h, g). (b) Hom(Kg, H)/Sg is a polytopal complex if and only if any complete g-partite subgraph of H is induced. This is the case if and only if ω(H) = g, i.e., the size of a largest clique in H is g.
SLIDE 58
Introduction The Cayley trick Hom-complexes Dissection complexes
The case Hom(Kg, H)
Proposition
The following diagram commutes: ⋆
i∈[g] ∆V(H)
⊃ Homt
+(Kg, H) ιL
− − → Hom(Kg, H) × 1
g
↓ π ↓ π ∆V(H) × ∆[g] ⊃ π Homt
+(Kg, H) ιπ L
− − − − → Hom(Kg, H)/Sg × 1
g
↓ π∆ ↓ π∆ ∆V(H) ⊃ Homt
+(Kg, H)/Sg
Hom(Kg, H)/Sg
SLIDE 59
Introduction The Cayley trick Hom-complexes Dissection complexes
Outline
Introduction The associahedron Dissections of polygons Results in this talk The Cayley trick Joins and projections . . . and the Cayley trick Hom-complexes Hom-complexes and the Cayley trick Symmetry classes The case Hom(Kg, H) Dissection complexes
SLIDE 60
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissection complexes
Let I(k, m) be the independence graph on the set of allowable diagonals: join two diagonals if they do not cross
Theorem
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
(c) For all k ≥ 3 and m ≥ 1, the simplicial complex T(k, m) is the (m − 2)-skeleton of a non-pure (m − 1)-dimensional polytopal complex ♦(k, m) whose cells are iterated cones
- ver cross-polytopes.
SLIDE 61
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissection complexes
Let I(k, m) be the independence graph on the set of allowable diagonals: join two diagonals if they do not cross
Theorem
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
(c) For all k ≥ 3 and m ≥ 1, the simplicial complex T(k, m) is the (m − 2)-skeleton of a non-pure (m − 1)-dimensional polytopal complex ♦(k, m) whose cells are iterated cones
- ver cross-polytopes.
SLIDE 62
Introduction The Cayley trick Hom-complexes Dissection complexes
Dissection complexes
Let I(k, m) be the independence graph on the set of allowable diagonals: join two diagonals if they do not cross
Theorem
(a) T(k, m) = skm−2 Homt
+
- Km−1, I(k, m)
- /Sm−1.
(b) D(k, m) = sk1 Hom
- Km−1, I(k, m)
- /Sm−1.
(c) For all k ≥ 3 and m ≥ 1, the simplicial complex T(k, m) is the (m − 2)-skeleton of a non-pure (m − 1)-dimensional polytopal complex ♦(k, m) whose cells are iterated cones
- ver cross-polytopes.
SLIDE 63
Introduction The Cayley trick Hom-complexes Dissection complexes
Fomin & Zelevinski’s question
T(4, 3) is the 1-skeleton of a polyhedral decomposition of ❘P2 into four squares and one octagon.
Question (Fomin & Zelevinski, 2005)
Is it true in general that T(k, m) is the (m − 2)-skeleton of a polyhedral manifold of dimension k + m − 5? Probably not:
Proposition
Let k ≥ 4 be even. T(k, 3) is the 1-skeleton of ♦(k, 3), a 2-dimensional polytopal complex that is the union of k/2 − 1 tori and Möbius strips, each one tesselated by 3k − 4 squares. But the tesselations of the torus boundaries above contain non-trivial torus knots, and it doesn’t seem possible to fill in a disk to separate the torus cell into two balls.
SLIDE 64
Introduction The Cayley trick Hom-complexes Dissection complexes
Fomin & Zelevinski’s question
T(4, 3) is the 1-skeleton of a polyhedral decomposition of ❘P2 into four squares and one octagon.
Question (Fomin & Zelevinski, 2005)
Is it true in general that T(k, m) is the (m − 2)-skeleton of a polyhedral manifold of dimension k + m − 5? Probably not:
Proposition
Let k ≥ 4 be even. T(k, 3) is the 1-skeleton of ♦(k, 3), a 2-dimensional polytopal complex that is the union of k/2 − 1 tori and Möbius strips, each one tesselated by 3k − 4 squares. But the tesselations of the torus boundaries above contain non-trivial torus knots, and it doesn’t seem possible to fill in a disk to separate the torus cell into two balls.
SLIDE 65
Introduction The Cayley trick Hom-complexes Dissection complexes
Fomin & Zelevinski’s question
T(4, 3) is the 1-skeleton of a polyhedral decomposition of ❘P2 into four squares and one octagon.
Question (Fomin & Zelevinski, 2005)
Is it true in general that T(k, m) is the (m − 2)-skeleton of a polyhedral manifold of dimension k + m − 5? Probably not:
Proposition
Let k ≥ 4 be even. T(k, 3) is the 1-skeleton of ♦(k, 3), a 2-dimensional polytopal complex that is the union of k/2 − 1 tori and Möbius strips, each one tesselated by 3k − 4 squares. But the tesselations of the torus boundaries above contain non-trivial torus knots, and it doesn’t seem possible to fill in a disk to separate the torus cell into two balls.
SLIDE 66
Introduction The Cayley trick Hom-complexes Dissection complexes
Fomin & Zelevinski’s question
T(4, 3) is the 1-skeleton of a polyhedral decomposition of ❘P2 into four squares and one octagon.
Question (Fomin & Zelevinski, 2005)
Is it true in general that T(k, m) is the (m − 2)-skeleton of a polyhedral manifold of dimension k + m − 5? Probably not:
Proposition
Let k ≥ 4 be even. T(k, 3) is the 1-skeleton of ♦(k, 3), a 2-dimensional polytopal complex that is the union of k/2 − 1 tori and Möbius strips, each one tesselated by 3k − 4 squares. But the tesselations of the torus boundaries above contain non-trivial torus knots, and it doesn’t seem possible to fill in a disk to separate the torus cell into two balls.
SLIDE 67
Introduction The Cayley trick Hom-complexes Dissection complexes
Fomin & Zelevinski’s question
T(4, 3) is the 1-skeleton of a polyhedral decomposition of ❘P2 into four squares and one octagon.
Question (Fomin & Zelevinski, 2005)
Is it true in general that T(k, m) is the (m − 2)-skeleton of a polyhedral manifold of dimension k + m − 5? Probably not:
Proposition
Let k ≥ 4 be even. T(k, 3) is the 1-skeleton of ♦(k, 3), a 2-dimensional polytopal complex that is the union of k/2 − 1 tori and Möbius strips, each one tesselated by 3k − 4 squares. But the tesselations of the torus boundaries above contain non-trivial torus knots, and it doesn’t seem possible to fill in a disk to separate the torus cell into two balls.
SLIDE 68
Introduction The Cayley trick Hom-complexes Dissection complexes
Open problems
- Invariants of (φ) Hom(+)(G, H)/SG: homotopy type, etc.
- So far, we are dealing with the An case. Can we also find
polytopal complexes for the Lie groups of type Bn and Dn?
- . . .
SLIDE 69
Introduction The Cayley trick Hom-complexes Dissection complexes
Open problems
- Invariants of (φ) Hom(+)(G, H)/SG: homotopy type, etc.
- So far, we are dealing with the An case. Can we also find
polytopal complexes for the Lie groups of type Bn and Dn?
- . . .
SLIDE 70
Introduction The Cayley trick Hom-complexes Dissection complexes
Open problems
- Invariants of (φ) Hom(+)(G, H)/SG: homotopy type, etc.
- So far, we are dealing with the An case. Can we also find
polytopal complexes for the Lie groups of type Bn and Dn?
- . . .