= 12 12 14 14 14 23 23 23 34 34 34 1 1 2 2 2 4 4 4 - - PowerPoint PPT Presentation
Plabic graphs and zonotopal tilings Pavel Galashin MIT galashin@mit.edu FPSAC 2018, Dartmouth College, July 19, 2018 1234 1234 1234 124 124 123 123 123 134 134 134 234 234 234 = 12 12 14 14 14 23 23 23 34 34 34 1 1 2
Pavel Galashin
MIT galashin@mit.edu
FPSAC 2018, Dartmouth College, July 19, 2018
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 1 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
level = 0 level = 1 level = 2 level = 3 level = 4 level = 5
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 1 3 3 5 2 3 5 1 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
level = 0 level = 1 level = 2 level = 3 level = 4 level = 5
1 3 3 5 1 3 5 2 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 1 3 3 5 2 3 5 1 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
level = 0 level = 1 level = 2 level = 3 level = 4 level = 5
1 3 3 5 1 3 5 2 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
1 2 1 5 2 3 3 4 4 5 1 3 3 5 1 2 1 5 2 3 3 4 4 5
level = 2
1 3 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
1 2 1 5 2 3 3 4 4 5 1 3 3 5 1 2 1 5 2 3 3 4 4 5
level = 2
1 3 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at level k of fine zonotopal tilings of Z(n, 3)
a (2, 5)-plabic graph
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Z(n, 3) for n = 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 2 / 33
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Definition (Minkowski sum)
A, B ⊂ Rd, A + B := {a + b | a ∈ A, b ∈ B}.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 4 / 33
Definition (Minkowski sum)
A, B ⊂ Rd, A + B := {a + b | a ∈ A, b ∈ B}.
Definition
Vector configuration: V = (v1, v2, . . . , vn), where vi ∈ Rd.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 4 / 33
Definition (Minkowski sum)
A, B ⊂ Rd, A + B := {a + b | a ∈ A, b ∈ B}.
Definition
Vector configuration: V = (v1, v2, . . . , vn), where vi ∈ Rd. Zonotope: ZV := [0, v1] + [0, v2] + · · · + [0, vn] ⊂ Rd.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 4 / 33
V = v1 v2 → ZV =
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 5 / 33
V = v1 v2 → ZV = V = v1 v2 v3 → ZV =
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 5 / 33
V = v1 v2 → ZV = V = v1 v2 v3 → ZV =
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 5 / 33
V = v1 v2 → ZV = V = v1 v2 v3 → ZV = V = v1 v2 v3 v4 → ZV = 0 v1 v2 v3 v4 v1 v2 v3 v4
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 5 / 33
Definition
Cyclic vector configuration: C(n, d) := (v1, v2, . . . , vn), where vi = (1, ri, r2
i , . . . , rd−1 i
) for some 0 < r1 < r2 < · · · < rn ∈ R.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 6 / 33
Definition
Cyclic vector configuration: C(n, d) := (v1, v2, . . . , vn), where vi = (1, ri, r2
i , . . . , rd−1 i
) for some 0 < r1 < r2 < · · · < rn ∈ R. C(4, 2) = v1 v2 v3 v4
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 6 / 33
Definition
Cyclic vector configuration: C(n, d) := (v1, v2, . . . , vn), where vi = (1, ri, r2
i , . . . , rd−1 i
) for some 0 < r1 < r2 < · · · < rn ∈ R. C(4, 2) = v1 v2 v3 v4
C(6, 3) =
v1 v2 v3 v4 v5 v6 z = 1
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 6 / 33
Definition
Cyclic vector configuration: C(n, d) := (v1, v2, . . . , vn), where vi = (1, ri, r2
i , . . . , rd−1 i
) for some 0 < r1 < r2 < · · · < rn ∈ R. Cyclic zonotope: Z(n, d) := ZC(n,d). C(4, 2) = v1 v2 v3 v4
C(6, 3) =
v1 v2 v3 v4 v5 v6 z = 1
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 6 / 33
Definition
A zonotopal tiling of ZV is a polyhedral subdivision ∆ of ZV into smaller zonotopes.
∆
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33
Definition
A zonotopal tiling of ZV is a polyhedral subdivision ∆ of ZV into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes.
∆
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33
Definition
A zonotopal tiling of ZV is a polyhedral subdivision ∆ of ZV into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. A piece ZV′ is a parallelotope if the vectors in V′ form a basis of Rd.
∆
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33
Definition
A zonotopal tiling of ZV is a polyhedral subdivision ∆ of ZV into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. A piece ZV′ is a parallelotope if the vectors in V′ form a basis of Rd.
∅ 4 34 234 1234 123 12 1 14 24 124
∆ Vert(∆)⊂ 2[n]
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33
Definition
A zonotopal tiling of ZV is a polyhedral subdivision ∆ of ZV into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. A piece ZV′ is a parallelotope if the vectors in V′ form a basis of Rd.
∅ 4 34 234 1234 123 12 1 14 24 124
v2 v2 v2 v2 ∆ Vert(∆)⊂ 2[n]
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33
Fact
Number of vertices in a fine zonotopal tiling of ZV equals the number Ind(V) of linearly independent subsets of V.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33
Fact
Number of vertices in a fine zonotopal tiling of ZV equals the number Ind(V) of linearly independent subsets of V. Ind(C(n, d)) = n
n 1
n d
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33
Fact
Number of vertices in a fine zonotopal tiling of ZV equals the number Ind(V) of linearly independent subsets of V. Ind(C(n, d)) = n
n 1
n d
V = v1 v2 v3 v4 ∆ =
∅ 4 34 234 1234 123 12 1 14 24 124 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33
Fact
Number of vertices in a fine zonotopal tiling of ZV equals the number Ind(V) of linearly independent subsets of V. Ind(C(n, d)) = n
n 1
n d
V = v1 v2 v3 v4 ∆ =
∅ 4 34 234 1234 123 12 1 14 24 124
Ind(V) = 4
4 1
4 2
| Vert(∆)| = 11.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33
Question
Which collections of subsets of [n] can appear as Vert(∆), where ∆ is a fine zonotopal tiling of Z(n, 2)?
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33
Question
Which collections of subsets of [n] can appear as Vert(∆), where ∆ is a fine zonotopal tiling of Z(n, 2)?
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa).
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33
Question
Which collections of subsets of [n] can appear as Vert(∆), where ∆ is a fine zonotopal tiling of Z(n, 2)?
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa). Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33
Question
Which collections of subsets of [n] can appear as Vert(∆), where ∆ is a fine zonotopal tiling of Z(n, 2)?
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa). Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
D ⊂ 2[n] is a strongly separated collection if all S, T ∈ D are strongly separated.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33
Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 10 / 33
Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
Proposition (Leclerc–Zelevinsky (1998))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 2), and maximal by inclusion strongly separated collections D ⊂ 2[n].
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 10 / 33
Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
Proposition (Leclerc–Zelevinsky (1998))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 2), and maximal by inclusion strongly separated collections D ⊂ 2[n].
Corollary (Leclerc–Zelevinsky (1998))
Purity phenomenon: every maximal by inclusion strongly separated collection D ⊂ 2[n] is also maximal by size: |D| = n
n 1
n 2
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 10 / 33
v1 v2 v3 v4 v5 v6 z = 1
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 11 / 33
v1 v2 v3 v4 v5 v6 z = 1
∅ 1 2 3 12 13 23 123 ∅ 1 2 3 12 13 23 123 ∅ 1 2 3 13 23 123
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 11 / 33
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Z(4, 3)
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 12 / 33
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Z(4, 3) Q: How many fine zonotopal tilings?
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 12 / 33
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa).
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa).
Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa).
Definition (G. (2017))
S, T ⊂ [n] are chord separated if there is no i < j < k < ℓ such that i, k ∈ S \ T and j, ℓ ∈ T \ S (or vice versa).
Strongly separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa).
Definition (G. (2017))
S, T ⊂ [n] are chord separated if there is no i < j < k < ℓ such that i, k ∈ S \ T and j, ℓ ∈ T \ S (or vice versa).
Strongly separated: Chord separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
1 2 3 4 5 6 7 8 9 2 4 5 1 7
S \ T T \ S
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33
Definition (Leclerc–Zelevinsky (1998))
S, T ⊂ [n] are strongly separated if there is no i < j < k such that i, k ∈ S \ T and j ∈ T \ S (or vice versa).
Definition (G. (2017))
S, T ⊂ [n] are chord separated if there is no i < j < k < ℓ such that i, k ∈ S \ T and j, ℓ ∈ T \ S (or vice versa).
Strongly separated: Chord separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
1 2 3 4 5 6 7 8 9 2 4 5 1 7
S \ T T \ S
When |S| = |T|, both definitions are due to Leclerc–Zelevinsky.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33
Strongly separated: Chord separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
1 2 3 4 5 6 7 8 9 2 4 5 1 7
S \ T T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 14 / 33
Strongly separated: Chord separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
1 2 3 4 5 6 7 8 9 2 4 5 1 7
S \ T T \ S
Proposition (Leclerc–Zelevinsky (1998))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 2), and maximal by inclusion strongly separated collections D ⊂ 2[n].
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 14 / 33
Strongly separated: Chord separated:
1 2 3 4 5 6 7 8 9 2 4 5 7 9
S \ T T \ S
1 2 3 4 5 6 7 8 9 2 4 5 1 7
S \ T T \ S
Proposition (Leclerc–Zelevinsky (1998))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 2), and maximal by inclusion strongly separated collections D ⊂ 2[n].
Theorem (G. (2017))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 3), and maximal by inclusion chord separated collections D ⊂ 2[n].
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 14 / 33
Chord separation: no i < j < k < ℓ such that i, k ∈ S \ T, j, ℓ ∈ T \ S or vice versa.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33
Chord separation: no i < j < k < ℓ such that i, k ∈ S \ T, j, ℓ ∈ T \ S or vice versa. The only two subsets of {1, 2, 3, 4} that are not chord separated:
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33
Chord separation: no i < j < k < ℓ such that i, k ∈ S \ T, j, ℓ ∈ T \ S or vice versa. The only two subsets of {1, 2, 3, 4} that are not chord separated: {1, 3} and {2, 4}.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33
Chord separation: no i < j < k < ℓ such that i, k ∈ S \ T, j, ℓ ∈ T \ S or vice versa. The only two subsets of {1, 2, 3, 4} that are not chord separated: {1, 3} and {2, 4}. There are exactly two maximal by inclusion chord separated collections D ⊂ 2[n].
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33
Chord separation: no i < j < k < ℓ such that i, k ∈ S \ T, j, ℓ ∈ T \ S or vice versa. The only two subsets of {1, 2, 3, 4} that are not chord separated: {1, 3} and {2, 4}. There are exactly two maximal by inclusion chord separated collections D ⊂ 2[n].
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Q: How many fine zonotopal tilings?
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33
Chord separation: no i < j < k < ℓ such that i, k ∈ S \ T, j, ℓ ∈ T \ S or vice versa. The only two subsets of {1, 2, 3, 4} that are not chord separated: {1, 3} and {2, 4}. There are exactly two maximal by inclusion chord separated collections D ⊂ 2[n].
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Q: How many fine zonotopal tilings? A: Two.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 1 ∅ 2 3 4 14 23 34 13 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 1 ∅ 2 3 4 14 23 34 13 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 13 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 12 124 ∅ 2 3 4 14 23 34 24 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 12 124 ∅ 2 3 4 14 23 34 24 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 12 124 ∅ 2 3 4 14 23 34 24 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 24 123 124 134 234 1234 12 124 ∅ 2 3 4 14 23 34 123 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33
Definition (Postnikov (2007))
A plabic graph is reduced if it contains:
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33
Definition (Postnikov (2007))
A plabic graph is reduced if it contains:
No closed strands
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33
Definition (Postnikov (2007))
A plabic graph is reduced if it contains:
No closed strands No strand intersects itself
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33
Definition (Postnikov (2007))
A plabic graph is reduced if it contains:
No closed strands No strand intersects itself No “bad double crossings”
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33
Definition (Postnikov (2007))
A plabic graph is reduced if it contains:
No closed strands No strand intersects itself No “bad double crossings” “Good double crossings” are OK!
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33
Definition (Postnikov (2007))
A plabic graph is reduced if it contains:
No closed strands No strand intersects itself No “bad double crossings” “Good double crossings” are OK!
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33
Definition (Postnikov (2007))
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i. 4 5 1 2 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33
Definition (Postnikov (2007))
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i. 4 5 1 2 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33
Definition (Postnikov (2007))
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i. 4 5 1 2 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33
Definition (Postnikov (2007))
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i. 4 5 1 2 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33
Definition (Postnikov (2007))
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i. 4 5 1 2 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33
Definition (Postnikov (2007))
A (k, n)-plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i. 4 5 1 2 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set:
1 2 1 5 2 3 3 4 4 5 1 3 3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
1 2 1 5 2 3 3 4 4 5 1 3 3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
2 3 3 4 4 5
3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
12 1 5
3 4 4 5 1 3 3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
1 2 1 5 23
4 5 13
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
1 2 1 5 2 3 34
1 3 3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
1 2 15 2 3 3 4 45 1 3 35
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
1 2 1 5 2 3 3 4 4 5 1 3 3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Postnikov (2007): each (k, n)-plabic graph has k(n − k) + 1 faces. Scott (2005): label each face of a (k, n)-plabic graph by a k-element set: include j in this set iff the face is to the left of the strand i → j.
1 2 1 5 2 3 3 4 4 5 1 3 3 5 1 2 1 5 2 3 3 4 4 5 1 3 3 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33
Conjecture (Leclecrc–Zelevinsky (1998), Scott (2005))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 22 / 33
Conjecture (Leclecrc–Zelevinsky (1998), Scott (2005))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1. Proved independently by Danilov–Karzanov–Koshevoy (2010) and Oh–Postnikov–Speyer (2011).
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 22 / 33
Conjecture (Leclecrc–Zelevinsky (1998), Scott (2005))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1. Proved independently by Danilov–Karzanov–Koshevoy (2010) and Oh–Postnikov–Speyer (2011).
Theorem (Oh–Postnikov–Speyer (2011))
The map G → Faces(G) is a bijection∗ between: (k, n)-plabic graphs, and maximal by inclusion chord separated collections D ⊂ [n]
k
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 22 / 33
Corollary (Oh–Postnikov–Speyer (2011))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33
Corollary (Oh–Postnikov–Speyer (2011))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1.
Theorem (G. (2017))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 3), and maximal by inclusion chord separated collections D ⊂ 2[n].
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33
Corollary (Oh–Postnikov–Speyer (2011))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1.
Theorem (G. (2017))
The map ∆ → Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z(n, 3), and maximal by inclusion chord separated collections D ⊂ 2[n].
Corollary (G. (2017))
Every maximal by inclusion chord separated collection D ⊂ 2[n] has size Ind(C(n, 3)) = n
n 1
n 2
n 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33
Corollary (Oh–Postnikov–Speyer (2011))
Every maximal by inclusion chord separated collection D ⊂ [n]
k
k(n − k) + 1. Luckily for us, n
n 1
n 2
n 3
n
(k(n − k) + 1).
Corollary (G. (2017))
Every maximal by inclusion chord separated collection D ⊂ 2[n] has size Ind(C(n, 3)) = n
n 1
n 2
n 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33
∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 1 3 3 5 2 3 5 1 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
level = 0 level = 1 level = 2 level = 3 level = 4 level = 5
1 3 3 5 1 3 5 2 3 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
S S a S b S c S a b S a c S b c S a b c S S a S b S c S a b S a c S b c S a b c S S a S b S c S a c S b c S a b c
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 25 / 33
S S a S b S c S a b S a c S b c S a b c S S a S b S c S a b S a c S b c S a b c S S a S b S c S a c S b c S a b c S S a S b S c S a b S a c S b c S a b c S S a S b S c S a b S a c S b c S a b c
z = |S| z = |S| + 1 z = |S| + 2 z = |S| + 3
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 25 / 33
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 1 ∅ 2 3 4 14 23 34 13 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 1 12 2 12 123 23 1 12 123 13 3 13 1 2 23 3 13 123 23
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 1 ∅ 2 3 4 14 23 34 13 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 14 124 1234 134 12 124 1234 123 1 12 124 14 13 134 1234 123 1 13 134 14 1 13 123 12
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 13 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 1 14 4 1 13 3 1 13 134 14 3 34 4 34 134 14 3 34 134 13
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 3 13 134 34 3 13 123 23 13 134 1234 123 23 123 1234 234 34 134 1234 234 3 23 234 34
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 13 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 26 / 33
Theorem (Postnikov (2007))
Any two (k, n)-plabic graphs are connected by a sequence of moves: (M1) (M2) (M3)
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 27 / 33
Theorem (Postnikov (2007))
Any two (k, n)-plabic graphs are connected by a sequence of moves: (M1) (M2) (M3) A flip of a fine zonotopal tiling of Z(n, 3) consists of replacing one tiling of Z(4, 3) with another.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 27 / 33
Theorem (Postnikov (2007))
Any two (k, n)-plabic graphs are connected by a sequence of moves: (M1) (M2) (M3) A flip of a fine zonotopal tiling of Z(n, 3) consists of replacing one tiling of Z(4, 3) with another.
Theorem (Ziegler (1993))
Any two fine zonotopal tilings of Z(n, 3) are connected by a sequence of flips.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 27 / 33
S S a S b S c S d S a b S a d S b c S c d S a b c S a b d S a c d S b c d S a b c d S S a S b S c S d S a b S a d S b c S c d S a b c S a b d S a c d S b c d S a b c d S S b S c S d S a d S b c S c d S a b c S a c d S b c d S a b c d S S a S b S c S d S a b S a c S a d S b c S c d S a b c S a b d S a c d S b c d S a b c d S S a S b S c S d S a b S a c S a d S b c S c d S a b c S a b d S a c d S b c d S a b c d (M1)
← − →
(M2)
← − →
(M3)
← − →
S S a S b S c S d S a b S a d S b c S b d S c d S a b c S a b d S a c d S b c d S a b c d S S a S b S c S d S a b S a d S b c S b d S c d S a b c S a b d S a c d S b c d S a b c d
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 28 / 33
S S c S b S a S b c S a c S a b S a b c S S c S b S a S b c S a c S a b S a b c S S c S b S a S a c S a b S a b c ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 1 2 3 4 5 1 2 1 5 2 3 3 4 4 5 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5 ∅ 3 4 5 1 5 2 3 3 4 4 5 1 4 5 2 3 4 3 4 5 1 2 3 4 1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 29 / 33
S S c S b S a S bc S ac S a b S a bc S S c S b S a S bc S ac S a b S a bc S S c S b S a S ac S a b S a bc ∅ 1 2
3
4 5 1 2 1 5 23
3 4
4 5 1 23 1 2 5 1 4 5 23 4
3 4 5
1 23 4 1 23 5 1 2 4 5 13 4 5 23 4 5 1 23 4 5 ∅ 1 2
3
4 5 1 2 1 5 23
3 4
4 5 1 23 1 2 5 1 4 5 23 4
3 4 5
1 23 4 1 23 5 1 2 4 5 13 4 5 23 4 5 1 23 4 5 ∅
3
4 5 1 5 23
3 4
4 5 1 4 5 23 4
3 4 5
1 23 4 1 2 4 5 13 4 5 23 4 5 1 23 4 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 29 / 33
1 2 1 5 23
3 4
4 5 13
3 5
1 2 1 5 23
3 4
4 5 13
3 5 4 5 1 2
3
∅ 1 2
3
4 5 1 2 1 5 23
3 4
4 5 1 23 1 2 5 1 4 5 23 4
3 4 5
1 23 4 1 23 5 1 2 4 5 13 4 5 23 4 5 1 23 4 5 ∅ 1 2
3
4 5 1 2 1 5 23
3 4
4 5 1 23 1 2 5 1 4 5 23 4
3 4 5
1 23 4 1 23 5 1 2 4 5 13 4 5 23 4 5 1 23 4 5 ∅
3
4 5 1 5 23
3 4
4 5 1 4 5 23 4
3 4 5
1 23 4 1 2 4 5 13 4 5 23 4 5 1 23 4 5
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 29 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at z = k of fine zonotopal tilings of Z(n, 3)
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 30 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at z = k of fine zonotopal tilings of Z(n, 3) Purity = ⇒ every (k, n)-plabic graph arises this way
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 30 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at z = k of fine zonotopal tilings of Z(n, 3) Purity = ⇒ every (k, n)-plabic graph arises this way Moves = horizontal sections of flips
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 30 / 33
Theorem (G. (2017))
(k, n)-plabic graphs
planar
dual
horizontal sections at z = k of fine zonotopal tilings of Z(n, 3) Purity = ⇒ every (k, n)-plabic graph arises this way Moves = horizontal sections of flips Strands = horizontal sections of pseudoplanes.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 30 / 33
Slides: http://math.mit.edu/~galashin/slides/FPSAC2018.pdf Pavel Galashin. Plabic graphs and zonotopal tilings.
Pavel Galashin and Alexander Postnikov. Purity and separation for oriented matroids arXiv preprint arXiv:1708.01329, 2017. Alexander Postnikov. Total positivity, Grassmannians, and networks. arXiv preprint math/0609764, 2006. Suho Oh, Alexander Postnikov, and David E. Speyer. Weak separation and plabic graphs.
Bernard Leclerc and Andrei Zelevinsky. Quasicommuting families of quantum Pl¨ ucker coordinates. In Kirillov’s seminar on representation theory, vol. 181 of Amer. Math. Soc. Transl., pages 85–108. Amer. Math. Soc., Providence, RI, 1998. G¨ unter M. Ziegler. Higher Bruhat orders and cyclic hyperplane arrangements. Topology, 32(2):259–279, 1993.
Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 32 / 33
∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 14 23 34 123 124 134 234 1234 ∅ 2 3 4 14 23 34 123 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234 ∅ 1 2 3 4 12 13 14 23 34 123 124 134 234 1234
Slides: http://math.mit.edu/~galashin/slides/FPSAC2018.pdf Pavel Galashin. Plabic graphs and zonotopal tilings.
Pavel Galashin and Alexander Postnikov. Purity and separation for oriented matroids arXiv preprint arXiv:1708.01329, 2017. Alexander Postnikov. Total positivity, Grassmannians, and networks. arXiv preprint math/0609764, 2006. Suho Oh, Alexander Postnikov, and David E. Speyer. Weak separation and plabic graphs.
Bernard Leclerc and Andrei Zelevinsky. Quasicommuting families of quantum Pl¨ ucker coordinates. In Kirillov’s seminar on representation theory, vol. 181 of Amer. Math. Soc. Transl., pages 85–108. Amer. Math. Soc., Providence, RI, 1998. G¨ unter M. Ziegler. Higher Bruhat orders and cyclic hyperplane arrangements. Topology, 32(2):259–279, 1993.