april 25, 2020 discrete math day of the northeastahmorales/talks/dmd.pdfbased on joint work with...
TRANSCRIPT
![Page 1: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/1.jpg)
Alejandro H. MoralesUMass Amherst
April 25, 2020Discrete Math Day of the NorthEast
Volumes and triangulations of flow polytopes of graphs
a+ 1 b+ 1
based on joint work with Karola Mészáros (Cornell)and Jessica Striker (NDSU)
slides available at people.math.umass.edu/∼ahmorales/talks/DMD.pdf
![Page 2: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/2.jpg)
Flow polytopes of graphs
graph G flow polytope FG poset volume
kn+1
?
1
n!
Catn := 1n+1
(2nn
)Euler numbers En
# standard tableauxstaircase shape
?
![Page 3: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/3.jpg)
Integral polytopesP a polytope in RN with integral vertices:
P is the convex hull of finitely many vertices v in ZN
P is the intersection of finitely many half spacesOR
![Page 4: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/4.jpg)
Integral polytopesP a polytope in RN with integral vertices:
P is the convex hull of finitely many vertices v in ZN
P is the intersection of finitely many half spacesOR
d-cube: convex hull of {0, 1}d
Cd ={
(x1, . . . , xd) 0 ≤ xi ≤ 1, i = 1, . . . , d}
![Page 5: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/5.jpg)
Volume of polytopes
Example:
standard simplex ∆n = {(x1, . . . , xn) |∑xi ≤ 1, xi ≥ 0}
∆2
euclidean volume
(normalized) volume 1
∆1 ∆3
1 12
16
normalized volume of P := dim(P )! · (euclidean volume of P )
![Page 6: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/6.jpg)
Volume of polytopes
Example:
normalized volume of P := dim(P )! · (euclidean volume of P )
euclidean volume 1
(normalized) volume d!
d-cube: convex hull of {0, 1}d
Cd ={
(x1, . . . , xd) 0 ≤ xi ≤ 1, i = 1, . . . , d}
![Page 7: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/7.jpg)
Volume of polytopes
Example:
normalized volume of P := dim(P )! · (euclidean volume of P )
euclidean volume 1
(normalized) volume d!
d-cube: convex hull of {0, 1}d
Cd ={
(x1, . . . , xd) 0 ≤ xi ≤ 1, i = 1, . . . , d}
![Page 8: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/8.jpg)
Lattice points of polytopes• #P ∩ ZN number of lattice points (discrete volume)
Example:
standard simplex ∆d = {(x1, . . . , xd) |∑d
i=1 xi ≤ 1, xi ≥ 0}
∆2
![Page 9: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/9.jpg)
Lattice points of polytopes• #P ∩ ZN number of lattice points (discrete volume)
LP (t) := #(tP ∩ ZN ) counts lattice points in t-dilation of P .
Example:
standard simplex ∆d = {(x1, . . . , xd) |∑d
i=1 xi ≤ 1, xi ≥ 0}
∆22
![Page 10: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/10.jpg)
Lattice points of polytopes• #P ∩ ZN number of lattice points (discrete volume)
LP (t) := #(tP ∩ ZN ) counts lattice points in t-dilation of P .
Example:
standard simplex ∆d = {(x1, . . . , xd) |∑d
i=1 xi ≤ 1, xi ≥ 0}
∆22
standard simplex t∆d = {(x1, . . . , xd) |∑d
i=1 xi ≤ t, xi ≥ 0}
![Page 11: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/11.jpg)
Lattice points of polytopes• #P ∩ ZN number of lattice points (discrete volume)
LP (t) := #(tP ∩ ZN ) counts lattice points in t-dilation of P .
Example:
standard simplex ∆d = {(x1, . . . , xd) |∑d
i=1 xi ≤ 1, xi ≥ 0}
∆22
standard simplex t∆d = {(x1, . . . , xd) |∑d
i=1 xi ≤ t, xi ≥ 0}
#(t∆d ∩ Zd) =(t+dd
)
![Page 12: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/12.jpg)
Flow polytopesG graph n+ 1 vertices |E| edges
a = (a1, a2, . . . , an,−∑ai), ai ∈ Zn
≥0
FG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
![Page 13: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/13.jpg)
Flow polytopesG graph n+ 1 vertices |E| edges
G
a = (a1, a2, . . . , an,−∑ai), ai ∈ Zn
≥0
FG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
a
a1 a2 a3 −a1 − a2 − a3
x14
x34
x13
x12 x23
x24x23 + x24 − x12 =a2
x12 + x13 + x14 =a1
x34 − x13 − x23 =a3
Example
![Page 14: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/14.jpg)
Flow polytopesG graph n+ 1 vertices |E| edges
G
a = (a1, a2, . . . , an,−∑ai), ai ∈ Zn
≥0
FG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
a
a1 a2 a3 −a1 − a2 − a3
x14
x34
x13
x12 x23
x24x23 + x24 − x12 =a2
x12 + x13 + x14 =a1
x34 − x13 − x23 =a3
Example
dimension of FG(a) is |E| − n
![Page 15: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/15.jpg)
Flow polytopesG graph n+ 1 vertices m edges
a = (1, 0, . . . , 0,−1)
FG(1, 0, . . . , 0,−1) is flows on G: netflow first vertex is 1,netflow last vertex −1, netflow other vertices is 0.
![Page 16: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/16.jpg)
Flow polytopesG graph n+ 1 vertices m edges
G
a = (1, 0, . . . , 0,−1)
FG(1, 0, . . . , 0,−1) is flows on G: netflow first vertex is 1,netflow last vertex −1, netflow other vertices is 0.
a
1 0 0 −1
x14
x34
x13
x12 x23
x24x23 + x24 − x12 = 0
x12 + x13 + x14 = 1
x34 − x13 − x23 = 0
Example
−x14 − x24 − x34 =−1
![Page 17: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/17.jpg)
Vertices of flow polytopes• vertices of flow polytopes of FG(1, 0, . . . , 0,−1) can be
viewed as unit flows on directed paths from vertex 1 ton+ 1 called routes.
1 −1
![Page 18: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/18.jpg)
Vertices of flow polytopes• vertices of flow polytopes of FG(1, 0, . . . , 0,−1) can be
viewed as unit flows on directed paths from vertex 1 ton+ 1 called routes.
1 −11 1 1
![Page 19: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/19.jpg)
Vertices of flow polytopes• vertices of flow polytopes of FG(1, 0, . . . , 0,−1) can be
viewed as unit flows on directed paths from vertex 1 ton+ 1 called routes.
1 −1
![Page 20: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/20.jpg)
Vertices of flow polytopes• vertices of flow polytopes of FG(1, 0, . . . , 0,−1) can be
viewed as unit flows on directed paths from vertex 1 ton+ 1 called routes.
1 −1
![Page 21: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/21.jpg)
Vertices of flow polytopes• vertices of flow polytopes of FG(1, 0, . . . , 0,−1) can be
viewed as unit flows on directed paths from vertex 1 ton+ 1 called routes.
1 −1
![Page 22: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/22.jpg)
Vertices of flow polytopes• vertices of flow polytopes of FG(1, 0, . . . , 0,−1) can be
viewed as unit flows on directed paths from vertex 1 ton+ 1 called routes.
1 −1
13
23
13
13
13
0
![Page 23: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/23.jpg)
Examples of flow polytopesFG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
ExampleG
1
x4
x1
x2
x3
x1 + x2 + x3 + x4 = 1
FG(1,−1) is a simplex−1
![Page 24: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/24.jpg)
Examples flow polytopesFG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
ExampleG
1
FG(1, 0− 1) is a product of simplicies ∆a ×∆b
−10
...
a+ 1 ...
b+ 1
![Page 25: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/25.jpg)
Examples flow polytopesFG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
ExampleG
1
FG(1, 0− 1) is a product of simplicies ∆a ×∆b
−10
...
a+ 1 ...
b+ 1
∆2 ×∆1
G
1 0 −1
![Page 26: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/26.jpg)
Examples flow polytopesFG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
ExampleG
1
FG(1, 0− 1) is a product of simplicies ∆a ×∆b
−10
...
a+ 1 ...
b+ 1
G
1 −10 0∆1 ×∆1 ×∆1
![Page 27: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/27.jpg)
Flow polytopes are "transcendental" tooflow polytopes have been related to:
• Jeffrey–Kirwan residues (Baldoni–Vergne 2009)• cluster algebras (Danilov–Karzanov–Koshevoy 2012)
• Toric geometry (Hille 2003)
![Page 28: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/28.jpg)
Flow polytopes are "transcendental" tooflow polytopes have been related to:
• generalized permutahedra (Mészáros-St. Dizier 2017)
• Schubert polynomials (Escobar-Mészáros 2018)(Fink-Mészáros-St. Dizier 2018)
• generalized permutahedra (Mészáros-St. Dizier 2017)• generalized permutahedra (Mészáros-St. Dizier 2017)• generalized permutahedra (Mészáros-St. Dizier 2017)
• diagonal harmonics (Mészáros-M-Rhoades 17, Liu-Mészáros-M 18)
1 1 1 −3
• Gelfand-Tsetlin polytopes (Liu-Mészáros-St. Dizier 2019)
![Page 29: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/29.jpg)
Flow polytopes are "transcendental" tooflow polytopes have been related to:
• Brändén-Huh’s Lorentzian polynomials (Mészáros-Setiabatra 2019)
• rational Catalan combinatorics(B-G-H-H-K-M-Y 2018, Yip 2019, Jang-Kim 2019)
• Alternating sign matrices (Mészáros-M-Striker 2019)
• juggling sequences (Harris-Insko-Omar 2015, B-H-H-M-S 2020)
![Page 30: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/30.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
![Page 31: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/31.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
![Page 32: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/32.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
![Page 33: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/33.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
![Page 34: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/34.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
Corollary If G is a planar graph thenvolumeFG(1, 0, . . . , 0,−1) = # linear extensions P .
By Stanley’s theory of order polytopes:
![Page 35: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/35.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
Corollary If G is a planar graph thenvolumeFG(1, 0, . . . , 0,−1) = # linear extensions P .
A linear extension of a poset P is an ordering of the posetelements compatible with the partial order.
By Stanley’s theory of order polytopes:
![Page 36: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/36.jpg)
Flow polytopes of planar graphsTheorem (Postnikov 13, Mészáros-M-Striker 19)If G is a planar graph then FG(1, 0, . . . , 0,−1) is equivalent toan order polytope of a certain poset P .
Corollary If G is a planar graph thenvolumeFG(1, 0, . . . , 0,−1) = # linear extensions P .
• Fk7(1, 0, 0, 0, 0, 0,−1) is not an order polytope.(Behrend-M-Panova 20+)
By Stanley’s theory of order polytopes:
![Page 37: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/37.jpg)
Flow polytopes of graphs
graph G flow polytope FG poset volume
kn+1
?
1
n!
Catn := 1n+1
(2nn
)Euler numbers En
# standard tableauxstaircase shape
?
![Page 38: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/38.jpg)
Examples flow polytopesFG(a) = {flows x(ε) ∈ R≥0, ε ∈ E(G) | netflow vertex i = ai}
Example
a
1 0 0 −1
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 0, . . . , 0,−1)
Fkn+1(1, 0, . . . , 0,−1) is called the Chan-Robbins-Yuen (CRYn) polytope
has 2n−1 vertices, dimension(n2
)
![Page 39: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/39.jpg)
Volume of the CRYn polytope
vn := volume(CRYn)
2 3 4 5 6 7n
vn 1 1 2 10 140 5880
![Page 40: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/40.jpg)
Volume of the CRYn polytope
vn := volume(CRYn)
2 3 4 5 6 7n
vn 1 1 2 10 140 5880
vnvn−1
1 2 5 14 42
![Page 41: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/41.jpg)
Volume of the CRYn polytope
vn := volume(CRYn)
2 3 4 5 6 7n
vn 1 1 2 10 140 5880
vnvn−1
1 2 5 14 42
• vn = Cat0Cat1 · · ·Catn−2 (Zeilberger 99)
Catn := 1n+1
(2nn
)are the Catalan numbers
![Page 42: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/42.jpg)
Volume of the CRYn polytope• vn = Cat0Cat1 · · ·Catn−2 (Zeilberger 99)
Catn := 1n+1
(2nn
)Catalan numbers (1, 1, 2, 5, 14, 42, . . .) count more than 200different objects
![Page 43: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/43.jpg)
Volume of the CRYn polytope• vn = Cat0Cat1 · · ·Catn−2 (Zeilberger 99)
Catn := 1n+1
(2nn
)Catalan numbers (1, 1, 2, 5, 14, 42, . . .) count more than 200different objects
![Page 44: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/44.jpg)
Volume of the CRYn polytope• vn = Cat0Cat1 · · ·Catn−2 (Zeilberger 99)
Catn := 1n+1
(2nn
)Catalan numbers (1, 1, 2, 5, 14, 42, . . .) count more than 200different objects
![Page 45: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/45.jpg)
Volume of the CRYn polytope• vn = Cat0Cat1 · · ·Catn−2 (Zeilberger 99)
Catn := 1n+1
(2nn
)Catalan numbers (1, 1, 2, 5, 14, 42, . . .) count more than 200different objects
. . . however, there is no combinatorial proof of formula for vn
![Page 46: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/46.jpg)
Flow polytopes of graphs
graph G flow polytope FG poset volume
kn+1
1
n!
Catn := 1n+1
(2nn
)Euler numbers En
# standard tableauxstaircase shape
Cat1Cat2 · · ·Catn−2Cat1Cat2 · · ·Catn−2?no poset
![Page 47: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/47.jpg)
Lattice points: Kostant’s partition functionlattice points of FG(a) are integral flows on G with netflow a
let KG(a) := #(FG(a) ∩ Zm) = LFG(a)(1)
Kkn+1(a) is called Kostant’s partition function.
![Page 48: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/48.jpg)
Lattice points: Kostant’s partition functionlattice points of FG(a) are integral flows on G with netflow a
let KG(a) := #(FG(a) ∩ Zm) = LFG(a)(1)
# of ways of writing a as an N-combination of vectorsei − ej for 1 ≤ i < j ≤ n+ 1
Kkn+1(a) =
Kkn+1(a) is called Kostant’s partition function.
![Page 49: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/49.jpg)
Lattice points: Kostant’s partition functionlattice points of FG(a) are integral flows on G with netflow a
let KG(a) := #(FG(a) ∩ Zm) = LFG(a)(1)
# of ways of writing a as an N-combination of vectorsei − ej for 1 ≤ i < j ≤ n+ 1
Kkn+1(a) =
1 0 −1
0
1 11 0 −1
1
0 0
(1, 0,−1) = e1 − e3 (1, 0,−1) = (e1 − e2) + (e2 − e3)
Kkn+1(a) is called Kostant’s partition function.
![Page 50: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/50.jpg)
Lattice points: Kostant’s partition functionlattice points of FG(a) are integral flows on G with netflow a
let KG(a) := #(FG(a) ∩ Zm) = LFG(a)(1)
# of ways of writing a as an N-combination of vectorsei − ej for 1 ≤ i < j ≤ n+ 1
Kkn+1(a) =
Kkn+1(a) is called Kostant’s partition function.
Generating function for Kkn+1(a):
∑a
KG(a)xa =∏
1≤i<j≤n+1
(1− xix−1j )−1.
![Page 51: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/51.jpg)
Lattice points: Kostant’s partition functionlattice points of FG(a) are integral flows on G with netflow a
let KG(a) := #(FG(a) ∩ Zm) = LFG(a)(1)
# of ways of writing a as an N-combination of vectorsei − ej for 1 ≤ i < j ≤ n+ 1
Kkn+1(a) =
1 0 −1
0
1 11 0 −1
1
0 0
(1, 0,−1) = e1 − e3 (1, 0,−1) = (e1 − e2) + (e2 − e3)
Formulas for weight multiplicities and tensor product multiplicities oftype A semisimple Lie algebras in terms of Kkn+1
(a).
Kkn+1(a) is called Kostant’s partition function.
![Page 52: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/52.jpg)
Lattice points: Kostant’s partition functionlattice points of FG(a) are integral flows on G with netflow a
let KG(a) := #(FG(a) ∩ Zm) = LFG(a)(1)
# of ways of writing a as an N-combination of vectorsei − ej for 1 ≤ i < j ≤ n+ 1
Kkn+1(a) =
Kkn+1(a) is called Kostant’s partition function.
". . . he said to me that in any good mathematical theory there should be atleast one “transcendental” element . . . should account for many of the subtletiesof the theory. In the Cartan-Weyl theory, he said that my partition function wasthe transcendental element."Bertram Kostant on profile of I. M. Gelfand (Notices of the AMS, Jan. 2013)
![Page 53: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/53.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
where ik is indeg(k)− 1
let KG(a) := #(FG(a) ∩ Zm)
![Page 54: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/54.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
Example
where ik is indeg(k)− 1
1 −1
let KG(a) := #(FG(a) ∩ Zm)
![Page 55: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/55.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
Example
where ik is indeg(k)− 1
1 −1
let KG(a) := #(FG(a) ∩ Zm)
![Page 56: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/56.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
Example
where ik is indeg(k)− 1
1 −1
0 −21 1 0 −21 11 2
1
volume = 2.
1
let KG(a) := #(FG(a) ∩ Zm)
![Page 57: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/57.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
where ik is indeg(k)− 1
Corollaryvolume(CRY n) = Kkn+1
(0, 0, 1, 2, . . . , n− 2,−(n−1
2
))
let KG(a) := #(FG(a) ∩ Zm)
![Page 58: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/58.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
where ik is indeg(k)− 1
Corollary
Zeilberger showed volume is Cat1 · · ·Catn−2 using thisinterpretation and the Morris constant term identity:
volume(CRY n) = Kkn+1(0, 0, 1, 2, . . . , n− 2,−
(n−1
2
))
let KG(a) := #(FG(a) ∩ Zm)
![Page 59: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/59.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
where ik is indeg(k)− 1
Corollary
Zeilberger showed volume is Cat1 · · ·Catn−2 using thisinterpretation and the Morris constant term identity:
constant term ofn∏
i=1
x−a+1i (1− xi)−b
∏1≤i<j≤n
(xi − xj)−c =
=n−1∏j=0
Γ(1 + c/2)Γ(a+ b− 1 + (n+ j − 1)c/2)
Γ(1 + (j + 1)c/2)Γ(a+ jc/2)Γ(b+ jc/2).
volume(CRY n) = Kkn+1(0, 0, 1, 2, . . . , n− 2,−
(n−1
2
))
let KG(a) := #(FG(a) ∩ Zm)
![Page 60: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/60.jpg)
Fundamental theorem volume of flow polytopes
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
where ik is indeg(k)− 1
Corollary
Zeilberger showed volume is Cat1 · · ·Catn−2 using thisinterpretation and the Morris constant term identity:
constant term ofn∏
i=1
x−a+1i (1− xi)−b
∏1≤i<j≤n
(xi − xj)−c =
=n−1∏j=0
Γ(1 + c/2)Γ(a+ b− 1 + (n+ j − 1)c/2)
Γ(1 + (j + 1)c/2)Γ(a+ jc/2)Γ(b+ jc/2).
volume(CRY n) = Kkn+1(0, 0, 1, 2, . . . , n− 2,−
(n−1
2
))
at a = b = c = 1 gives Cat1 · · ·Catn−2.
let KG(a) := #(FG(a) ∩ Zm)
![Page 61: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/61.jpg)
Zeilberger’s entire paper
![Page 62: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/62.jpg)
Flow polytopes of graphs
graph G flow polytope FG poset volume
kn+1
1
n!
Catn := 1n+1
(2nn
)Euler numbers En
# standard tableauxstaircase shape
Cat1Cat2 · · ·Catn−2Cat1Cat2 · · ·Catn−2?no poset
![Page 63: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/63.jpg)
Flow polytopes of graphs
graph G flow polytope FG poset volume
kn+1
1
n!
Catn := 1n+1
(2nn
)Euler numbers En
# standard tableauxstaircase shape
Cat1Cat2 · · ·Catn−2Cat1Cat2 · · ·Catn−2?no poset ?
![Page 64: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/64.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
![Page 65: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/65.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
Example
1 −1
![Page 66: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/66.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
Example
1 −1
0 −21 1 0 −21 11 2
11
volume = KG(0, 1, 1,−2) = 2.
![Page 67: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/67.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
Example
1 −1
0 −21 1 0 −21 11 2
11
Postnikov–Stanley gave a recursive triangulation of FG withsimplices indexed by integer flows in a similar flow polytope.
volume = KG(0, 1, 1,−2) = 2.
![Page 68: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/68.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
Example
1 −1
0 −21 1 0 −21 11 2
11
Postnikov–Stanley gave a recursive triangulation of FG withsimplices indexed by integer flows in a similar flow polytope.
volume = KG(0, 1, 1,−2) = 2.
G0 G1 G2
a i b a i ba i b ore f fe
e+ f e+ f
![Page 69: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/69.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
Example
1 −1
0 −21 1 0 −21 11 2
11
Postnikov–Stanley gave a recursive triangulation of FG withsimplices indexed by integer flows in a similar flow polytope.Question 1:
volume = KG(0, 1, 1,−2) = 2.
Can we describe this triangulation explicitly? i.e. what simplexcorresponds to each integer flow?
![Page 70: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/70.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
![Page 71: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/71.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
ProofBoth sides are the volume of FG(10 · · · 0− 1) ≡ FGr (10 · · · 0− 1).
![Page 72: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/72.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).Example:
1 −1
0 −21 1
0 −21 1
1 2
11
G
G
volume = 2
![Page 73: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/73.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).Example:
1 −1
0 −21 1
0 −21 1
1 2
11
Gr
1 −1
G
G
volume = 2
![Page 74: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/74.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).Example:
1 −1
0 −21 1
0 −21 1
1 2
11
1−10 10
−10 101
Gr
1 −1
GExample:
G Gr
volume = 2
![Page 75: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/75.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
• two distinct flow polytopes have the same number oflattice points!
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
![Page 76: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/76.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
• two distinct flow polytopes have the same number oflattice points!
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
0 1 2 110 −2−3
G Gr
Example
![Page 77: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/77.jpg)
Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
• two distinct flow polytopes have the same number oflattice points!
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
a+ 1 b+ 1...
...a −a
G a+ 1b+ 1
b
...... −b
Gr
0 0
Example
(a+bb
)= #(a∆b ∩ Zb) = #(b∆a ∩ Za) =
(a+ba
)
![Page 78: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/78.jpg)
Bijection between lattice points KG and KGr
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik, i′k is indegree −1 vertex k in G and Gr.
Question 2:Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−
∑k ik)
and lattice points of FGr (0, i′2, i′3, . . . ,−
∑k i′k)?
• KG(a) := #(FG(a) ∩ Zm)
![Page 79: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/79.jpg)
Summary
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
Question 1:
Question 2:Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−
∑k ik)
and lattice points of FGr (0, i′2, i′3, . . . ,−
∑k i′k)?
Can we describe this triangulation explicitly? i.e. what simplexcorresponds to each integer flow?
• let KG(a) := #(FG(a) ∩ Zm)
![Page 80: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/80.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
![Page 81: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/81.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths• for a route P with vertex v, Pv and vP are the subpaths ending and
starting at v.
setup
![Page 82: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/82.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
![Page 83: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/83.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
Example: two framings of G
![Page 84: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/84.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• routes P and Q are coherent if at shared vertex v, Pv,Qv havesame relative order as vP, vQ in the framing.
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
![Page 85: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/85.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• routes P and Q are coherent if at shared vertex v, Pv,Qv havesame relative order as vP, vQ in the framing.
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
P
Q
X
![Page 86: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/86.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• routes P and Q are coherent if at shared vertex v, Pv,Qv havesame relative order as vP, vQ in the framing.
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
P
Q
×
![Page 87: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/87.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• routes P and Q are coherent if at shared vertex v, Pv,Qv havesame relative order as vP, vQ in the framing.
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
Example: two framings of G
P
Q
× P
Q
X
![Page 88: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/88.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
• recall vertices of FG are indexed by routes/paths
• a framing of a graph is an ordering of the incoming and outgoingedges on each vertex 2, 3, . . . , n.
Example:
• routes P and Q are coherent if at shared vertex v, Pv,Qv havesame relative order as vP, vQ in the framing.
• for a route P with vertex v, Pv and vP are the subpaths ending andstarting at v.
setup
• a clique C is a maximal collection of pairwise coherent routes
![Page 89: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/89.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
Theorem (Danilov-Karzanov-Koshevoy 12)Given a framed graph G, the simplices {∆C} whose verticesare routes in cliques C give a unimodular triangulation of FG.
![Page 90: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/90.jpg)
DKK triangulation of flow polytopesIn 2012 Danilov-Karzanov-Koshevoy gave another triangulation of FG
characterizing the vertices of each simplex
Theorem (Danilov-Karzanov-Koshevoy 12)Given a framed graph G, the simplices {∆C} whose verticesare routes in cliques C give a unimodular triangulation of FG.
Example: FG
∆C
∆C′
![Page 91: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/91.jpg)
Summary
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
Question 1:Can we describe the Postnikov-Stanley triangulation explicitly? i.e. whatsimplex corresponds to each integer flow?
let KG(a) := #(FG(a) ∩ Zm)
![Page 92: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/92.jpg)
Relation between triangulationsQuestion 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
![Page 93: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/93.jpg)
Relation between triangulationsQuestion 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
![Page 94: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/94.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
![Page 95: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/95.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
![Page 96: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/96.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
0
![Page 97: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/97.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
0
![Page 98: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/98.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
0
![Page 99: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/99.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
0
![Page 100: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/100.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
1
![Page 101: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/101.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 1
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
2
![Page 102: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/102.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
Example:
0 −21 11 2
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
00
0
0
![Page 103: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/103.jpg)
Relation between triangulations
Theorem (Mészaros-M-Striker 19)Given a framing, the DKK triangulation is a PS triangulation andthere is a bijection between the cliques and the integer flows inFG(0, i2, i3, . . .): Φ : C 7→ fl.
fl((u, v)) = # times (u, v) appears in {Pv | P ∈ C} −1
Example:
0 −21 1
1
1
Question 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?
• for a route P with vertex v, Pv is the subpaths ending at v.
00
0
0
![Page 104: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/104.jpg)
Summary
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−
∑k ik),
CorollaryKG(0, i2, i3, . . . ,−
∑k ik) = KGr (0, i′2, i
′3, . . . ,−
∑k i′k)
Gr reverse of G, ik (i′k) is indegree −1 vertex k in G (Gr).
Question 2:Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−
∑k ik)
and lattice points of FGr (0, i′2, i′3, . . . ,−
∑k i′k)?
let KG(a) := #(FG(a) ∩ Zm)
![Page 105: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/105.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
![Page 106: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/106.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
![Page 107: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/107.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,
Φr : C 7→ fl′,fl′((u, v)) = # times (v, u) appears in {vP | P ∈ C} −1
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
![Page 108: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/108.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
Theorem (M 2020+)
fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,
Φr : C 7→ fl′,fl′((u, v)) = # times (v, u) appears in {vP | P ∈ C} −1
Φr ◦ Φ−1 is a desired bijection.
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
![Page 109: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/109.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
Theorem (M 2020+)
fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,
Φr : C 7→ fl′,fl′((u, v)) = # times (v, u) appears in {vP | P ∈ C} −1
Φr ◦ Φ−1 is a desired bijection.
Example:
Φ
0 −21 11 2
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
00
0 0
![Page 110: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/110.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
Theorem (M 2020+)
fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,
Φr : C 7→ fl′,fl′((u, v)) = # times (v, u) appears in {vP | P ∈ C} −1
Φr ◦ Φ−1 is a desired bijection.
Example:
−10 10
1ΦrΦ
0 −21 11 2
mirror
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
00
0 0 0 0
0 0 0
![Page 111: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/111.jpg)
Bijection between lattice points KG and KGr
Q2: Is there a bijection between lattice points of FG(0, i2, i3, . . . ,−∑
k ik)and lattice points of FGr (0, i′2, i
′3, . . . ,−
∑k i′k)?
Theorem (M 2020+)
fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,
Φr : C 7→ fl′,fl′((u, v)) = # times (v, u) appears in {vP | P ∈ C} −1
Φr ◦ Φ−1 is a desired bijection.
Example:
ΦrΦ
0 −21 1
1
1
−10 10
• for a route P with vertex v, Pv and vP are subpaths ending andstarting at v.
000
0 0 0
0 0 10
mirror
![Page 112: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/112.jpg)
Summary• FG(a) flow polytope of a graph G netflow a
• volumes and lattice points important in representation theory
![Page 113: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/113.jpg)
Summary
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik).
• let KG(a) := #(FG(a) ∩ Zm)
• FG(a) flow polytope of a graph G netflow a
• volumes and lattice points important in representation theory
• if G is planar then volumeFG(1, 0, . . . , 0,−1) is thenumber of linear extensions of a poset P .
• let KG(a) := #(FG(a) ∩ Zm)• let KG(a) := #(FG(a) ∩ Zm)kn+1
![Page 114: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/114.jpg)
Summary
Theorem (Stanley-Postnikov 09, Baldoni-Vergne 09)
volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑
k ik).
• We give an explicit bijection between the integer flows ofFG(0, i2, i3, . . .) and the simplices of a DKK triangulation.
• let KG(a) := #(FG(a) ∩ Zm)
• FG(a) flow polytope of a graph G netflow a
• volumes and lattice points important in representation theory
• if G is planar then volumeFG(1, 0, . . . , 0,−1) is thenumber of linear extensions of a poset P .
• let KG(a) := #(FG(a) ∩ Zm)• let KG(a) := #(FG(a) ∩ Zm)
• The bijection depends on a framing of G and has interestingsymmetry properties.
kn+1
![Page 115: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/115.jpg)
Gracias
Panta Rhei = everything flows (Heraclitus)
• (with K. Mészáros, J. Striker) On flow polytopes, orderpolytopes, and certain faces of the alternating sign matrixpolytope, arxiv:1510.03357v2, Discrete andComputational Geometry, Volume 62 (2019) 128–163
• Triangulations of flow polytopes, in preparation
![Page 116: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/116.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
p
![Page 117: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/117.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
![Page 118: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/118.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
![Page 119: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/119.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
• encode choices with a bipartite noncrossing tree
![Page 120: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/120.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
Example:
• encode choices with a bipartite noncrossing tree
e
e′
f
f ′
e
e′f
f ′
![Page 121: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/121.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
Example:
• encode choices with a bipartite noncrossing tree
e
e′
f
f ′
e
e′f
f ′
e
e′
f
f ′
![Page 122: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/122.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
Example:
• encode choices with a bipartite noncrossing tree
e
e′
f
f ′
e
e′f
f ′
e
e′
f
f ′
e
e′
f
f ′
![Page 123: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/123.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
Example:
• encode choices with a bipartite noncrossing tree
e
e′
f
f ′
e
e′f
f ′
e
e′
f
f ′
![Page 124: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/124.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
• encode choices with a bipartite noncrossing tree
e
e′f
f ′
e
e′
f
f ′
Example:
![Page 125: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/125.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
• encode choices with a bipartite noncrossing tree
e
e′f
f ′
e
e′
f
f ′
Example:
![Page 126: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/126.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
• encode choices with a bipartite noncrossing tree
e
e′f
f ′
e
e′
f
f ′
Example:
• bookkeep the noncrossing trees as an integer flow
![Page 127: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/127.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
• encode choices with a bipartite noncrossing tree• bookkeep the noncrossing trees as an integer flow
e
e′
f
f ′
Example:
0 −21 11 2
![Page 128: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/128.jpg)
Postnikov-Stanley triangulation of flow polytopesG0 G1 G2
a i b a i ba i bor
p q q−pp
p−qq
q ≥ pq < p
FG0≡ FG1
∪ FG2
p
Subdivision Lemma
and the intersection FG1∩ FG2
is lower dimensional.
• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.
• encode choices with a bipartite noncrossing tree• bookkeep the noncrossing trees as an integer flow
e
e′
f
f ′
Example:
0 −21 1
1
1
![Page 129: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/129.jpg)
Proof sketch: correspondence
G0 G1 G2
a i b a i ba i bor
e f fee+ f e+ f
In subdivision view new edges as sum/path of original edges
0 −21 1
1
1
![Page 130: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/130.jpg)
Proof sketch: correspondence
G0 G1 G2
a i b a i ba i bor
e f fee+ f e+ f
In subdivision view new edges as sum/path of original edges
Example:
0 −21 1
1
1
e
e′f
f ′
e
e′
f
f ′k
e+ f
e′ + f
e′ + f ′k
g g
![Page 131: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/131.jpg)
Proof sketch: correspondence
G0 G1 G2
a i b a i ba i bor
e f fee+ f e+ f
In subdivision view new edges as sum/path of original edges
Example:
0 −21 1
1
1
e
e′f
f ′
e
e′
f
f ′
g
e+ f ′
e′ + f ′k
k
e+ f
e′ + f
e′ + f ′
e+ f
e+ f ′ + k
e′ + f ′ + kk
g g g + k
![Page 132: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/132.jpg)
Proof sketch: correspondence
G0 G1 G2
a i b a i ba i bor
e f fee+ f e+ f
In subdivision view new edges as sum/path of original edges
Example:
0 −21 1
1
1
e
e′f
f ′
e
e′
f
f ′
g
e+ f ′
e′ + f ′k
k
e+ f
e′ + f
e′ + f ′
e+ f
e+ f ′ + k
e′ + f ′ + kk
g g g + k
![Page 133: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/133.jpg)
Connection to diagonal harmonics
a
1 1 1 −3
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 1, . . . , 1,−n)
![Page 134: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/134.jpg)
Connection to diagonal harmonics
a
1 1 1 −3
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 1, . . . , 1,−n)
Fkn+1(1, 1, . . . , 1,−n) is called the Tesler polytope
![Page 135: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/135.jpg)
Connection to diagonal harmonics
a
1 1 1 −3
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 1, . . . , 1,−n)
Fkn+1(1, 1, . . . , 1,−n) is called the Tesler polytope
has n! vertices, dimension(n2
)
![Page 136: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/136.jpg)
Connection to diagonal harmonics
a
1 1 1 −3
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 1, . . . , 1,−n)
Fkn+1(1, 1, . . . , 1,−n) is called the Tesler polytope
has n! vertices, dimension(n2
)• a weighted sum over lattice points gives Hilbert series of
the space of diagonal harmonics (Haglund 2011)
![Page 137: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/137.jpg)
Connection to diagonal harmonics
a
1 1 1 −3
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 1, . . . , 1,−n)
Fkn+1(1, 1, . . . , 1,−n) is called the Tesler polytope
has n! vertices, dimension(n2
)• a weighted sum over lattice points gives Hilbert series of
the space of diagonal harmonics (Haglund 2011)
• Connection was suggested by François Bergeron
![Page 138: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/138.jpg)
Connection to diagonal harmonics
a
1 1 1 −3
x14
x34
x13
x12 x23
x24
G is the complete graph kn+1
a = (1, 1, . . . , 1,−n)
Fkn+1(1, 1, . . . , 1,−n) is called the Tesler polytope
has n! vertices, dimension(n2
)
Theorem (Mészáros, M, Rhoades 2014)
volume equals (n2)!∏n−2
i=1 (2i+1)n−i−1· Cat1Cat2 · · ·Catn−1
• a weighted sum over lattice points gives Hilbert series ofthe space of diagonal harmonics (Haglund 2011)
![Page 139: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/139.jpg)
Other results: Lidskii volume formula
volumeFG(a1, . . . , an) =∑j
(m− n
j1, . . . , jn
)aj11 · · · ajnn
×KG(j1 − o1, . . . , jn − on, 0)
ov = outdeg(v)− 1
Theorem (Baldoni-Vergne 08, Postnikov-Stanley 08)graph G with m edges, n+ 1 vertices, ai ≥ 0
![Page 140: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/140.jpg)
Other results: Lidskii volume formula
volumeFG(a1, . . . , an) =∑j
(m− n
j1, . . . , jn
)aj11 · · · ajnn
×KG(j1 − o1, . . . , jn − on, 0)
ov = outdeg(v)− 1
Theorem (Baldoni-Vergne 08, Postnikov-Stanley 08)graph G with m edges, n+ 1 vertices, ai ≥ 0
• Original proof uses Jeffrey–Kirwan iterated residues
![Page 141: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/141.jpg)
Other results: Lidskii volume formula
volumeFG(a1, . . . , an) =∑j
(m− n
j1, . . . , jn
)aj11 · · · ajnn
×KG(j1 − o1, . . . , jn − on, 0)
ov = outdeg(v)− 1
Theorem (Baldoni-Vergne 08, Postnikov-Stanley 08)graph G with m edges, n+ 1 vertices, ai ≥ 0
• Original proof uses Jeffrey–Kirwan iterated residues
• give a proof using polytope subdivisions (Mészáros-M 2019)
![Page 142: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/142.jpg)
Other results: Lidskii volume formula
volumeFG(a1, . . . , an) =∑j
(m− n
j1, . . . , jn
)aj11 · · · ajnn
×KG(j1 − o1, . . . , jn − on, 0)
ov = outdeg(v)− 1
Theorem (Baldoni-Vergne 08, Postnikov-Stanley 08)graph G with m edges, n+ 1 vertices, ai ≥ 0
• Original proof uses Jeffrey–Kirwan iterated residues
• give a proof using polytope subdivisions (Mészáros-M 2019)
Corollary:KG(j1 − o1, . . . , jn − on, 0) are mixed volumes and so arelog-concave in j1, . . . , jn.
![Page 143: April 25, 2020 Discrete Math Day of the NorthEastahmorales/talks/DMD.pdfbased on joint work with Karola Mészáros (Cornell) and Jessica Striker (NDSU) slides available atpeople.math.umass.edu](https://reader036.vdocument.in/reader036/viewer/2022070903/5f6bc92fb5f69347df387e68/html5/thumbnails/143.jpg)
Other results: Lidskii volume formula
volumeFG(a1, . . . , an) =∑j
(m− n
j1, . . . , jn
)aj11 · · · ajnn
×KG(j1 − o1, . . . , jn − on, 0)
ov = outdeg(v)− 1
Theorem (Baldoni-Vergne 08, Postnikov-Stanley 08)graph G with m edges, n+ 1 vertices, ai ≥ 0
• Original proof uses Jeffrey–Kirwan iterated residues
• give a proof using polytope subdivisions (Mészáros-M 2019)
• define combinatorial objects like parking functions that index thevolume of FG(a) (B-G-H-H-K-M-Y 2019)
1 1 −221
12 1
2
21