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

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

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

?

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

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}

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 )

Volume of polytopes

Example:


euclidean volume 1

(normalized) volume d!


Cd ={

(x1, . . . , xd) 0 ≤ xi ≤ 1, i = 1, . . . , d}

Volume of polytopes

Example:


euclidean volume 1

(normalized) volume d!


Cd ={

(x1, . . . , xd) 0 ≤ xi ≤ 1, i = 1, . . . , d}

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


LP (t) := #(tP ∩ ZN ) counts lattice points in t-dilation of P .

Example:


i=1 xi ≤ 1, xi ≥ 0}

∆22



Example:


i=1 xi ≤ 1, xi ≥ 0}

∆22

standard simplex t∆d = {(x1, . . . , xd) |∑d

i=1 xi ≤ t, xi ≥ 0}



Example:


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

)

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}


G

a = (a1, a2, . . . , an,−∑ai), ai ∈ Zn

≥0


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


G

a = (a1, a2, . . . , an,−∑ai), ai ∈ Zn

≥0


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

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.

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

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



1 −11 1 1



1 −1



1 −1



1 −1



1 −1

13

23

13

13

13

0

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

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


ExampleG

1


−10

...

a+ 1 ...

b+ 1

∆2 ×∆1

G

1 0 −1


ExampleG

1


−10

...

a+ 1 ...

b+ 1

G

1 −10 0∆1 ×∆1 ×∆1

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)


• 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)


• 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)

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:



A linear extension of a poset P is an ordering of the posetelements compatible with the partial order.




• Fk7(1, 0, 0, 0, 0, 0,−1) is not an order polytope.(Behrend-M-Panova 20+)




kn+1

?

1

n!

Catn := 1n+1

(2nn

)Euler numbers En


?


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

)

Volume of the CRYn polytope

vn := volume(CRYn)

2 3 4 5 6 7n

vn 1 1 2 10 140 5880


vn := volume(CRYn)

2 3 4 5 6 7n

vn 1 1 2 10 140 5880

vnvn−1

1 2 5 14 42


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

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


Catn := 1n+1

(2nn



Catn := 1n+1

(2nn



Catn := 1n+1

(2nn


. . . however, there is no combinatorial proof of formula for vn



kn+1

1

n!

Catn := 1n+1

(2nn

)Euler numbers En


Cat1Cat2 · · ·Catn−2Cat1Cat2 · · ·Catn−2?no poset

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.



# of ways of writing a as an N-combination of vectorsei − ej for 1 ≤ i < j ≤ n+ 1

Kkn+1(a) =





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) =


Generating function for Kkn+1(a):

∑a

KG(a)xa =∏

1≤i<j≤n+1

(1− xix−1j )−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) =


". . . 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)

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)



∑k ik),

Example


1 −1




∑k ik),

Example


1 −1




∑k ik),

Example


1 −1

0 −21 1 0 −21 11 2

1

volume = 2.

1




∑k ik),


Corollaryvolume(CRY n) = Kkn+1

(0, 0, 1, 2, . . . , n− 2,−(n−1

2

))




∑k ik),


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

))




∑k ik),


Corollary


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

))




∑k ik),


Corollary


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.


Zeilberger’s entire paper



kn+1

1

n!

Catn := 1n+1

(2nn

)Euler numbers En


Cat1Cat2 · · ·Catn−2Cat1Cat2 · · ·Catn−2?no poset



kn+1

1

n!

Catn := 1n+1

(2nn

)Euler numbers En


Cat1Cat2 · · ·Catn−2Cat1Cat2 · · ·Catn−2?no poset ?

Fundamental theorem + symmetryTheorem (Stanley-Postnikov 09, Baldoni-Vergne 09)

volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),

Example

1 −1


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.


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.


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


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?


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).


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)


ProofBoth sides are the volume of FG(10 · · · 0− 1) ≡ FGr (10 · · · 0− 1).


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑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


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)


1 −1

0 −21 1

0 −21 1

1 2

11

Gr

1 −1

G

G

volume = 2


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)


1 −1

0 −21 1

0 −21 1

1 2

11

1−10 10

−10 101

Gr

1 −1

GExample:

G Gr

volume = 2


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)

• two distinct flow polytopes have the same number oflattice points!



volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)



0 1 2 110 −2−3

G Gr

Example


volumeFG(1, 0, . . . , 0,−1) = KG(0, i2, i3, . . . ,−∑

k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)



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

)

Bijection between lattice points KG and KGr


∑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)

Summary


∑k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)


Question 1:


∑k ik)


∑k i′k)?

Can we describe this triangulation explicitly? i.e. what simplexcorresponds to each integer flow?

• let KG(a) := #(FG(a) ∩ Zm)

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



• 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:


setup

Example: two framings of G





Example:

• routes P and Q are coherent if at shared vertex v, Pv,Qv havesame relative order as vP, vQ in the framing.


setup





Example:



setup

P

Q

X





Example:



setup

P

Q

×





Example:



setup

Example: two framings of G

P

Q

× P

Q

X





Example:



setup

• a clique C is a maximal collection of pairwise coherent routes



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.



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′

Summary


∑k ik),

Question 1:Can we describe the Postnikov-Stanley triangulation explicitly? i.e. whatsimplex corresponds to each integer flow?


Relation between triangulationsQuestion 1:Can we describe the Postnikov-Stanley (PS) triangulation explicitly? i.e.what simplex corresponds to each integer flow?

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.

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?




Example:

0 −21 1






Example:

0 −21 1




0



Example:

0 −21 1




0



Example:

0 −21 1




0



Example:

0 −21 1




0



Example:

0 −21 1




1



Example:

0 −21 1




2



Example:

0 −21 11 2




00

0

0




Example:

0 −21 1

1

1



00

0

0

Summary


∑k ik),


∑k ik) = KGr (0, i′2, i

′3, . . . ,−

∑k i′k)



∑k ik)


∑k i′k)?



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.




′3, . . . ,−

∑k i′k)?

fl((u, v)) =# times (u, v) appears in {Pv | P ∈ C} −1Φ : C 7→ fl,





′3, . . . ,−

∑k i′k)?


Φr : C 7→ fl′,fl′((u, v)) = # times (v, u) appears in {vP | P ∈ C} −1





′3, . . . ,−

∑k i′k)?

Theorem (M 2020+)



Φr ◦ Φ−1 is a desired bijection.





′3, . . . ,−

∑k i′k)?

Theorem (M 2020+)




Example:

Φ

0 −21 11 2


00

0 0




′3, . . . ,−

∑k i′k)?

Theorem (M 2020+)




Example:

−10 10

1ΦrΦ

0 −21 11 2

mirror


00

0 0 0 0

0 0 0




′3, . . . ,−

∑k i′k)?

Theorem (M 2020+)




Example:

ΦrΦ

0 −21 1

1

1

−10 10


000

0 0 0

0 0 10

mirror

Summary• FG(a) flow polytope of a graph G netflow a

• volumes and lattice points important in representation theory

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


• 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

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


• 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

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

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


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.


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma



• fix a framing of G, for each vertex i = 2, . . . , n iteratesubdivision lemma until you get rid of vertex i.


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




• encode choices with a bipartite noncrossing tree


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




Example:


e

e′

f

f ′

e

e′f

f ′


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




Example:


e

e′

f

f ′

e

e′f

f ′

e

e′

f

f ′


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




Example:


e

e′

f

f ′

e

e′f

f ′

e

e′

f

f ′

e

e′

f

f ′


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




Example:


e

e′

f

f ′

e

e′f

f ′

e

e′

f

f ′


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma





e

e′f

f ′

e

e′

f

f ′

Example:


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma





e

e′f

f ′

e

e′

f

f ′

Example:


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma





e

e′f

f ′

e

e′

f

f ′

Example:

• bookkeep the noncrossing trees as an integer flow


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




• encode choices with a bipartite noncrossing tree• bookkeep the noncrossing trees as an integer flow

e

e′

f

f ′

Example:

0 −21 11 2


a i b a i ba i bor

p q q−pp

p−qq

q ≥ pq < p

FG0≡ FG1

∪ FG2

p

Subdivision Lemma




• 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

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


G0 G1 G2

a i b a i ba i bor

e f fee+ f e+ f


Example:

0 −21 1

1

1

e

e′f

f ′

e

e′

f

f ′k

e+ f

e′ + f

e′ + f ′k

g g


G0 G1 G2

a i b a i ba i bor

e f fee+ f e+ f


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


G0 G1 G2

a i b a i ba i bor

e f fee+ f e+ f


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

Connection to diagonal harmonics

a

1 1 1 −3

x14

x34

x13

x12 x23

x24


a = (1, 1, . . . , 1,−n)


a

1 1 1 −3

x14

x34

x13

x12 x23

x24


a = (1, 1, . . . , 1,−n)

Fkn+1(1, 1, . . . , 1,−n) is called the Tesler polytope


a

1 1 1 −3

x14

x34

x13

x12 x23

x24


a = (1, 1, . . . , 1,−n)


has n! vertices, dimension(n2

)


a

1 1 1 −3

x14

x34

x13

x12 x23

x24


a = (1, 1, . . . , 1,−n)



)• a weighted sum over lattice points gives Hilbert series of

the space of diagonal harmonics (Haglund 2011)


a

1 1 1 −3

x14

x34

x13

x12 x23

x24


a = (1, 1, . . . , 1,−n)



)• a weighted sum over lattice points gives Hilbert series of

the space of diagonal harmonics (Haglund 2011)

• Connection was suggested by François Bergeron


a

1 1 1 −3

x14

x34

x13

x12 x23

x24


a = (1, 1, . . . , 1,−n)



)

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)

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



(m− n

j1, . . . , jn

)aj11 · · · ajnn

×KG(j1 − o1, . . . , jn − on, 0)

ov = outdeg(v)− 1


• Original proof uses Jeffrey–Kirwan iterated residues



(m− n

j1, . . . , jn

)aj11 · · · ajnn

×KG(j1 − o1, . . . , jn − on, 0)

ov = outdeg(v)− 1



• give a proof using polytope subdivisions (Mészáros-M 2019)



(m− n

j1, . . . , jn

)aj11 · · · ajnn

×KG(j1 − o1, . . . , jn − on, 0)

ov = outdeg(v)− 1




Corollary:KG(j1 − o1, . . . , jn − on, 0) are mixed volumes and so arelog-concave in j1, . . . , jn.



(m− n

j1, . . . , jn

)aj11 · · · ajnn

×KG(j1 − o1, . . . , jn − on, 0)

ov = outdeg(v)− 1




• 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

Documents

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