The Chan-Robbins-Yuen polytope : ( b ij ) R n 2 | doubly-stochastic - - PowerPoint PPT Presentation

Open problem: volumes of flow polytopes Alejandro H. Morales LaCIM, Universit´ e du Qu´ ebec ` a Montr´ eal June 23, 2014

joint with: Karola M´ esz´ aros, Jessica Striker; Drew Armstrong, Karola M´ esz´ aros, and Brendon Rhoades;

Stanley@ vol

vol

Karola M´ esz´ aros

= convex hull n × n permutation matrices The Chan-Robbins-Yuen polytope:

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2

= convex hull n × n permutation matrices The Chan-Robbins-Yuen polytope:

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• 2n−1 vertices,

dimension n

2

= convex hull n × n permutation matrices The Chan-Robbins-Yuen polytope:

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• 2n−1 vertices,

dimension n

2

• 1

1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0

CRY3

= convex hull n × n permutation matrices The Chan-Robbins-Yuen polytope:

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• 2n−1 vertices,

dimension n

2

• 1

1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0

CRY3

1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1

From CRYn to a flow polytope

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• K4

a b c d e f

From CRYn to a flow polytope

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• K4

a b c d e f a + b + c =1

a b c

a b c 1

From CRYn to a flow polytope

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• K4

a b c d e f

b c

a + b + c =1 d + e − a =0 d e a

a d e

1

From CRYn to a flow polytope

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• K4

a b c d e f

a d e

d + e − a =0

b c

a + b + c =1 f d b f − b − d =0

b d f

1

From CRYn to a flow polytope

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• K4

a b c d e f

a b c e d f

f − b − d =0 d + e − a =0 a + b + c =1 −c − e − f =−1 c e f 1 −1

From CRYn to a flow polytope

CRYn :=

• (bij) ∈ Rn2 | doubly-stochastic matrix, bij = 0, i − j ≥ 2
• K4

a b c d e f

a b c e d f

f − b − d =0 d + e − a =0 a + b + c =1 −c − e − f =−1 c e f 1 −1 −c − e − f =−1 f e c

c e f

• Correspondence CRYn and flows in complete graph Kn+1 with

netflow: 1 first vertex, −1 last vertex, 0 other vertices. b a d

Volume of the CRYn polytope

vn := volume(CRYn) 2 3 4 5 6 7 n vn 1 1 2 10 140 5880

Volume of the CRYn polytope

vn := volume(CRYn) 2 3 4 5 6 7 n vn 1 1 2 10 140 5880 v2n v2n−2 2 70 1 10 5880

Volume of the CRYn polytope

vn := volume(CRYn) 2 3 4 5 6 7 n vn 1 1 2 10 140 5880 v2n v2n−2 2 70 vn vn−1 1 2 5 14 42 1 10 5880 1 10 5880

Volume of the CRYn polytope

vn := volume(CRYn) 2 3 4 5 6 7 n vn 1 1 2 10 140 5880 v2n v2n−2 2 70

• vn = Cat0Cat1 · · · Catn−2

(Zeilberger 99) vn vn−1 1 2 5 14 42 (conjecture Chan-Robbins-Yuen 99) 1 10 5880 1 10 5880

CRYn: permutation matrices vertices: 1 −1 flow polytope complete graph

CRYn: permutation matrices

• 1. vertices: alternating sign

matrices vertices:

• 3. type D analogue of CRYn

Variants

• 2. change netflow from

(1, 0, . . . , 0, −1) to (1, 1, . . . , 1, −n) 1 −1 flow polytope complete graph

Alternating sign matrices

• rows and columns sum to 1

permutation matrices alternating sign matrices

• entries are 0, 1

1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 0 0

Alternating sign matrices

• rows and columns sum to 1
• nonzero entries in rows and

columns alternate in sign permutation matrices

• rows and columns sum to 1

alternating sign matrices

• entries are 0, 1
• entries are 0, 1, −1

1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1

First enumerated by Zeilberger 92

• 1. The CRY polytope of ASMs

CRYASM

n

= convex hull n × n ASMs

• 1. The CRY polytope of ASMs

The polytope CRY′

n of ASMs is an order polytope as defined by

Stanley 86. (M´ esz´ aros-M-Striker 13+)

CRYASM

n

= convex hull n × n ASMs

• 1. The CRY polytope of ASMs

The polytope CRY′

n of ASMs is an order polytope as defined by

Stanley 86. (M´ esz´ aros-M-Striker 13+)

Example

.3 .4 .1 .2 .6 .1 .1 .7 .2 .1 .1 .9 .8

CRYASM

n

= convex hull n × n ASMs

• 1. The CRY polytope of ASMs

The polytope CRY′

n of ASMs is an order polytope as defined by

Stanley 86. (M´ esz´ aros-M-Striker 13+)

Example

.3 .4 .1 .2 .6 .1 .1 .7 .2 .1 .1 .9 .8 .2 .3 .7 .8 .9 .8 .8 .7 .3 .2 .1 .2 .7 .3 corner sums complement

CRYASM

n

= convex hull n × n ASMs

• 1. The CRY polytope of ASMs

The polytope CRY′

n of ASMs is an order polytope as defined by

Stanley 86. (M´ esz´ aros-M-Striker 13+)

CRYASM

n

= convex hull n × n ASMs In EC1

• 1. The CRY polytope of ASMs

volume = f(n−1,n−2,...,1) = #SY T(δn−1) Catn vertices P

CRYASM

n

= convex hull n × n ASMs

Question CRYn: CRYASM

n

: permutation matrices alternating sign matrices vertices: vertices:

1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1

What can we learn about CRYn from CRYASM

n

?

• 2. The Tesler polytope

volume = n−2

i=0 Cati

CRYn: flow polytope complete graph 1 −1

Theorem (Zeilberger 99):

CRY3 :

• 2n−1 vertices
• dimension

n

2

• 2. The Tesler polytope

volume = n−2

i=0 Cati

CRYn: flow polytope complete graph 1 −1 flow polytope complete graph different nettflow 1

Theorem (Zeilberger 99):

CRY3 : T3 :

• 2n−1 vertices
• dimension

n

2

• 1

1 −3

• 2. The Tesler polytope

volume = n−2

i=0 Cati

CRYn: flow polytope complete graph 1 −1 flow polytope complete graph different nettflow 1

Theorem (Zeilberger 99):

CRY3 : T3 :

• 2n−1 vertices
• dimension

n

2

• 1

1 −3

• lattice points are

Tesler matrices

a c b x y z a b c x y z

• 2. The Tesler polytope

volume = n−2

i=0 Cati

CRYn: flow polytope complete graph 1 −1 flow polytope complete graph different nettflow 1

Theorem (Zeilberger 99):

CRY3 : T3 :

• 2n−1 vertices
• dimension

n

2

• 1

1 −3

• lattice points are

Tesler matrices

1 0 1 0 1 2 1 1 1 2

• 2. The Tesler polytope

volume = n−2

i=0 Cati

CRYn: flow polytope complete graph 1 −1 flow polytope complete graph different nettflow 1

Theorem (Zeilberger 99):

CRY3 : T3 :

• 2n−1 vertices
• dimension

n

2

• 1

1 −3

• dimension

n

2

• n! vertices
• lattice points are

Tesler matrices

1 0 1 0 1 2 1 1 1 2

• 2. The Tesler polytope

volume = n−2

i=0 Cati

CRYn: flow polytope complete graph 1 −1 flow polytope complete graph different nettflow 1

Theorem vol = f(n−1,n−2,...,1) · n−1

i=0 Cati

Theorem (Zeilberger 99):

CRY3 : T3 :

• 2n−1 vertices
• dimension

n

2

• 1

1 −3

(Armstrong-M´ esz´ aros-M-Rhoades 14+)

• dimension

n

2

• n! vertices
• lattice points are

Tesler matrices

1 0 1 0 1 2 1 1 1 2

• 3. The type D CRY polytope

CRYn: Second generalization CRYn: flow polytope complete graph 1 −1

Theorem (Zeilberger 99):

CRY3 :

• 2n−1 vertices
• dimension

n

2

• volume = n−2

i=0 Cati

• 3. The type D CRY polytope

CRYn: Second generalization CRYn: flow polytope complete graph 1 −1 flow polytope complete signed graph 2

Theorem (Zeilberger 99):

CRY3 : CRYD

3 :

• 2n−1 vertices
• dimension

n

2

• volume = n−2

i=0 Cati

• 3. The type D CRY polytope

CRYn: Second generalization CRYn: flow polytope complete graph 1 −1 flow polytope complete signed graph 2

Theorem (Zeilberger 99):

CRY3 : CRYD

3 :

• 2n−1 vertices
• dimension

n

2

• 3n − 2n vertices
• dimension n2 − 1

volume = n−2

i=0 Cati

• 3. The type D CRY polytope

CRYn: Second generalization CRYn: flow polytope complete graph 1 −1 flow polytope complete signed graph 2

Conjecture: (M´ esz´ aros-M 12) volume = 2(n−1)2 · n−1

i=0 Cati

Theorem (Zeilberger 99):

CRY3 : CRYD

3 :

• 2n−1 vertices
• dimension

n

2

• 3n − 2n vertices
• dimension n2 − 1

volume = n−2

i=0 Cati

Happy birthday Richard!

CRY3 CRYASM

3

T3 CRYD

2

Why product of Catalan? conjecture volume How do they fit together?