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 Stanley@ vol 0 vol 0 June 23, 2014 joint with: Karola M´ esz´ aros, Jessica Striker; Drew Armstrong, Karola M´ esz´ aros, and Brendon Rhoades; Karola M´ esz´ aros

The Chan-Robbins-Yuen polytope : ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := = convex hull n × n permutation matrices 0

The Chan-Robbins-Yuen polytope : ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := = convex hull n × n permutation matrices 0 • 2 n − 1 vertices, � n � dimension 2

The Chan-Robbins-Yuen polytope : ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := = convex hull n × n permutation matrices 0 • 2 n − 1 vertices, � n � dimension 2 CRY 3 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0

The Chan-Robbins-Yuen polytope : ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := = convex hull n × n permutation matrices 0 • 2 n − 1 vertices, � n � dimension 2 CRY 3 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0

From CRY n to a flow polytope ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := a b c K 4 d e f

From CRY n to a flow polytope ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := a + b + c =1 a b c a b c K 4 c d e b f a 1

From CRY n to a flow polytope ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := a + b + c =1 a b c a K 4 c d + e − a =0 d e d e e b f a d 1 0

From CRY n to a flow polytope ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := a + b + c =1 a b c b K 4 c d + e − a =0 d e d e b b f f f − b − d =0 a d d f 1 0 0

From CRY n to a flow polytope ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := a + b + c =1 a b c c K 4 c d + e − a =0 d e e e b f f f − b − d =0 a d f − c − e − f = − 1 1 0 0 − 1

From CRY n to a flow polytope ( b ij ) ∈ R n 2 | doubly-stochastic matrix, b ij = 0 , i − j ≥ 2 � � CRY n := a + b + c =1 a b c c c K 4 c c d + e − a =0 d e e e e e b b f f f f − b − d =0 a a d d f f − c − e − f = − 1 − c − e − f = − 1 1 0 0 − 1 • Correspondence CRY n and flows in complete graph K n +1 with netflow: 1 first vertex, − 1 last vertex, 0 other vertices.

Volume of the CRY n polytope v n := volume( CRY n ) 2 3 4 5 6 7 n 1 1 2 10 140 5880 v n

Volume of the CRY n polytope v n := volume( CRY n ) 2 3 4 5 6 7 n 1 1 1 2 10 10 140 5880 5880 v n v 2 n 2 70 v 2 n − 2

Volume of the CRY n polytope v n := volume( CRY n ) 2 3 4 5 6 7 n 1 1 1 1 2 10 10 10 140 5880 5880 5880 v n v 2 n 2 70 v 2 n − 2 v n 1 2 5 14 42 v n − 1

Volume of the CRY n polytope v n := volume( CRY n ) 2 3 4 5 6 7 n 1 1 1 1 2 10 10 10 140 5880 5880 5880 v n v 2 n 2 70 v 2 n − 2 v n 1 2 5 14 42 v n − 1 (conjecture Chan-Robbins-Yuen 99) • v n = Cat 0 Cat 1 · · · Cat n − 2 (Zeilberger 99)

CRY n : vertices: permutation matrices flow polytope complete graph 1 − 1

Variants CRY n : vertices: 1. vertices: alternating sign permutation matrices matrices 2. change netflow from flow polytope complete graph (1 , 0 , . . . , 0 , − 1) to (1 , 1 , . . . , 1 , − n ) 1 − 1 3. type D analogue of CRY n

Alternating sign matrices permutation matrices alternating sign matrices • entries are 0 , 1 • rows and columns sum to 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0

Alternating sign matrices permutation matrices alternating sign matrices • entries are 0 , 1 • entries are 0 , 1 , − 1 • rows and columns sum to 1 • rows and columns sum to 1 • nonzero entries in rows and columns alternate in sign First enumerated by Zeilberger 92 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0

1. The CRY polytope of ASMs CRY ASM = convex hull n × n ASMs n 0

1. The CRY polytope of ASMs CRY ASM = convex hull n × n ASMs n 0 The polytope CRY ′ n of ASMs is an order polytope as defined by Stanley 86. (M´ esz´ aros-M-Striker 13+)

1. The CRY polytope of ASMs CRY ASM = convex hull n × n ASMs n 0 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 . 7 . 2 . 1 . 6 0 . 8 . 1 . 1 0 0 . 9 . 1

1. The CRY polytope of ASMs CRY ASM = convex hull n × n ASMs n 0 The polytope CRY ′ n of ASMs is an order polytope as defined by Stanley 86. (M´ esz´ aros-M-Striker 13+) Example . 7 . 3 . 2 . 8 . 3 . 4 . 1 . 2 . 2 . 7 . 7 . 7 . 2 . 1 . 6 . 8 . 8 . 1 0 . 8 . 1 . 3 . 3 . 1 corner . 2 . 9 complement 0 0 . 9 . 1 sums

1. The CRY polytope of ASMs CRY ASM = convex hull n × n ASMs n 0 The polytope CRY ′ n of ASMs is an order polytope as defined by Stanley 86. (M´ esz´ aros-M-Striker 13+) In EC1

1. The CRY polytope of ASMs CRY ASM = convex hull n × n ASMs n 0 P Cat n vertices volume = f ( n − 1 ,n − 2 ,..., 1) = # SY T ( δ n − 1 )

CRY n : CRY ASM : n vertices: vertices: permutation matrices alternating sign matrices 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 Question What can we learn about CRY n from CRY ASM ? n

2. The Tesler polytope CRY n : flow polytope complete graph CRY 3 : 1 0 0 − 1 • 2 n − 1 vertices � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

2. The Tesler polytope CRY n : flow polytope complete graph flow polytope complete graph different nettflow CRY 3 : T 3 : 1 0 0 − 1 1 1 1 − 3 • 2 n − 1 vertices � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

2. The Tesler polytope CRY n : flow polytope complete graph flow polytope complete graph different nettflow x y b CRY 3 : T 3 : z a c 1 0 0 − 1 1 1 1 − 3 x y a b c z • lattice points are Tesler matrices • 2 n − 1 vertices � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

2. The Tesler polytope CRY n : flow polytope complete graph flow polytope complete graph different nettflow 0 1 0 CRY 3 : T 3 : 1 2 1 0 0 − 1 1 1 1 1 − 3 0 1 2 1 0 1 • lattice points are Tesler matrices • 2 n − 1 vertices � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

2. The Tesler polytope CRY n : flow polytope complete graph flow polytope complete graph different nettflow 0 1 0 CRY 3 : T 3 : 1 2 1 0 0 − 1 1 1 1 1 − 3 0 1 2 1 0 1 • lattice points are Tesler matrices • 2 n − 1 vertices • n ! vertices � n � n � � • dimension • dimension 2 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

2. The Tesler polytope CRY n : flow polytope complete graph flow polytope complete graph different nettflow 0 1 0 CRY 3 : T 3 : 1 2 1 0 0 − 1 1 1 1 1 − 3 0 1 2 1 0 1 • lattice points are Tesler matrices • 2 n − 1 vertices • n ! vertices � n � n � � • dimension • dimension 2 2 Theorem (Armstrong-M´ esz´ aros-M-Rhoades 14+) Theorem (Zeilberger 99): vol = f ( n − 1 ,n − 2 ,..., 1) · � n − 1 volume = � n − 2 i =0 Cat i i =0 Cat i

3. The type D CRY polytope CRY n : Second generalization CRY n : flow polytope complete graph CRY 3 : 1 − 1 • 2 n − 1 vertices � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

3. The type D CRY polytope CRY n : Second generalization CRY n : flow polytope complete graph flow polytope complete signed graph CRY D CRY 3 : 3 : 1 − 1 2 • 2 n − 1 vertices � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i

3. The type D CRY polytope CRY n : Second generalization CRY n : flow polytope complete graph flow polytope complete signed graph CRY D CRY 3 : 3 : 1 − 1 2 • 3 n − 2 n vertices • 2 n − 1 vertices • dimension n 2 − 1 � n � • dimension 2 Theorem (Zeilberger 99): volume = � n − 2 i =0 Cat i