The classification of Zamolodchikov periodic quivers

Pavel Galashin

MIT

galashin@mit.edu

March 30, 2016

Joint work with Pavlo Pylyavskyy

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Part 1: Statement

The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4

5

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4

5

Pavel Galashin (MIT) Z-periodic quivers classification March 30, 2016 3 / 37

The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4

5

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Recurrent quivers

w1 v

w2

. . .

wk

w ′k

. . .

w ′2

u w ′1

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4

5

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4

5 The T -system associated with Q is periodic.

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T -system

a b

c d

e f

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T -system

a b

c d

e f

−→

b+ca b

c c+bfd

c+fe f

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T -system

b+ca

b+ca

+ c+bfd

b

b+ca

c+fe

+ c+bfd

c

c+bfd

c+fe

c+fe

+ c+bfd

f

b+ca b

c c+bfd

c+fe f

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ADE Dynkin diagrams: bad definitions

Name Picture h

An n + 1

Dn 2n − 2

E6 12

E7 18

E8 30

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Tensor product

D5 ⊗ A3

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Example

1 1

1 1

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4

5 The T-system associated with Q is periodic.

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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Tropical T -system

a b

c d

e f

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Tropical T -system

a b

c d

e f

max(b, c)− a b

c max(c , b + f )− d

max(c , f )− e f

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Example

0 1

0 0

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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Fixed point

a b a

b c b

a b a

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Fixed point

a b a

b c b

a b a

a2 = b + b; b2 = a2 + c ; c2 = b2 + b2.

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Fixed point

a b a

b c b

a b a

a2 = b + b; b2 = a2 + c ; c2 = b2 + b2.

a =

√

4 + 2√2; b = 2 +

√2; c = 2 + 2

√2.

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2 Q has a strictly subadditive labeling.

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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Strictly subadditive labeling

a b a

b c b

a b a

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Strictly subadditive labeling

a b a

b c b

a b a

2a > max(b, b); 2b > max(a + a, c); 2c > max(b + b, b + b).

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Strictly subadditive labeling

2 3 2

3 4 3

2 3 2

2a > max(b, b); 2b > max(a + a, c); 2c > max(b + b, b + b).

a = 2; b = 3; c = 4.

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ADE Dynkin diagrams: good definitions

a b c b a

2a > b; 2b > a + c ; 2c > b + b.

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ADE Dynkin diagrams: good definitions

4 6 7 6 4

2a > b; 2b > a + c ; 2c > b + b.

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ADE Dynkin diagrams: good definitions

1√3 2

√3 1

2a > b; 2b > a + c ; 2c > b + b.

2a =b

cos(π/h); 2b =

a + c

cos(π/h); 2c =

b + b

cos(π/h).

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1

2 Q has a strictly subadditive labeling.

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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The result

Theorem

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1 Q is an admissible ADE bigraph.

2 Q has a strictly subadditive labeling.

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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Admissible ADE bigraphs

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Admissible ADE bigraphs

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Admissible ADE bigraphs

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Admissible ADE bigraphs

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Admissible ADE bigraphs

Pavel Galashin (MIT) Z-periodic quivers classification March 30, 2016 19 / 37

The result

Theorem (G.-Pylyavskyy, 2016)

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1 Q is an admissible ADE bigraph.

2 Q has a strictly subadditive labeling.

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

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The result

Theorem (G.-Pylyavskyy, 2016)

Let Q be a bipartite recurrent quiver. Then the following are equivalent.

1 Q is an admissible ADE bigraph.

2 Q has a strictly subadditive labeling.

3 Q has a fixed point.

4 The tropical T -system is periodic for any initial value.

5 The T-system associated with Q is periodic.

In all cases, both the T -system and its tropicalization have perioddividing

h + h′.

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Part 2: Stembridge’s Classification

Tensor products

D5 ⊗ A3

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Twists

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Twists

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Twists

A5 × A5.

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The case (Am−1D)n

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The case (Am−1D)n

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The case (ADm−1)n

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The case EE n

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11 exceptional bigraphs

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11 exceptional bigraphs

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11 exceptional bigraphs

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Part 3: Proof

Twists

a b

c d

e f

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Twists

a b

c d

e f

µi−→

a b

ae+bfd

ae+bfc

e f

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Twists

a b

c d

e f

µi−→

a b

ae+bfd

ae+bfc

e f

(µiµi+1)3 = id !

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Plan of the proof

(4) tropical T -system is periodic

(2) strictly subadditive labelling property

(3) fixed point property

(5) T -system is periodic

(1) Q is an admissible ADE bigraph

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Subadditive function

2 3 2

3 4 3

2 3 2

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Plan of the proof

(4) tropical T -system is periodic

(2) strictly subadditive labelling property

(3) fixed point property

(5) T -system is periodic

(1) Q is an admissible ADE bigraph

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A2n−1 ↔ Dn+1 duality

b c

2a

b c

a

b c

a

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A2n−1 ↔ Dn+1 duality

b c

2a

b c

b max(b, 0)− c

max(2b, 0) − 2a

b max(b, 0)− c

a

b c

a

max(b, 0) − a

b max(b, 0)− c

max(b, 0) − a

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(Am−1D)n ↔ A2n−1 ⊗ Dm+1 duality

v+1v

−

1

v2 vm vm+2 v2m

v2m+1

v3m−1

u1

u+2 u

+m u

+m+2 u

+2m

v+2m+1

v+3m−1

v−

3m−1

v−

2m+1

u−

2m u−

m+2 u−

m u−

2

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Bibliography

Sergey Fomin and Andrei Zelevinsky.Y -systems and generalized associahedra.Ann. of Math. (2), 158(3):977–1018, 2003.

Bernhard Keller.The periodicity conjecture for pairs of Dynkin diagrams.Ann. of Math. (2), 177(1):111–170, 2013.

John R. Stembridge.Admissible W -graphs and commuting Cartan matrices.Adv. in Appl. Math., 44(3):203–224, 2010.

Pavel Galashin and Pavlo PylyavskyyThe classification of Zamolodchikov periodic quivers.arXiv:1603.03942 (2016).

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Thank you!