# Parking functions in types C and B

## Full text

(1)

### Parking functions in types C and B

Robin Sulzgruber joint work with Marko Thiel

Universit¨at Wien

Ellwangen, 23rd March 2015

(2)

### Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

(3)

### Outline

Regions of the Shi arrangement

Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

(4)

### Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

(5)

### Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths

Vertically labelled paths

(6)

### Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

(7)

Parking functions

### Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x ∈V :hx, αi=k}. Letsα,k be the reflection throughHα,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

(8)

Parking functions

### Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x ∈V :hx, αi=k}. Letsα,k be the reflection throughHα,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

(9)

Parking functions

### Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x∈V :hx, αi=k}.

Letsα,k be the reflection throughHα,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

(10)

Parking functions

### Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x∈V :hx, αi=k}.

Letsα,k be thereflectionthrough Hα,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

(11)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α}for all α∈Φ, sα,0(Φ)⊆Φ for allα ∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ+ denote the positive roots.

We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

The root system of typeC2 in V =R2.

(12)

Parking functions

### Roots

Definition

Acrystallographic root systemof V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, sα,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ+ denote thepositive roots. We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

The root system of typeC2 in V =R2.

(13)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ,

sα,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ+ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

The root system of typeC2 in V =R2.

(14)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, sα,0(Φ)⊆Φ for allα∈Φ,

2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ+ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

The root system of typeC2 in V =R2.

(15)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, sα,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

Let Φ+ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

The root system of typeC2 in V =R2.

(16)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, sα,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

Let Φ+ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

(17)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, sα,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

LetΦ+ denote the positive roots.

We only consider irreducible root systems.

Example

2e1

e2+e1 2e2

e2−e1

The root system of typeC2 in V =R2.

(18)

Parking functions

### Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, sα,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

Let Φ+ denote the positive roots.

Example

2e1

e2+e1 2e2

e2−e1

(19)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn. The finite torus is

Zn/(2n+ 1)Zn.

(20)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding corootas

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn. The finite torus is

Zn/(2n+ 1)Zn.

(21)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define thecoroot latticeas Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn. The finite torus is

Zn/(2n+ 1)Zn.

(22)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torusas T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn. The finite torus is

Zn/(2n+ 1)Zn.

(23)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn. The finite torus is

Zn/(2n+ 1)Zn.

(24)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn.

The finite torus is Zn/(2n+ 1)Zn.

(25)

Parking functions

### The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ+

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ+={ej±ei : 1≤i <j ≤n}

∪ {2ei : 1≤i ≤n}.

The coroot lattice is Qˇ =Zn. The finite torus is

Zn/(2n+ 1)Zn.

(26)

Parking functions

### Classical parking functions

Definition

A classical parking function is an integer vector

f = (f1,f2, . . . ,fn) with nonnegative entries such that there exists a permutation σ∈Sn with

fσ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1)

σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

(27)

Parking functions

### Classical parking functions

Definition

Aclassical parking function is an integer vector

f = (f1,f2, . . . ,fn) with nonnegative entries such that there exists a permutation σ∈Sn with

fσ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1)

σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

(28)

Parking functions

### Classical parking functions

Definition

A classical parking function is an integer vector

f = (f1,f2, . . . ,fn) with nonnegative entries such that there exists a permutation σ∈Sn with

fσ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1)

σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

(29)

Parking functions

### Classical parking functions

Definition

A classical parking function is an integer vector

f = (f1,f2, . . . ,fn) with nonnegative entries such that there exists a permutation σ∈Sn with

fσ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1) σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

(30)

Parking functions

### Classical parking functions

Definition

A classical parking function is an integer vector

f = (f1,f2, . . . ,fn) with nonnegative entries such that there exists a permutation σ∈Sn with

fσ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1) σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

(31)

Parking functions

### Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zn/(2n+ 1)Zn is the finite torus of typeCn. Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where −n ≤fi ≤n.

(32)

Parking functions

### Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zn/(2n+ 1)Zn is the finite torus of typeCn. Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where −n ≤fi ≤n.

(33)

Parking functions

### Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zn/(2n+ 1)Zn is the finite torus of typeCn.

Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where −n ≤fi ≤n.

(34)

Parking functions

### Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zn/(2n+ 1)Zn is the finite torus of typeCn. Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where−n≤fi ≤n.

(35)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dnand a permutation σ∈Sn such that

σi < σi+1

for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(36)

Parking functions

### Vertically labelled Dyck paths

Definition

Avertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(37)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli arise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(38)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(39)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i =1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(40)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(41)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(42)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(43)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(44)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3

The rises of π are i = 1,3,5,6.

Let σ=3417256.

The pair (π, σ) is a vertical labelling.

(45)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(46)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(47)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(48)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(49)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(50)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(51)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i =1,3,5,6.

Let σ=3417256.

The pair (π, σ) is a vertical labelling.

(52)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(53)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(54)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

(55)

Parking functions

### Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ Dn and a permutation σ∈Sn such that

σi < σi+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

(56)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(57)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(58)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(59)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(60)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(61)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(62)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(63)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(64)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(65)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

(66)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,0,

0

,

4

,

4

,

1

)

(67)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,0,0,

4

,

4

,

1

)

(68)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,

4

,0,0,

4

,

4

,

1

)

(69)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,

4

,0,0,

4

,

4

,1)

(70)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,4,0,0,

4

,

4

,1)

(71)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,4,0,0,4,

4

,1)

(72)

Parking functions

### From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,4,0,0,4,4,1)

(73)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5. Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(74)

Parking functions

### Vertically labelled lattice paths

Definition

Avertically labelled lattice path is a pair(π, σ)of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5. Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(75)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5. Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(76)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i =2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(77)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(78)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(79)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(80)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(81)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

(82)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

(83)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

(84)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

(85)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

(86)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

(87)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

(88)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3 6

The rises of π are i =2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

(89)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

(90)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

(91)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

(92)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

(93)

Parking functions

### Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ Ln from (0,0) to (n,n) and a signed permutationσ ∈Hn such that

σi < σi+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

(94)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0

Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(95)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(96)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0

Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(97)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(98)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(99)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(100)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(101)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(102)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(103)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

(104)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,

4

,

4

,

−4

,

−4

,

4

)

(105)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,

4

,

4

,

−4

,−4,

4

)

(106)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,

4

,

4

,−4,−4,

4

)

(107)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,4,

4

,−4,−4,

4

)

(108)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,4,4,−4,−4,

4

)

(109)

Parking functions

### From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,4,4,−4,−4,4)

(110)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e1,2e2,e2+e1,e2−e1}.

(111)

Shi regions

### The Shi arrangement

Definition

We define theShi arrangementof the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e1,2e2,e2+e1,e2−e1}.

(112)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e ,2e ,e +e ,e −e }.

(113)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e1,2e2,e2+e1,e2−e1}.

(114)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e ,2e ,e +e ,e −e }.

(115)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e1,2e2,e2+e1,e2−e1}.

(116)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e ,2e ,e +e ,e −e }.

(117)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e1,2e2,e2+e1,e2−e1}.

(118)

Shi regions

### The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

ShiΦ ={Hα,k :α∈Φ+,k = 0,1}.

The connected components of

V − [

H∈ShiΦ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ+={2e ,2e ,e +e ,e −e }.

Updating...

## References

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