[PPT] - On (rational) Shi tableaux Robin Sulzgruber 78 th S eminaire PowerPoint Presentation

SLIDE 1

On (rational) Shi tableaux

Robin Sulzgruber 78th S´ eminaire Lotharingien de Combinatoire March 26th–29th 2017 • Ottrott • France

Robin Sulzgruber On (rational) Shi tableaux March 2017 1 / 33

SLIDE 2

Setting the stage

Robin Sulzgruber On (rational) Shi tableaux March 2017 2 / 33

SLIDE 3

Setting the stage

Definition Let V be a Euclidean vector space, α ∈ V a non-zero vector and k ∈ Z. Define the affine hyperplane Hα,k =

x ∈ V : x, α = k
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 2 / 33

SLIDE 4

Setting the stage

Definition Let V be a Euclidean vector space, α ∈ V a non-zero vector and k ∈ Z. Define the affine hyperplane Hα,k =

x ∈ V : x, α = k
.

Define the reflection in Hα,k as sα,k(x) = x + 2k − x, α α, α α .

Robin Sulzgruber On (rational) Shi tableaux March 2017 2 / 33

SLIDE 5

Setting the stage

Definition Let V be a Euclidean vector space, α ∈ V a non-zero vector and k ∈ Z. Define the affine hyperplane Hα,k =

x ∈ V : x, α = k
.

Define the reflection in Hα,k as sα,k(x) = x + 2k − x, α α, α α . Definition An irreducible crystallographic root system is a finite subset Φ ⊆ V with some properties.

Robin Sulzgruber On (rational) Shi tableaux March 2017 2 / 33

SLIDE 6

Setting the stage

Definition Let V be a Euclidean vector space, α ∈ V a non-zero vector and k ∈ Z. Define the affine hyperplane Hα,k =

x ∈ V : x, α = k
.

Define the reflection in Hα,k as sα,k(x) = x + 2k − x, α α, α α . Definition An irreducible crystallographic root system is a finite subset Φ ⊆ V with some properties. The Weyl group of Φ is the group generated by the reflections sα,0 for α ∈ Φ+.

Robin Sulzgruber On (rational) Shi tableaux March 2017 2 / 33

SLIDE 7

The root system of type A2

Robin Sulzgruber On (rational) Shi tableaux March 2017 3 / 33

SLIDE 8

The root system of type A2

α2 = e2 − e3 α1 + α2 = e1 − e3 α1 = e1 − e2

Robin Sulzgruber On (rational) Shi tableaux March 2017 3 / 33

SLIDE 9

The root system of type A2

Hα2,0 Hα1,0 Hα1+α2,0 α2 = e2 − e3 α1 + α2 = e1 − e3 α1 = e1 − e2

Robin Sulzgruber On (rational) Shi tableaux March 2017 3 / 33

SLIDE 10

The root system of type A2

Hα2,0 Hα1,0 Hα1+α2,0 The reflections sα1,0, sα2,0 generate the symmetric group S3.

Robin Sulzgruber On (rational) Shi tableaux March 2017 3 / 33

SLIDE 11

The affine Weyl group

Robin Sulzgruber On (rational) Shi tableaux March 2017 4 / 33

SLIDE 12

The affine Weyl group

Definition The affine arrangement of Φ is defined as Aff =

Hα,k : α ∈ Φ+, k ∈ Z
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 4 / 33

SLIDE 13

The affine Weyl group

Definition The affine arrangement of Φ is defined as Aff =

Hα,k : α ∈ Φ+, k ∈ Z
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 4 / 33

SLIDE 14

The affine Weyl group

Definition The affine arrangement of Φ is defined as Aff =

Hα,k : α ∈ Φ+, k ∈ Z
.

The regions of the affine arrangement are called alcoves.

Robin Sulzgruber On (rational) Shi tableaux March 2017 4 / 33

SLIDE 15

The affine Weyl group

Definition The affine arrangement of Φ is defined as Aff =

Hα,k : α ∈ Φ+, k ∈ Z
.

The regions of the affine arrangement are called alcoves. The affine Weyl group W is the group generated by all reflections in the hyperplanes of Aff. It acts simply transitively on the set of alcoves.

Robin Sulzgruber On (rational) Shi tableaux March 2017 4 / 33

SLIDE 16

The Shi arrangement

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 17

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 18

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 19

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Theorem (Shi 1987, 1997) The Shi arrangement has (h + 1)r regions and 1 |W |

r

i=1

(di + h) dominant regions.

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 20

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Theorem (Shi 1987, 1997) The Shi arrangement has (h + 1)r regions and 1 |W |

r

i=1

(di + h) dominant regions. (n + 1)n−1

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 21

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Theorem (Shi 1987, 1997) The Shi arrangement has (h + 1)r regions and 1 |W |

r

i=1

(di + h) dominant regions. (n + 1)n−1 = 42 = 16

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 22

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Theorem (Shi 1987, 1997) The Shi arrangement has (h + 1)r regions and 1 |W |

r

i=1

(di + h) dominant regions. (n + 1)n−1 = 42 = 16 1 n + 1 2n n

Robin Sulzgruber

On (rational) Shi tableaux March 2017 5 / 33

SLIDE 23

The Shi arrangement

Definition The Shi arrangement is defined as Shi =

Hα,k : α ∈ Φ+, k ∈ {0, 1}
.

Theorem (Shi 1987, 1997) The Shi arrangement has (h + 1)r regions and 1 |W |

r

i=1

(di + h) dominant regions. (n + 1)n−1 = 42 = 16 1 n + 1 2n n

=

6 · 5 · 4 4 · 3 · 2 · 1 = 5

Robin Sulzgruber On (rational) Shi tableaux March 2017 5 / 33

SLIDE 24

The m-Shi arrangement

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 25

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 26

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 27

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Theorem (Athanasiadis 2004, Yoshinaga 2004) The Shi arrangement has (mh + 1)r regions and 1 |W |

r

i=1

(di + mh) dominant regions.

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 28

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Theorem (Athanasiadis 2004, Yoshinaga 2004) The Shi arrangement has (mh + 1)r regions and 1 |W |

r

i=1

(di + mh) dominant regions. (mn + 1)n−1

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 29

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Theorem (Athanasiadis 2004, Yoshinaga 2004) The Shi arrangement has (mh + 1)r regions and 1 |W |

r

i=1

(di + mh) dominant regions. (mn + 1)n−1 = 49

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 30

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Theorem (Athanasiadis 2004, Yoshinaga 2004) The Shi arrangement has (mh + 1)r regions and 1 |W |

r

i=1

(di + mh) dominant regions. (mn + 1)n−1 = 49 1 mn + 1 mn + n n

Robin Sulzgruber

On (rational) Shi tableaux March 2017 6 / 33

SLIDE 31

The m-Shi arrangement

Definition The m-Shi arrangement is defined as Shim =

Hα,k : α ∈ Φ+, −m < k ≤ m
.

Theorem (Athanasiadis 2004, Yoshinaga 2004) The Shi arrangement has (mh + 1)r regions and 1 |W |

r

i=1

(di + mh) dominant regions. (mn + 1)n−1 = 49 1 mn + 1 mn + n n

= 12

Robin Sulzgruber On (rational) Shi tableaux March 2017 6 / 33

SLIDE 32

Walls and floors

Robin Sulzgruber On (rational) Shi tableaux March 2017 7 / 33

SLIDE 33

Walls and floors

Definition A hyperplane Hα,k is called wall of an alcove if it supports a facet of the alcove.

Robin Sulzgruber On (rational) Shi tableaux March 2017 7 / 33

SLIDE 34

Walls and floors

Definition A hyperplane Hα,k is called wall of an alcove if it supports a facet of the alcove. A wall is called floor if it separates the alcove from the fundamental alcove.

Robin Sulzgruber On (rational) Shi tableaux March 2017 7 / 33

SLIDE 35

Walls and floors

Definition A hyperplane Hα,k is called wall of an alcove if it supports a facet of the alcove. A wall is called floor if it separates the alcove from the fundamental alcove.

[4, 2, 0]

Robin Sulzgruber On (rational) Shi tableaux March 2017 7 / 33

SLIDE 36

The height of a hyperplane

Robin Sulzgruber On (rational) Shi tableaux March 2017 8 / 33

SLIDE 37

The height of a hyperplane

Definition Define the height of a hyperplane Hα,k as |ht(α) − hk|.

Robin Sulzgruber On (rational) Shi tableaux March 2017 8 / 33

SLIDE 38

The height of a hyperplane

Definition Define the height of a hyperplane Hα,k as |ht(α) − hk|.

[1, 2, 3] 13 10 7 4 1 2 5 8 11 14 14 11 8 5 2 1 4 7 10 13 7 4 1 2 5 Robin Sulzgruber On (rational) Shi tableaux March 2017 8 / 33

SLIDE 39

Shi alcoves

Robin Sulzgruber On (rational) Shi tableaux March 2017 9 / 33

SLIDE 40

Shi alcoves

Theorem (Shi 1987, Athanasiadis 2005, Thiel 2015) The regions of the m-Shi arrangement are in bijection with alcoves whose floors have height less than mh + 1.

Robin Sulzgruber On (rational) Shi tableaux March 2017 9 / 33

SLIDE 41

Shi alcoves

Theorem (Shi 1987, Athanasiadis 2005, Thiel 2015) The regions of the m-Shi arrangement are in bijection with alcoves whose floors have height less than mh + 1.

[4, 2, 0] [1, −1, 6] [2, 0, 4] [1, 0, 5] [1, 2, 3] [2, 1, 3] [2, 3, 1] [3, 1, 2] [1, 3, 2] [3, 2, 1] [−1, 3, 4] [0, 1, 5] [0, 2, 4] [−2, 5, 3] [−1, 4, 3] [0, 4, 2] 13 10 7 4 5 8 11 14 14 11 8 5 4 7 10 13 10 7 4 5 8 1 2 2 1 1 2

Robin Sulzgruber On (rational) Shi tableaux March 2017 9 / 33

SLIDE 42

Robin Sulzgruber On (rational) Shi tableaux March 2017 10 / 33

SLIDE 43

[7, 2, −3] [4, −1, 3] [5, 0, 1] [4, 0, 2] [1, −4, 9] [2, −3, 7] [1, −3, 8] [4, 2, 0] [5, 1, 0] [5, 3, −2] [6, 1, −1] [4, 3, −1] [6, 2, −2] [1, −1, 6] [2, −2, 6] [2, 0, 4] [3, −2, 5] [1, 0, 5] [3, −1, 4] [1, 2, 3] [2, 1, 3] [2, 3, 1] [3, 1, 2] [1, 3, 2] [3, 2, 1] [−1, 0, 7] [0, −2, 8] [0, −1, 7] [1, 5, 0] [2, 4, 0] [3, 4, −1] [−2, 2, 6] [−1, 1, 6] [−1, 3, 4] [0, 1, 5] [−2, 3, 5] [0, 2, 4] [−2, 5, 3] [−1, 4, 3] [−1, 6, 1] [0, 4, 2] [−2, 6, 2] [0, 5, 1] [−4, 6, 4] [−3, 4, 5] [−3, 5, 4] [−5, 8, 3] [−4, 7, 3] [−3, 7, 2] 19 16 13 10 7 8 11 14 17 20 20 17 14 11 8 7 10 13 16 19 16 13 10 7 8 11 14 4 1 2 5 5 2 1 4 4 1 2 5

Robin Sulzgruber On (rational) Shi tableaux March 2017 10 / 33

SLIDE 44

Inverse Shi alcoves

Robin Sulzgruber On (rational) Shi tableaux March 2017 11 / 33

SLIDE 45

Inverse Shi alcoves

Theorem (Fishel, Vazirani 2010) The regions of the m-Shi arrangement are in bijection with the alcoves inside the simplex bounded by the hyperplanes of height mh + 1.

Robin Sulzgruber On (rational) Shi tableaux March 2017 11 / 33

SLIDE 46

Inverse Shi alcoves

Theorem (Fishel, Vazirani 2010) The regions of the m-Shi arrangement are in bijection with the alcoves inside the simplex bounded by the hyperplanes of height mh + 1.

[−2, 2, 6] [1, 5, 0] [0, 1, 5] [1, 0, 5] [1, 2, 3] [2, 1, 3] [3, 1, 2] [2, 3, 1] [1, 3, 2] [3, 2, 1] [0, 4, 2] [2, 0, 4] [0, 2, 4] [4, −1, 3] [−1, 4, 3] [−1, 3, 4] 16 13 10 7 1 2 5 8 11 14 14 11 8 5 2 1 7 10 13 16 7 1 2 5 8 11 4 4 4

Robin Sulzgruber On (rational) Shi tableaux March 2017 11 / 33

SLIDE 47

Robin Sulzgruber On (rational) Shi tableaux March 2017 12 / 33

SLIDE 48

[−5, 2, 9] [−2, 5, 3] [3, −2, 5] [−2, 3, 5] [1, 8, −3] [−3, 1, 8] [1, −3, 8] [−2, 2, 6] [2, −2, 6] [6, −2, 2] [2, 6, −2] [−2, 6, 2] [6, 2, −2] [1, 5, 0] [5, 1, 0] [0, 1, 5] [5, 0, 1] [1, 0, 5] [0, 5, 1] [1, 2, 3] [2, 1, 3] [3, 1, 2] [2, 3, 1] [1, 3, 2] [3, 2, 1] [−3, 4, 5] [5, −3, 4] [−3, 5, 4] [1, −1, 6] [−1, 1, 6] [−1, 6, 1] [4, 2, 0] [2, 4, 0] [0, 4, 2] [2, 0, 4] [4, 0, 2] [0, 2, 4] [4, −1, 3] [−1, 4, 3] [3, 4, −1] [−1, 3, 4] [4, 3, −1] [3, −1, 4] [0, 7, −1] [−1, 0, 7] [0, −1, 7] [7, −4, 3] [−4, 7, 3] [−4, 3, 7] 22 19 16 13 10 4 1 2 5 8 11 14 17 17 14 11 8 5 2 1 4 10 13 16 19 22 10 4 1 2 5 8 11 14 17 7 7 7

Robin Sulzgruber On (rational) Shi tableaux March 2017 12 / 33

SLIDE 49

A rational analogue

Robin Sulzgruber On (rational) Shi tableaux March 2017 13 / 33

SLIDE 50

A rational analogue

Definition Let p be a positive integer relatively prime to the Coxeter number h. An alcove is called p-stable if its inverse lies inside the simplex bounded by the hyperplanes of height p.

Robin Sulzgruber On (rational) Shi tableaux March 2017 13 / 33

SLIDE 51

A rational analogue

Definition Let p be a positive integer relatively prime to the Coxeter number h. An alcove is called p-stable if its inverse lies inside the simplex bounded by the hyperplanes of height p. Theorem (Thiel 2015) The number of p-stable alcoves equals pr. The number of dominant p-stable alcoves equals 1 |W |

r

i=1

(p + ei).

Robin Sulzgruber On (rational) Shi tableaux March 2017 13 / 33

SLIDE 52

Robin Sulzgruber On (rational) Shi tableaux March 2017 14 / 33

SLIDE 53

[−3, 2, 7] [0, 5, 1] [1, 0, 5] [0, 1, 5] [0, 2, 4] [2, 0, 4] [4, 0, 2] [2, 4, 0] [0, 4, 2] [4, 2, 0] [−2, 3, 5] [5, −2, 3] [−2, 5, 3] [3, 2, 1] [2, 3, 1] [1, 3, 2] [2, 1, 3] [3, 1, 2] [1, 2, 3] [3, −1, 4] [−1, 3, 4] [−1, 4, 3] [1, 6, −1] [−1, 1, 6] [1, −1, 6] 13 10 7 4 1 2 8 11 14 17 17 14 11 8 2 1 4 7 10 13 13 10 7 4 1 2 8 5 5 5 Robin Sulzgruber On (rational) Shi tableaux March 2017 14 / 33

SLIDE 54

Robin Sulzgruber On (rational) Shi tableaux March 2017 15 / 33

SLIDE 55

[−3, 2, 7] [3, −1, 4] [1, 0, 5] [2, 0, 4] [0, 2, 4] [0, 1, 5] [−2, 3, 5] [−1, 1, 6] [−1, 3, 4] [−2, 2, 6] [4, 0, 2] [5, −2, 3] [4, −1, 3] [3, 2, 1] [3, 1, 2] [1, 3, 2] [2, 1, 3] [2, 3, 1] [1, 2, 3] [0, 5, 1] [0, 4, 2] [−1, 4, 3] [1, 6, −1] [2, 4, 0] [1, 5, 0] 16 13 10 7 4 1 2 5 8 11 14 17 17 14 11 8 5 2 1 4 7 10 13 16 13 10 7 4 1 2 5 8 11

Robin Sulzgruber On (rational) Shi tableaux March 2017 15 / 33

SLIDE 56

Shi tableaux

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 57

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 58

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

The Shi tableau of w is the collection of the numbers tmh+1(α, w) for α ∈ Φ+.

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 59

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

The Shi tableau of w is the collection of the numbers tmh+1(α, w) for α ∈ Φ+.

[4, 2, 0] [2, 0, 4] [1, 2, 3] [0, 2, 4] [0, 4, 2] Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 60

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

The Shi tableau of w is the collection of the numbers tmh+1(α, w) for α ∈ Φ+.

[4, 2, 0] [2, 0, 4] [1, 2, 3] [0, 2, 4] [0, 4, 2]

w = [4, 2, 0]

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 61

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

The Shi tableau of w is the collection of the numbers tmh+1(α, w) for α ∈ Φ+.

[4, 2, 0] [2, 0, 4] [1, 2, 3] [0, 2, 4] [0, 4, 2]

w = [4, 2, 0] t4(α1, w) = 1

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 62

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

The Shi tableau of w is the collection of the numbers tmh+1(α, w) for α ∈ Φ+.

[4, 2, 0] [2, 0, 4] [1, 2, 3] [0, 2, 4] [0, 4, 2]

w = [4, 2, 0] t4(α1, w) = 1 t4(α2, w) = 1

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 63

Shi tableaux

Definition (Fishel, Tzanaki, Vazirani 2011) Let w(A◦) be a dominant Shi alcove and α ∈ Φ+. Define tmh+1(α, w) as the number of Shi hyperplanes

f the form Hα,k that separate w(A◦) and A◦.

The Shi tableau of w is the collection of the numbers tmh+1(α, w) for α ∈ Φ+.

[4, 2, 0] [2, 0, 4] [1, 2, 3] [0, 2, 4] [0, 4, 2]

w = [4, 2, 0] t4(α1, w) = 1 t4(α2, w) = 1 t4(α1 + α2, w) = 1

Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

SLIDE 64

Rational Shi tableaux