ADE Platonic solids Root systems Kostant’s game
Amazing Diagrams Everywhere
Sira Gratz
Sira Gratz ADE ADE Platonic solids Overview Root systems The diagrams Kostant’s game Overview
1 ADE diagrams, and some first examples
2 Quiver representations
3 Cluster algebras
Sira Gratz ADE We will call these diagrams the ADE diagrams.
ADE Platonic solids Overview Root systems The diagrams Kostant’s game The list
Name Diagram An, n ≥ 1 • • • ... • • •
• • ... • •
Dn, n ≥ 4 • • • • • •
E6 • • • • • • •
E7 • • • • • • • •
E8 •
Sira Gratz ADE ADE Platonic solids Overview Root systems The diagrams Kostant’s game The list
Name Diagram An, n ≥ 1 • • • ... • • •
• • ... • •
Dn, n ≥ 4 • • • • • •
E6 • • • • • • •
E7 • • • • • • • •
E8 • We will call these diagrams the ADE diagrams.
Sira Gratz ADE ADE Platonic solids Overview Root systems The diagrams Kostant’s game A sample of classifications
ADE diagrams show up in the classification of Platonic solids finite subgroups of SO(3) complex simple Lie algebras connected graphs with least eigenvalue −2 quivers of finite representation type cluster algebras of finite type etc.
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Platonic solids
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Kepler’s Solar System
Sira Gratz ADE A platonic solid is a regular convex polyhedron.
All the faces of a platonic solid are the same regular polygon, all the faces meet at the same angle, and the number of edges incident with each vertex is the same.
ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Definition
Platonic solids 3 A polyhedron is a region in R bounded by planes; it has two-dimensional faces, which meet in one-dimensional edges, which meet in vertices.
Sira Gratz ADE All the faces of a platonic solid are the same regular polygon, all the faces meet at the same angle, and the number of edges incident with each vertex is the same.
ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Definition
Platonic solids 3 A polyhedron is a region in R bounded by planes; it has two-dimensional faces, which meet in one-dimensional edges, which meet in vertices. A platonic solid is a regular convex polyhedron.
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Definition
Platonic solids 3 A polyhedron is a region in R bounded by planes; it has two-dimensional faces, which meet in one-dimensional edges, which meet in vertices. A platonic solid is a regular convex polyhedron.
All the faces of a platonic solid are the same regular polygon, all the faces meet at the same angle, and the number of edges incident with each vertex is the same.
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game List
https://commons.wikimedia.org/wiki/Category: Set_of_Platonic_solids;_green;_animated
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Hosohedron and dihedron
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Duality
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license. https://creativecommons.org/licenses/by-sa/4.0/deed.en
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Duality
tetrahedron ←→ tetrahedron cube ←→ octahedron dodecahedron ←→ icosahedron regular dihedron (n vertices) ←→ regular hosohedron (n faces)
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Duality
tetrahedron ←→ tetrahedron cube ←→ octahedron dodecahedron ←→ icosahedron regular n-dihedron ←→ regular n-hosohedron
Sira Gratz ADE These are precisely the finite subgroups of SO(3).
ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Classification
ADE diagram Rotational symmetries of the . . . An “directed” regular n-hosohedron and n-dihedron Dn regular n-hosohedron and n-dihedron E6 tetrahedron E7 cube and octahedron E8 dodecahedron and icosahedron
Sira Gratz ADE ADE An ancient classification Platonic solids The polyhedra Root systems Classification Kostant’s game Classification
ADE diagram Subgroup of SO(3) An cyclic group Cn Dn dihedral group Dn E6 tetrahedral group E7 octahedral group E8 icosahedral group These are precisely the finite subgroups of SO(3).
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Reflections
Definition n Let α ∈ R . The reflection sα is the reflection along the n hyperplane perpendicular to α: For β ∈ R we have (β, α) s (β) = β − 2 α α (α, α)
Sira Gratz ADE (β,α) −2 (α,α) α
sα(β)
(β,α) |α|
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game
The reflection sα
β
θ α
Recall: (β, α) = |β||α| cos(θ), and, in particular, |α|2 = (α, α).
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game
The reflection sα
(β,α) −2 (α,α) α
β sα(β)
θ α
(β,α) |α|
Recall: (β, α) = |β||α| cos(θ), and, in particular, |α|2 = (α, α).
Sira Gratz ADE We call an element α ∈ Φ a root of Φ.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Definition
Definition n A finite set Φ of non-trivial vectors in R is a (crystallographic) root system if it satisfies the following n Φ spans R The only scalar multiples of α ∈ Φ that belong to Φ are ±α For all α ∈ Φ we have sα(Φ) = Φ If α, β ∈ Φ, then sα(β) = β − mα with m ∈ Z.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Definition
Definition n A finite set Φ of non-trivial vectors in R is a (crystallographic) root system if it satisfies the following n Φ spans R The only scalar multiples of α ∈ Φ that belong to Φ are ±α For all α ∈ Φ we have sα(Φ) = Φ If α, β ∈ Φ, then sα(β) = β − mα with m ∈ Z.
We call an element α ∈ Φ a root of Φ.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Example
π π 3 3
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game
This file is licensed under the Creative Commons Attribution 3.0 Unported license. https://creativecommons.org/licenses/by/3.0/deed.en
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Possible angles
Question Let Φ be a (crystallographic) root system, and let α, β ∈ Φ. What are the possible values for the angle θ between α and β?
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Possible angles
Question Let Φ be a (crystallographic) root system, and let α, β ∈ Φ. What are the possible values for the angle θ between α and β?
(α, β) (β, α) (α, β)2 2 2 = 4 = 4 cos2(θ) ∈ (α, α) (β, β) |α|2|β|2 Z | {z } | {z } ∈Z ∈Z
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Possible angles
Question Let Φ be a (crystallographic) root system, and let α, β ∈ Φ. What are the possible values for the angle θ between α and β?
2 4 cos (θ) ∈ Z; cos(θ) ∈ [−1, 1] √ √ 1 2 3 ⇒ cos(θ) ∈ {0, ± , ± , ± , ±1} 2 2 2
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Possible angles
Question Let Φ be a (crystallographic) root system, and let α, β ∈ Φ. What are the possible values for the angle θ between α and β?
√ √ 1 2 3 cos(θ) ∈ {0, ± 2 , ± 2 , ± 2 , ±1} π π 2π π 3π π 5π ⇔ θ ∈ { 2 , 3 , 3 , 4 , 4 , 6 , 6 , 0, π}
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Simple roots
Definition n Let Φ ⊆ R be a root system. A subset ∆ ⊆ Φ is called a set of simple roots of Φ, if the following axioms are satisfied. n ∆ = {α1, . . . , αn} is a basis of R Every β ∈ Φ can be written as
n X β = mi αi i=1
such that for all i = 1,..., n we have either mi ∈ Z≥0 or mi ∈ Z≤0.
Sira Gratz ADE We set
Φ+ = {positive roots of Φ}; Φ− = {negative roots of Φ}.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Positive and negative roots Definition
Let Φ be a root system, and ∆ = {α1, . . . , αn} ⊆ Φ a choice of simple roots of Φ. If in the expression
n X β = mi αi i=1
we have mi ∈ Z≥0 for all i = 1,..., n, we say that β is positive, and else, if mi ∈ Z≤0 for all i = 1,..., n we say that it is negative.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Positive and negative roots Definition
Let Φ be a root system, and ∆ = {α1, . . . , αn} ⊆ Φ a choice of simple roots of Φ. If in the expression
n X β = mi αi i=1
we have mi ∈ Z≥0 for all i = 1,..., n, we say that β is positive, and else, if mi ∈ Z≤0 for all i = 1,..., n we say that it is negative.
We set
Φ+ = {positive roots of Φ}; Φ− = {negative roots of Φ}.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Positive and negative roots
Note that we have Φ = Φ+ t Φ−. Furthermore, Φ− = −Φ+ = {−β | β ∈ Φ+}.
Sira Gratz ADE α2 α1 + α2
−α1 α1
−α1 − α2 −α2
+ − ∆ = {α1, α2}; Φ = {α1, α2, α1 + α2}; Φ = {−α1, −α2, −α1 − α2}
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Example
π π 3 3
Sira Gratz ADE α2 α1 + α2
−α1 α1
−α1 − α2 −α2
+ − Φ = {α1, α2, α1 + α2}; Φ = {−α1, −α2, −α1 − α2}
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Example
α2
α1
Sira Gratz ADE
∆ = {α1, α2}; ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Example
α2 α1 + α2
−α1 α1
−α1 − α2 −α2
+ − ∆ = {α1, α2}; Φ = {α1, α2, α1 + α2}; Φ = {−α1, −α2, −α1 − α2}
Sira Gratz ADE The possible angles between two distinct simple roots thus are π 2π 3π 5π , , , 2 3 4 6
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Recall: The possible angles between two distinct roots in a root system are: π π 2π π 3π π 5π , , , , , , , π 2 3 3 4 4 6 6
Sira Gratz ADE The possible angles between two distinct simple roots thus are π 2π 3π 5π , , , 2 3 4 6
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Recall: The possible angles between two distinct roots in a root system are: π π 2π π 3π π 5π , , , , , , , π 2 3 3 4 4 6 6 π Furthermore, the angle between two simple roots must be ≥ 2 :
β sα(β) = β − mα
α
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Recall: The possible angles between two distinct roots in a root system are: π π 2π π 3π π 5π , , , , , , , π 2 3 3 4 4 6 6 The possible angles between two distinct simple roots thus are π 2π 3π 5π , , , 2 3 4 6
Sira Gratz ADE The vertices of Γ(Φ) are labelled by ∆ Given v, w ∈ ∆, we draw no edge between v and w, if v ⊥ w; 2π a single edge if they make an angle of 3 ; 3π a double edge if they make an angle of 4 ; 5π a triple edge if they make an angle of 6 . If two simple roots, that are joined by an edge, have different length, we draw an arrow from the longer root to the shorter root.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Definition Let Φ be a root system with fixed set of simple roots ∆. We associate to this a diagram Γ(Φ) as follows:
Sira Gratz ADE Given v, w ∈ ∆, we draw no edge between v and w, if v ⊥ w; 2π a single edge if they make an angle of 3 ; 3π a double edge if they make an angle of 4 ; 5π a triple edge if they make an angle of 6 . If two simple roots, that are joined by an edge, have different length, we draw an arrow from the longer root to the shorter root.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Definition Let Φ be a root system with fixed set of simple roots ∆. We associate to this a diagram Γ(Φ) as follows: The vertices of Γ(Φ) are labelled by ∆
Sira Gratz ADE If two simple roots, that are joined by an edge, have different length, we draw an arrow from the longer root to the shorter root.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Definition Let Φ be a root system with fixed set of simple roots ∆. We associate to this a diagram Γ(Φ) as follows: The vertices of Γ(Φ) are labelled by ∆ Given v, w ∈ ∆, we draw no edge between v and w, if v ⊥ w; 2π a single edge if they make an angle of 3 ; 3π a double edge if they make an angle of 4 ; 5π a triple edge if they make an angle of 6 .
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game A diagram from a root system
Definition Let Φ be a root system with fixed set of simple roots ∆. We associate to this a diagram Γ(Φ) as follows: The vertices of Γ(Φ) are labelled by ∆ Given v, w ∈ ∆, we draw no edge between v and w, if v ⊥ w; 2π a single edge if they make an angle of 3 ; 3π a double edge if they make an angle of 4 ; 5π a triple edge if they make an angle of 6 . If two simple roots, that are joined by an edge, have different length, we draw an arrow from the longer root to the shorter root.
Sira Gratz ADE • •
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Example
α2 α1 + α2
2π 3 −α1 α1
−α1 − α2 −α2
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Example
α2 α1 + α2
2π 3 −α1 α1
−α1 − α2 −α2
• •
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Irreducible root systems
The diagram Γ(Φ) does not depend on the choice of simple roots. Definition We call a root system Φ irreducible, if the associated diagram Γ(Φ) is connected.
Sira Gratz ADE If Φ is an irreducible (crystallographic) root system, then the diagram Γ(Φ) is a Dynkin diagram.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Classification of irreducible root systems
Theorem The irreducible (crystallographic) root systems are classified by the Dynkin diagrams.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Classification of irreducible root systems
Theorem The irreducible (crystallographic) root systems are classified by the Dynkin diagrams.
If Φ is an irreducible (crystallographic) root system, then the diagram Γ(Φ) is a Dynkin diagram.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game The Dynkin diagrams
Name Diagram An, n ≥ 1 • • • ... • • Bn, n ≥ 2 • • • ... • • Cn, n ≥ 3 • • • ... • • • • • ... • • Dn, n ≥ 4 • • • • • •
E6 • • • • • • •
E7 • • • • • • • •
Sira Gratz ADE They are associated to those root systems where all roots have the same length.
ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game
Observation The ADE diagrams are precisely the simply laced Dynkin diagrams.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game
Observation The ADE diagrams are precisely the simply laced Dynkin diagrams. They are associated to those root systems where all roots have the same length.
Sira Gratz ADE ADE Reflections Platonic solids Definitions Root systems Classification Kostant’s game Some numerology
Type # roots # positive roots # simple roots 2 n2+n An n + n 2 n 2 2 Bn 2n n n 2 2 Cn 2n n n Dn 2n(n − 1) n(n − 1) n E6 72 36 6 E7 126 63 7 E8 240 120 8 F4 48 24 4 G2 12 6 2
Sira Gratz ADE We say that the vertex v is unhappy if 1 X |v| < |w|; 2 w neighbour of v
happy if 1 X |v| = |w|; 2 w neighbour of v
manic if 1 X |v| > |w|. 2 w neighbour of v
ADE The rules Platonic solids Example Root systems Classification Kostant’s game The set-up
Let Γ be a connected, finite, simple graph, and give each vertex v of Γ a positive number |v|.
Sira Gratz ADE happy if 1 X |v| = |w|; 2 w neighbour of v
manic if 1 X |v| > |w|. 2 w neighbour of v
ADE The rules Platonic solids Example Root systems Classification Kostant’s game The set-up
Let Γ be a connected, finite, simple graph, and give each vertex v of Γ a positive number |v|. We say that the vertex v is unhappy if 1 X |v| < |w|; 2 w neighbour of v
Sira Gratz ADE manic if 1 X |v| > |w|. 2 w neighbour of v
ADE The rules Platonic solids Example Root systems Classification Kostant’s game The set-up
Let Γ be a connected, finite, simple graph, and give each vertex v of Γ a positive number |v|. We say that the vertex v is unhappy if 1 X |v| < |w|; 2 w neighbour of v
happy if 1 X |v| = |w|; 2 w neighbour of v
Sira Gratz ADE ADE The rules Platonic solids Example Root systems Classification Kostant’s game The set-up
Let Γ be a connected, finite, simple graph, and give each vertex v of Γ a positive number |v|. We say that the vertex v is unhappy if 1 X |v| < |w|; 2 w neighbour of v
happy if 1 X |v| = |w|; 2 w neighbour of v
manic if 1 X |v| > |w|. 2 w neighbour of v
Sira Gratz ADE The steps: Find an unhappy vertex v. Replace the number |v| of v by X |w| − |v| w neighbour of v
The goal : Have none of the vertices be unhappy!
ADE The rules Platonic solids Example Root systems Classification Kostant’s game
Starting point: Let Γ be a connected, finite, simple graph, and give each vertex of Γ the number 0. Now upset the balance by changing the number of one vertex to 1.
Sira Gratz ADE The goal : Have none of the vertices be unhappy!
ADE The rules Platonic solids Example Root systems Classification Kostant’s game
Starting point: Let Γ be a connected, finite, simple graph, and give each vertex of Γ the number 0. Now upset the balance by changing the number of one vertex to 1. The steps: Find an unhappy vertex v. Replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE ADE The rules Platonic solids Example Root systems Classification Kostant’s game
Starting point: Let Γ be a connected, finite, simple graph, and give each vertex of Γ the number 0. Now upset the balance by changing the number of one vertex to 1. The steps: Find an unhappy vertex v. Replace the number |v| of v by X |w| − |v| w neighbour of v
The goal : Have none of the vertices be unhappy!
Sira Gratz ADE 1
2 2 1
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
•
• • •
•
Assign 1 to one vertex, and 0s to all others.
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
0
0 0 0
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
0
0 0 0
1
Find an unhappy vertex v: 1 X |v| < |w| 0 < 1 (0 + 0 + 1) = 1 2 2 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| (1 + 0 + 0) − 0 = 1 w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
0
1 0 0
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
0
1 0 0
1
Find an unhappy vertex v: 1 X |v| < |w| 0 < 1 2 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| 1 − 0 =(1 + 0) − 0 = 1 w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
1 1 0
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
1 1 0
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
1 1 0
1
Find an unhappy vertex v: 1 X |v| < |w| 1 < 1 (1 + 1 + 1) = 3 2 2 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| (1 + 1 + 1) − 1 = 2 w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
2 1 0
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
2 1 0
1
Find an unhappy vertex v: 1 X |v| < |w| 0 < 1 2 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| 1 − 0 = 1 w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
2 1 1
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
2 1 1
1
Find an unhappy vertex v: 1 X |v| < |w| 1 < 1 (1 + 2) = 3 2 2 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| (2 + 1) − 1 = 2 w neighbour of v
Sira Gratz ADE 1
2 2 1
1
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
2 2 1
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE ADE The rules Platonic solids Example Root systems Classification Kostant’s game Example: D4
1
2 2 1
1
Find an unhappy vertex v: 1 X |v| < |w| 2 w neighbour of v
and replace the number |v| of v by X |w| − |v| w neighbour of v
Sira Gratz ADE In fact: Theorem Kostant’s game ends on the simple connected graph Γ if and only if Γ is an ADE diagram.
ADE The rules Platonic solids Example Root systems Classification Kostant’s game Kostant’s game ends for Dynkin diagrams
Exercise Convince yourself that the game ends for all ADE diagrams.
Sira Gratz ADE ADE The rules Platonic solids Example Root systems Classification Kostant’s game Kostant’s game ends for Dynkin diagrams
Exercise Convince yourself that the game ends for all ADE diagrams.
In fact: Theorem Kostant’s game ends on the simple connected graph Γ if and only if Γ is an ADE diagram.
Sira Gratz ADE