lecture 15: dynkin diagrams and subgroups of lie...

Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Lecture 15: Dynkin Diagrams andsubgroups of Lie groups

Daniel Bump

May 26, 2020


The rank two root systems

The rank two root systems are, in the Cartan classification A2,C2, G2 and A1 × A1. Lie groups representing these are SL(3),Sp(4), G2 (the automorphism group of the octonions andSL(2)× SL(2).

We will study general root systems by finding rank two rootsystems inside them, so let us take a closer look at the rank tworoot systems.

We will denote by {α1, · · · , αr} the simple roots. (In this sectionr = 2.) We will also introduce α0, the negative of the highestroot which we may call the affine root.

We proved if αi, αj are simple roots then 〈αi, αj〉 6 0. Thisremains true if we include α0


The A2 root system

α1

α2

α0

The shaded area is the positive Weyl chamber C+. The weightlattice is indicated as lighter dots. The root lattice has index 3 inthe SU(3) weight lattice. Positive roots are red.

If all roots have the same length, the root system is calledsimply-laced. The A2 root system is simply-laced.


The C2 root system

α1

α2

α0

This is the Sp(4) root system. The simple roots are α1 = (1,−1)and α2 = (0, 2). The root lattice has index two in the Sp(4)weight lattice, which we are identifying with Z2.


The B2 root system

α1

α2

α0

The SO(5) or spin(5) root system is accidentally isomorphic tothe Sp(4) root system.

The SO(5) weight lattice is Z2. The spin(5) weight lattice is

Z2 ⊕(Z2 + ( 1

2 ,12))


The G2 root system

α1

α2

α0

This time the root lattice equals the weight lattice.


The A1 × A1 root lattice

α1

α1

This is the reducible root system for SU(2)× SU(2). There is noaffine root.


The Dynkin diagram

The Dynkin diagram is a graph whose vertices are the simpleroots. Draw an edge connecting αi to αj if they are notorthogonal.

For the extended Dynkin diagram, we add a node for α0.

We often use a dashed line for connections of α0. Here is theextended Dynkin diagram for A3:

α1 α2 α3

α0 α1 = (1,−1, 0, 0)α2 = (0, 1,−1, 0)α3 = (0, 0, 1,−1)α0 = (−1, 0, 0, 1)


Double and triple bonds

If αi and αj have different lengths, we connect them by:

a double bond if their root lengths are in the ratio√

2;a triple bond if their root lengths are in the ratio

√3.

The triple bond only occurs with G2. Here are the angles of theroots:

bond angle exampleno bond π

2 SU(2)× SU(2)

single bond 2π3 SU(3)

double bond 3π4 Sp(4)

triple bond 5π6 G2


The direction of the arrow

If the roots are connected by a double or triple bond, they havedifferent lengths. We draw an arrow from the long root to theshort root.

Here are the extended Dynkin diagram of type Bn and Cn:

α1 α2 α3 αn−2 αn−1 αn

α0

α0 α1 α2 α3 αn−2 αn−1 αn


What we learn from Dynkin diagrams

The Dynkin diagram shows the relations between the simpleroots.

The extended Dynkin diagrams adds the affine root.

From the Dynkin diagram we may read off:

Generators and relations for the Weyl group;All Levi subgroups;

From the extended Dynkin diagram we may read off:

Generators and relations for the affine Weyl group;More general Lie subgroups


Type Dn

α1 α2 α3 αn−3 αn−2

αn−1

αn

α0

The group D4 = spin(8) is particularly interesting. Here is itsextended Dynkin diagram:

α0

α1

α2

α3

α4


Triality

Another use of the Dynkin diagram is to make manifest theouter automorphisms of a Lie group. Symmetries of the Dynkindiagram may be realized as automorphisms of the group in itssimply-connected form.

The D4 Dynkin diagram has an automorphism of degree 3.

α1α2

α3

α4

This is an automorphism of the simply-connected group spin(8)or the adjoint form PGSO(8).


Triality (continued)

The group spin(2k) has two irreducible representations ofdegree 2k−1 called the spin representations. It also has anirreducible representation of degree 2k, the standardrepresentation. If k = 4, then 2k = 2k−1 = 8. Thus spin(8) hasthree irreducible representations of degree 8. These arepermuted by triality.

The reason is that the center of spin(8) is Z2 × Z2. Triality actson the center and the kernel Z2 of the homomorphismspin(8)→ SO(8) is not invariant under triality.

The fixed subgroup of this automorphism is the exceptionalgroup G2, the automorphism of the octonions.


Exceptional groups

We will at least give the extended Dynkin diagrams for theexceptional types G2, F4, E6, E7 and E8. Here is G2:

α0 α1 α2

There are two conventions for the ordering of the roots, due toDynkin and Bourbaki. They differ in the exceptional groups. Weare following Bourbaki.


The exceptional group F4

α0 α1 α2 α3 α3

The group F4 is the next exceptional group. It is theautomorphism group of a 27-dimensional (nonassociative)Jordan algebra discovered by A. A. Albert that is also closelyrelated to the exceptional groups E6,E7 and E8. The exceptionalgroup G2 is a subgroup.


The exceptional groups E6, E7 and E8

α1 α3 α4 α5 α6

α2

α0

α0 α1

α2

α3 α4 α5 α6 α7

α1 α3 α4 α5 α6 α7 α8

α2

α0


Levi subgroups

One application of the Dynkin diagram and extended Dynkindiagram is to envision embeddings of Lie groups. Manymaximal subgroups can be visualized instantly.

The easiest case is that of a Levi subgroup. Let us choose asubset S of the simple roots and consider the complex Liealgebra generated by

X±α, α ∈ S.

This is a Levi subgroup of the complex Lie group GC. (If wewant we can intersect it with the compact Lie group G.)


Levi decomposition of parabolics

Levi subgroups appear as Levi decompositions of parabolicsubgroups. A subgroup P containing the Borel subgroup B (ofGC) whose Lie algebra is

tC ⊕⊕

α∈Φ+

Xα

is called a parabolic subgroup. It is a semidirect product of anormal unipotent group and a parabolic subgroup.


For example, let G = GL(4), S = {α1, α3}. Let

P =

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗∗ ∗

.

This has a decomposition P = MU with U normal:

M =

∗ ∗∗ ∗

∗ ∗∗ ∗

, U =

1 ∗ ∗1 ∗ ∗

11

.

The subgroup M is a Levi subgroup. The group U is called theunipotent radical of P.


Levi subgroups from Dynkin diagrams

Starting from the Dynkin diagram of G, if we erase one or morenodes, we obtain the Dynkin diagram of a Levi subgroup. In theabove example, the Dynkin diagram of GL(4) is of Type A3.

α1 α2 α3

After selecting S = {α1, α3}, that is, erasing the middle node,we obtain the Dynkin diagram of the Levi subgroupGL(2)×GL(2), of type A1 × A1:

α1 α3

All Levi subgroups can be determined easily from the Dynkindiagram. A Levi subgroup may or may not be a maximalsubgroup.


Subgroups from the extended Dynkin diagram

If we erase one node of the extended Dynkin diagram, wetypically obtain the Dynkin diagram of a subgroup that is often amaximal subgroup.

Here is the extended Dynkin diagram of SO(9) (Type B4):

α1 α2 α3 α4

α0

Erasing the root α4 gives the Dynkin diagram of type D4 and wehave obtained the embedding SO(8)→ SO(9);


Review: convex sets of roots and Lie subalgebras

In Lecture 7 we considered a subset S of Φ ∪ {0} such that

α, β ∈ S, α+ β ∈ Φ ∪ {0} ⇒ α+ β ∈ S. (∗)

We will call such a set convex. Then

gC,S =⊕

α∈S

Xα

is closed under the bracket, so it is a Lie subalgebra of gC. Weare denoting tC = X0 even though 0 is not a root.Since [gα, gβ] ⊆ gα+β, the complex vector space gC,S is acomplex Lie algebra. It is not contained in g, only gC.


Examples: G2

We consider two convex sets of roots of Φ ∪ {0} in the caseG = G2. These two convex sets are root systems.

α1

α2

3α1 + 2α2

α0

First, we can take S = {±α1,±(3α1 + 2α2), )}. Note that theroots α1 and 3α1 + 2α2 are orthogonal. The Lie algebra in thiscase is of Type A1 × A1.


Another G2 example

The other convex set of roots is the set of long roots:

α1

α2

3α1 + 2α2

α0

This root system is of type A2. We see from theseconsiderations that the (complex) G2 Lie algebra has Liesubalgebras of Types A1 × A1 and A2, so G2 should contain Liesubgroups isomorphic to SU(2)× SU(2) and SU(3).


The G2 examples from the extended Dynkin diagram

We can predict these subgroups of types SU(2)× SU(2) andSU(3) by looking at the Extended Dynkin diagram.

α0 α1 α2

Eliminating α1 produces a Dynkin diagram of type A1 × A1.Eliminating α2 produces A2.


More orthogonal embeddings

We have seen that the extended Dynkin diagram explains theembedding SO(2n)→ SO(2n + 1). But what about theembedding SO(2n + 1)→ SO(2n + 2)?

For this embedding root spaces of SO(2n + 1) are not mappedto a single root space of SO(2n + 2) but instead to a sum of tworoot spaces. We imagine the Dynkin diagram of type Dn+1folded onto the Dynkin diagram of type Bn:

α1 α2 α3 αn−2 αn−1

α1 α2 α3 αn−2 αn−1

αn

αn+1

αn

}


Root folding

Root folding refers to a map from one Dynkin diagram toanother that may be 2-1 or (in one example) 3-1. The foldedDynkin diagram is then the Dynkin diagram of the other.

We saw at the Dynkin diagram of Dn+1 can be folded into theDynkin diagram of Bn, explaining the embedding of SO(2n + 1)into SO(2n + 2). Here is another example. We may fold theDynkin diagram of D4 into G2, showing that G2 is subgroup ofspin(8).

α2 α1

α2

α1

α4

α3

}


Maximal subgroups of Lie groups

Maximal subgroups of Lie groups were classified by Dynkin. Hemissed a few, for Seitz and his student Testerman found somenew maximal subgroups of exceptional groups. There areseven maximal subgroups of E8 that are isomorphic to SL(2).

If H is a subgroup of G, a basic problem is to compute thebranching rule that describes how irreducible representations ofG decompose into irreducibles when restricted to H. In somecases, one may find a general description of the branchingrules. In other cases, one still wants to have an efficientalgorithm to decompose any particular given representation.

As we will demonstrate, Sage knows all of the maximalsubgroups of Lie groups up to rank 8, and is able to computethe branching rules efficiently.


Subgroups associated to representations

Subgroups of orthogonal and symplectic groups cansometimes be recognized as follows. Start with arepresentation π : G→ GL(n) of some Lie group. The image ofπ might be a maximal subgroup of GL(n). On the other hand ifπ is self-contragredient, it will never be maximal, for its imagewill be contained in either O(n) or Sp(n).

The Frobenius-Schur indicator that recognizes whether theimage of π is contained in (n) or Sp(n). If G is compact, this is

ε(π) =

∫

Gχπ(g2) dg.

If it is +1, the representation is orthogonal; if it is −1 therepresentation is symplectic. It is 0 if the representation is notself-contragredient.


Example: the embedding of SU(3) into SO(8)

The adjoint representation of any semisimple Lie group isorthogonal, since the Killing form

B(x, y) = tr(Ad(x) Ad(y))

is then known to be nondegenerate, and is obviouslysymmetric. Thus it is known in advance that ε(Ad) = 1.

The Ad : SL(3)→ SO(8) thus factors through the orthogonalgroup O(8). This SL(3) is indeed a maximal subgroup of SO(8).


A Sage session

In Sage, the WeylCharacterRing is a class for the irreduciblerepresentations of a Lie group. We can create the WeylCharacterRing of type D4. Sage will tell you the maximalsubgroups and give you the syntax of a branching rule that canuse to branch representations.

sage: D4=WeylCharacterRing("D4",style="coroots")sage: D4.maximal_subgroups()B3:branching_rule("D4","B3","symmetric")A2:branching_rule("D4","A2(1,1)","plethysm")A1xC2: ...A1xA1xA1xA1: ...

I’ve omitted the A1 × C2 and A1 × A1 × A1 × A1 branching rulessince they don’t fit on a line.


Example of branching

The branching rule we are interested in is the A2 plethysm, sowe implement that. As our guinea pig we take a moderatelylarge representation of D4.

sage: A2=WeylCharacterRing("A2",style="coroots")sage: br=branching_rule("D4","A2(1,1)","plethysm")sage: rep=D4(1,2,1,1)sage: rep.degree()25725

This will work for much larger representations than this one.This branching rule is fast even for representations of spin(8)with degrees into the millions.


Get ready, get set, branch

sage: rep.branch(A2,rule=br)A2(0,0) + 10*A2(0,3) + 11*A2(1,1) + 10*A2(3,0) +22*A2(1,4) + 10*A2(0,6) + 24*A2(2,2) + 22*A2(4,1)+ 24*A2(2,5) + 12*A2(1,7) + 3*A2(0,9) +30*A2(3,3) + 10*A2(6,0) + 24*A2(5,2) + 14*A2(3,6)+ 6*A2(2,8) + 2*A2(1,10) + 23*A2(4,4) +12*A2(7,1) + 14*A2(6,3) + 4*A2(4,7) + A2(3,9) +9*A2(5,5) + 3*A2(9,0) + 6*A2(8,2) + 4*A2(7,4) +A2(6,6) + 2*A2(10,1) + A2(9,3)

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