Lecture 15: Dynkin Diagrams and Subgroups of Lie Groups

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

Lecture 15: Dynkin Diagrams and subgroups of Lie groups

Daniel Bump

May 26, 2020 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

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 and SL(2) × SL(2).

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

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

We proved if αi, αj are simple roots then hαi, αji 6 0. This remains true if we include α0 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The A2 root system

α2

α1

α0

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

If all roots have the same length, the root system is called simply-laced. The A2 root system is simply-laced. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The C2 root system

α2

α0

α1

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

The B2 root system

α2

α0 α1

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

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

2 2 1 1 Z ⊕ Z + ( 2 , 2 ) Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The G2 root system

α2

α1

α0

This time the root lattice equals the weight lattice. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The A1 × A1 root lattice

α1

This is the reducible root system for SU(2) × SU(2). There is no afﬁne root. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The Dynkin diagram

The Dynkin diagram is a graph whose vertices are the simple roots. Draw an edge connecting αi to αj if they are not orthogonal.

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

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

α0 α1 = (1, −1, 0, 0) α2 = (0, 1, −1, 0) α3 = (0, 0, 1, −1) = (−1 0 0 1) α1 α2 α3 α0 , , , Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

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 the roots:

bond angle example π no bond 2 SU(2) × SU(2)

2π single bond 3 SU(3)

3π double bond 4 Sp(4)

5π triple bond 6 G2 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The direction of the arrow

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

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

α0

α1 α2 α3 αn 2 αn 1 αn − −

α0 α1 α2 α3 αn 2 αn 1 αn − − Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

What we learn from Dynkin diagrams

The Dynkin diagram shows the relations between the simple roots.

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

Type Dn

α0 αn 1 −

α1 α2 α3 αn 3 αn 2 − −

αn

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

α0

α1 α3 α2

α4 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Triality

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

The D4 Dynkin diagram has an automorphism of degree 3.

α4

α2 α1

α3

This is an automorphism of the simply-connected group spin(8) or the adjoint form PGSO(8). Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Triality (continued)

The group spin(2k) has two irreducible representations of degree 2k−1 called the spin representations. It also has an irreducible representation of degree 2k, the standard representation. If k = 4, then 2k = 2k−1 = 8. Thus spin(8) has three irreducible representations of degree 8. These are permuted by triality.

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

The ﬁxed subgroup of this automorphism is the exceptional group G2, the automorphism of the octonions. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Exceptional groups

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

α0 α1 α2

There are two conventions for the ordering of the roots, due to Dynkin and Bourbaki. They differ in the exceptional groups. We are following Bourbaki. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The exceptional group F4

α0 α1 α2 α3 α3

The group F4 is the next exceptional group. It is the automorphism group of a 27-dimensional (nonassociative) Jordan algebra discovered by A. A. Albert that is also closely related to the exceptional groups E6, E7 and E8. The exceptional group G2 is a subgroup. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The exceptional groups E6, E7 and E8

α0

α2

α1 α3 α4 α5 α6

α2

α0 α1 α3 α4 α5 α6 α7

α2

α0 α1 α3 α4 α5 α6 α7 α8 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Levi subgroups

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

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

X±α, α ∈ S.

This is a Levi subgroup of the complex Lie group GC. (If we want we can intersect it with the compact Lie group G.) Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Levi decomposition of parabolics

Levi subgroups appear as Levi decompositions of parabolic subgroups. A subgroup P containing the Borel subgroup B (of

GC) whose Lie algebra is

tC ⊕ Xα + αM∈Φ is called a parabolic subgroup. It is a semidirect product of a normal unipotent group and a parabolic subgroup. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

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

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ P =   .  ∗ ∗    ∗ ∗     This has a decompositionP = MU with Unormal:

∗ ∗ 1 ∗ ∗ ∗ ∗ 1 ∗ ∗ M =   , U =   .  ∗ ∗  1       ∗ ∗  1         The subgroup M is a Levi subgroup. The group U is called the unipotent radical of P. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Levi subgroups from Dynkin diagrams

Starting from the Dynkin diagram of G, if we erase one or more nodes, we obtain the Dynkin diagram of a Levi subgroup. In the above 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 subgroup GL(2) × GL(2), of type A1 × A1:

α1 α3

All Levi subgroups can be determined easily from the Dynkin diagram. A Levi subgroup may or may not be a maximal subgroup. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Subgroups from the extended Dynkin diagram

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

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

α0

α1 α2 α3 α4

Erasing the root α4 gives the Dynkin diagram of type D4 and we have obtained the embedding SO(8) → SO(9); Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

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 = Xα α∈S M

is closed under the bracket, so it is a Lie subalgebra of gC. We are denoting tC = X0 even though 0 is not a root. Since [gα, gβ] ⊆ gα+β, the complex vector space gC,S is a complex Lie algebra. It is not contained in g, only gC. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Examples: G2

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

3α1 + 2α2

α2

α1

α0

First, we can take S = {±α1, ±(3α1 + 2α2), )}. Note that the roots α1 and 3α1 + 2α2 are orthogonal. The Lie algebra in this case is of Type A1 × A1. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Another G2 example

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

3α1 + 2α2

α2

α1

α0

This root system is of type A2. We see from these considerations that the (complex) G2 Lie algebra has Lie subalgebras of Types A1 × A1 and A2, so G2 should contain Lie subgroups isomorphic to SU(2) × SU(2) and SU(3). Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

The G2 examples from the extended Dynkin diagram

We can predict these subgroups of types SU(2) × SU(2) and SU(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. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

More orthogonal embeddings

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

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

αn α1 α2 α3 αn 2 αn 1 − −

αon+1

α1 α2 α3 αn 2 αn 1 αn − − Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Root folding

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

We saw at the Dynkin diagram of Dn+1 can be folded into the Dynkin diagram of Bn, explaining the embedding of SO(2n + 1) into SO(2n + 2). Here is another example. We may fold the Dynkin diagram of D4 into G2, showing that G2 is subgroup of spin(8). α1

α2 α 3 ) α4

α2 α1 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Maximal subgroups of Lie groups

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

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

As we will demonstrate, Sage knows all of the maximal subgroups of Lie groups up to rank 8, and is able to compute the branching rules efﬁciently. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Subgroups associated to representations

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

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

2 ε(π) = χπ(g ) dg. ZG If it is +1, the representation is orthogonal; if it is −1 the representation is symplectic. It is 0 if the representation is not self-contragredient. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

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

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

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

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

The Ad : SL(3) → SO(8) thus factors through the orthogonal group O(8). This SL(3) is indeed a maximal subgroup of SO(8). Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

A Sage session

In Sage, the WeylCharacterRing is a class for the irreducible representations of a Lie group. We can create the Weyl CharacterRing of type D4. Sage will tell you the maximal subgroups and give you the syntax of a branching rule that can use 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 rules since they don’t ﬁt on a line. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

Example of branching

The branching rule we are interested in is the A2 plethysm, so we implement that. As our guinea pig we take a moderately large 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. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage

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)