# Group Theory: Birdtracks, Lie's, and Exceptional Groups

Predrag Cvitanović
Pages: 280
https://www.jstor.org/stable/j.ctt7rnjq

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Acknowledgments
(pp. xi-xiv)
4. Chapter One Introduction
(pp. 1-4)

This monograph offers a derivation of all classical and exceptional semisimple Lie algebras through a classification of “primitive invariants.” Using somewhat unconventional notation inspired by the Feynman diagrams of quantum field theory, the invariant tensors are represented by diagrams; severe limits on what simple groups could possibly exist are deduced by requiring that irreducible representations be of integer dimension. The method provides the full Killing-Cartan list of all possible simple Lie algebras, but fails to prove the existence ofF₄,E₆,E₇ andE₈.

One simple quantum field theory question started this project; what is the group-theoretic factor for the...

5. Chapter Two A preview
(pp. 5-13)

The theory of Lie groups presented here had mutated greatly throughout its genesis. It arose from concrete calculations motivated by physical problems; but as it was written, the generalities were collected into introductory chapters, and the applications receded later and later into the text.

As a result, the first seven chapters are largely a compilation of definitions and general results that might appear unmotivated on first reading. The reader is advised to work through the examples, section 2.2 and section 2.3 in this chapter, jump to the topic of possible interest (such as the unitary groups, chapter 9, or the...

6. Chapter Three Invariants and reducibility
(pp. 14-26)

Basic group-theoretic notions are introduced: groups, invariants, tensors, the diagrammatic notation for invariant tensors.

The key results are the construction of projection operators from invariant matrices, the Clebsch-Gordan coefficients rep of projection operators (4.18), the invariance conditions (4.35) and the Lie algebra relations (4.47).

The basic idea is simple: a hermitian matrix can be diagonalized. If this matrix is an invariant matrix, it decomposes the reps of the group into direct sums of lower-dimensional reps. Most of computations to follow implement the spectral decomposition

${\text{M}} = {\lambda _1}{{\mathbf{P}}_1} + {\lambda _2}{{\mathbf{P}}_2} + \cdots + {\lambda _r}{{\mathbf{P}}_r},$

which associates with each distinct root λiof invariant matrix M a projection operator (3.48):...

7. Chapter Four Diagrammatic notation
(pp. 27-41)

Some aspects of the representation theory of Lie groups are the subject of this monograph. However, it is not written in the conventional tensor notation but instead in terms of an equivalent diagrammatic notation. We shall refer to this style of carrying out group-theoretic calculations asbirdtracks(and so do reputable journals [51]). The advantage of diagrammatic notation will become self-evident, I hope. Two of the principal benefits are that it eliminates “dummy indices,” and that it does not force group-theoretic expressions into the 1-dimensional tensor format (both being means whereby identical tensor expressions can be made to look totally...

8. Chapter Five Recouplings
(pp. 42-48)

Clebsches discussed in section 4.2 project a tensor in$V^p \otimes \bar V^q$onto a subspace λ. In practice one usually reduces a tensor step by step, decomposing a 2-particle state at each step. While there is some arbitrariness in the order in which these reductions are carried out, the final result is invariant and highly elegant: any group-theoretical invariant quantity can be expressed in terms of Wigner 3- and 6-jcoefficients.

We denote the clebsches for μ ⊗ ν → λ by

Here λ,μ,ν are rep labels, and the corresponding tensor indices are suppressed. Furthermore, if μ and ν are irreducible reps,...

9. Chapter Six Permutations
(pp. 49-59)

The simplest example of invariant tensors is the products of Kronecker deltas. On tensor spaces they represent index permutations. This is the way in which the symmetric groupSp, the group of permutations ofpobjects, enters into the theory of tensor reps. In this chapter, I introduce birdtracks notation for permutations, symmetrizations and antisymmetrizations and collect a few results that will be useful later on. These are the (anti)symmetrization expansion formulas (6.10) and (6.19), Levi-Civita tensor relations (6.28) and (6.30), the characteristic equations (6.50), and the invariance conditions (6.54) and (6.56). The theory of Young tableaux (or plethysms) is...

10. Chapter Seven Casimir operators
(pp. 60-75)

The construction of invariance groups, developed elsewhere in this monograph, is self-contained, and none of the material covered in this chapter is necessary for understanding the remainder of the monograph. We have argued in section 5.2 that all relevant group-theoretic numbers are given by vacuum bubbles (reduced matrix elements, 3n-jcoefficients,etc.), and we have described the algorithms for their evaluation. That is all that is really needed in applications.

However, one often wants to cross-check one’s calculation against the existing literature. In this chapter we discuss why and how one introduces casimirs (or Dynkin indices), we construct independent Casimir...

11. Chapter Eight Group integrals
(pp. 76-81)

In this chapter we discuss evaluation of group integrals of form

$\int {dg{G_a}^b{G_c}^d \ldots {G^e}_f{G^g}_h} ,$(8.1)

where${G_a}^b$is a [n×n] defining matrix rep ofg∈ 𝓖,Gis the matrix rep of the action ofgon the conjugate vector space, which we write as in (3.12),

${G^a}_b = {({G^\dag })_b}^a,$

and the integration is over the entire range ofg. As always, we assume that 𝓖 is a compact Lie group, and${G_a}^b$is unitary. Such integrals are of import for certain quantum field theory calculations, and the chapter should probably be skipped by a reader not interested in such applications. The...

12. Chapter Nine Unitary groups
(pp. 82-117)
P. Cvitanović, H. Elvang and A. D. Kennedy

U(n) is the group of all transformations that leave invariant the norm$\bar qq = \delta _b^a q^b q_a$of a complex vectorq. ForU(n) there are no other invariant tensors beyond those constructed of products of Kronecker deltas. They can be used to decompose the tensor reps ofU(n). For purely covariant or contravariant tensors, the symmetric group can be used to construct the Young projection operators. In sections. 9.1–9.2 we show how to do this for 2- and 3-index tensors by constructing the appropriate characteristic equations.

For tensors with more indices it is easier to construct the Young projection operators directly from...

13. Chapter Ten Orthogonal groups
(pp. 118-131)

Orthogonal groupSO(n) is the group of transformations that leaves invariant a symmetric quadratic form (q,q) =gμνqμqν:

If (q,q) is an invariant, so is its complex conjugate (q,q)*=gμνqμqν, and

is also an invariant tensor. The matrix$A_\mu ^\nu = g_{\mu \sigma } g^{\sigma \nu }$must be proportional to unity, as otherwise its characteristic equation would decompose the definingn-dimensional rep. A convenient normalization is

As the indices can be raised and lowered at will, nothing is gained by keeping the arrows. Our convention will be to perform all contractions with metric tensors with upper indices and omit the arrows and the open dots:

All...

14. Chapter Eleven Spinors
(pp. 132-147)
P. Cvitanović and A. D. Kennedy

In chapter 10 we have discussed the tensor reps of orthogonal groups. However, the spinor reps ofSO(n) also play a fundamental role in physics, both as reps of space-time symmetries (Pauli spin matrices, Dirac gamma matrices, fermions inD-dimensional supergravities), and as reps of internal symmetries (SO(10) grand unified theory, for example). In calculations of radiative corrections, the QED spin traces can easily run up to traces of products of some twelve gamma matrices [195], and efficient evaluation algorithms are of great practical importance. A most straightforward algorithm would evaluate such a trace in some$11!! = 11 \cdot 9 \cdot 7 \cdot 5 \cdot 3 \simeq 10,000$steps. Even computers...

15. Chapter Twelve Symplectic groups
(pp. 148-150)

Symplectic groupSp(n) is the group of all transformations that leave invariant a skew symmetric (p,q) =fabpaqb:

The birdtrack notation is motivated by the need to distinguish the first and the second index: it is a special case of the birdtracks for antisymmetric tensors of even rank (6.57). If (p,q) is an invariant, so is its complex conjugate (p,q)*=fbapaqb, and

is also an invariant tensor. The matrix$A_a^b = f_{ac} f^{cb}$must be proportional to unity, as otherwise its characteristic equation would decompose the definingn-dimensional rep. A convenient normalization is

Indices can be raised and lowered at will, so the arrows...

16. Chapter Thirteen Negative dimensions
(pp. 151-154)
P. Cvitanović and A. D. Kennedy

A cursory examination of the expressions for the dimensions and the Dynkin indices listed in tables 7.3 and 7.5, and in the tables of chapter 9, chapter 10, and chapter 12, reveals intriguing symmetries under substitutionn→ –n. This kind of symmetry is best illustrated by the reps ofSU(n); if λ stands for a Young tableau withpboxes, and$\bar \lambda$for the transposed tableau obtained by flipping λ across the diagonal (i.e., exchanging symmetrizations and antisymmetrizations), then the dimensions of the correspondingSU(n) reps are related by

$SU(n):\quad d_\lambda (n) = ( - 1)^p d_{\bar \lambda } ( - n)$. (13.1)

This is evident from the standard recipe...

17. Chapter Fourteen Spinors’ symplectic sisters
(pp. 155-161)
P. Cvitanović and A. D. Kennedy

Dirac discovered spinors in his search for a vectorial quantity that could be interpreted as a “square root” of the Minkowski 4-momentum squared,

$]p_1 \gamma _1 + p_2 \gamma _2 + p_3 \gamma _3 + p_4 \gamma _4 )^2 = - p_1^2 - p_2^2 - p_3^2 = p_4^2.$

What happens if one extends a Minkowski 4-momentum (p₁,p₂,p₃,p₄) into fermionic, Grassmann dimensions$(p_{ - n} ,p_{ - n + 1} , \ldots ,p_{ - 2} ,p_{ - 1} ,p_2 , \ldots ,p_{n - 1} ,p_n )$?

The Grassmann sectorpμanticommute and the gamma-matrix relatives in the Grassmann dimensions have to satisfy the Heisenberg algebra commutation relation,

$[\gamma _\mu ,\gamma _\nu ] = f_{\mu \nu } 1,$,

instead of the Clifford algebra anticommutator condition (11.2), with the bilinear invariantfμν= –fνμskew-symmetric in the Grassmann dimensions.

In chapter 12, we showed that the symplectic groupSp(n) is the invariance group of a skew-symmetric bilinear symplectic...

18. Chapter Fifteen SU(n) family of invariance groups
(pp. 162-169)

SU(n) preserves the Levi-Civita tensor, in addition to the Kronecker δ of section 9.10. This additional invariant induces nontrivial decompositions ofU(n) reps. In this chapter, we show how the theory ofSU(2) reps (the quantum mechanics textbooks’ theory of angular momentum) is developed by birdtracking; thatSU(3) is the unique group with the Kronecker delta and a rank-3 antisymmetric primitive invariant; thatSU(4) is isomorphic toSO(6); and that forn≥ 4, onlySU(n) has the Kronecker δ and rank-nantisymmetric tensor primitive invariants.

ForSU(2), we can construct an additional invariant matrix that would appear to induce...

19. Chapter Sixteen G₂ family of invariance groups
(pp. 170-179)

In this chapter, we begin the construction of all invariance groups that possess a symmetric quadratic and an antisymmetric cubic invariant in the defining rep. The resulting classification is summarized in figure 16.1. We find that the cubic invariant must satisfy either the Jacobi relation (16.7) or the alternativity relation (16.11). In the former case, the invariance group can be any semisimple Lie group in its adjoint rep; we pursue this possibility in the next chapter. The latter case is developed in this chapter; we find that the invariance group is eitherSO(3) or the exceptional Lie groupG₂. The...

20. Chapter Seventeen E₈ family of invariance groups
(pp. 180-189)

In this chapter we continue the construction of invariance groups characterized by a symmetric quadratic and an antisymmetric cubic primitive invariant. In the preceding chapter we proved that the cubic invariant must either satisfy the alternativity relation (16.11), or the Jacobi relation (4.48), and showed that the first case hasSO(3) andG₂ as the only interesting solutions.

Here we pursue the second possibility and determine all invariance groups that preserve a symmetric quadratic (4.28) and an antisymmetric cubic primitive invariant (4.46),

with the cubic invariant satisfying the Jacobi relation (4.48)

Enumerate all Lie algebras defined...

21. Chapter Eighteen E₆ family of invariance groups
(pp. 190-209)

In this chapter, we determine all invariance groups whose primitive invariant tensors are$\delta _b^a$and fully symmetricdabc,dabc. The reduction of$V \otimes V$space yields a rule for evaluation of the loop contraction of fourd-invariants (18.9). The reduction of$V \otimes \bar V$yields the first Diophantine condition (18.13) on the allowed dimensions of the defining rep. The reduction ofVVVtensors is straightforward, but the reduction ofAVspace yields the second Diophantine condition (d₄ in table 18.4) and limits the defining rep dimension ton≤ 27. The solutions of the two Diophantine conditions form...

22. Chapter Nineteen F₄ family of invariance groups
(pp. 210-217)

In this chapter we classify and construct all invariance groups whose primitive invariant tensors are a symmetric bilineardab, and a symmetric trilineardabc, satisfying the relation (19.16).

Take as primitives a symmetric quadratic invariantdaband a symmetric cubic invariantdabc. As explained in chapter 12, we can usedabto lower all indices. In the birdtrack notation, we drop the open circles denoting symmetric 2-index invariant tensordab, and we drop arrows on all lines:

The definingn-dimensional rep is by assumption irreducible, so

Were (19.3) nonvanishing, we could use to project out a 1-dimensional subspace, violating the...

23. Chapter Twenty E₇ family and its negative-dimensional cousins
(pp. 218-228)

Parisi and Sourlas [269] have suggested that a Grassmann vector space of dimensionncan be interpreted as an ordinary vector space of dimension –n. As we have seen in chapter 13, semisimple Lie groups abound with examples in which ann→ –nsubstitution can be interpreted in this way. An early example was Penrose’s binors [280], reps ofSU(2) =Sp(2) constructed asSO(–2), and discussed here in chapter 14. This is a special case of a general relation betweenSO(n) andSp(–n) established in chapter 13; if symmetrizations and antisymmetrizations are interchanged, reps ofSO(n)...

24. Chapter Twenty-One Exceptional magic
(pp. 229-236)

The study of invariance algebras as pursued in chapters 16–20 might appear a rather haphazard affair. Given a set of primitives, one derives a set of Diophantine equations, constructs the family of invariance algebras, and moves onto the next set of primitives. However, a closer scrutiny of the Diophantine conditions leads to a surprise: most of these equations are special cases of one and the same Diophantine equation, and they magically arrange all exceptional families into a triangular array I call the Magic Triangle.

Our construction of invariance algebras has generated a series of Diophantine conditions that we now...

25. Appendix A. Recursive decomposition
(pp. 237-238)
26. Appendix B. Properties of Young projections
(pp. 239-242)
H. Elvang and P. Cvitanović
27. Bibliography
(pp. 243-258)
28. Index
(pp. 259-263)