version: 26 january 1984
GROUP THEORY

Part I

Predrag Cvitanovic, NORDITA, Copenhagen

CONTENTS
(sections marked with * are not included in part I)


INTRODUCTION
A. Prefatory appologies
B. Alternative titles
C. Acknowledgements

A PREVIEW
A. Basic concepts
B. An example

INVARIANTS AND REDUCIBILITY
A. Defining representation
B. Infinitesimal transformations
C. Invariants
D. Projection operators
E. Clebsch-Gordan coefficients 
F. Decomposition of representations
G. Further invariants

TENSOR REPRESENTATIONS
A. Tensors
B. Index population control
C. Birdrtracks
D. Several representations
E. Infinitesimal transformations
F. Invariant tensors
G. Clebsches in birdtracks
H. Lie algebra in birdtracks
I. Couplings and recouplings
J. Wigner 3n-j coefficients
K. Wigner-Eckhart theorem
L. Irrelevancy of clebsches

PERMUTATIONS
A. Permutations in birdtracks
B. Symmetrization
C. Antisymmetrization
D. Levi-Civita tensor
E. Determinants
F. Characteristic equations
G. Fully (anti)symmetric tensors
H. Young tableaux, Dynkin labels

UNITARY GROUPS
A. Two-quark states
B. Quark-antiquark states
C. Quark-gluon states
D. Two-gluon states
E. Three-quark states
F. Four-quark states
G. Dynkin labels
H. Kronecker products of Young tableaux


ORTHOGONAL GROUPS
A. Two-quark states
B. Quark-gluon states
C. Two-gluon states
D. Three-quark states
E. Gravity tensors
F. Dynkin labels


SPINORS
A. Spinograpy
B. Fierzing around
C. Fierz coefficients
D. 6j coefficients
E. Exemplary evaluations
F. Invariance of $\gamma$-matrices 
G. Handedness
H. Kahane algorithm

*SYMPLECTIC GROUPS
A. Two-quark states
B. Quark-gluon states
C. Dynkin labels

NEGATIVE DIMENSIONS
A. SU(n) = SU(-n) <remeber bar} 
B. SO(n) = Sp(-n) <remeber bar}


*SPINSTERS
This chapter is based on P. Cvitanovic and A.D.
Kennedy, Phys. Scripta 26, 5 (1982).


CASIMIR OPERATORS
A. Casimirs and Lie algebra
B. Independent casimirs
C. Casimir operators
D. Quadratic, cubic Casimir operators
Dynkin indices
E. Quartic casimirs
F. Evaluation of casimirs


*QCD COLOR WEIGHTS
A. Feynman rules
B. Couplings
C. Vanishing weights
D. Color bases
E. Vacuum weights
F. Gauge sets (based on P. Cvitanovic, P.G. Lauwers and P.N.
Scharbach, Nucl. Phys. B186, 165 (1981)).

*COUNTING COLORS
A. Coloring with red, blue, and green
B. Double-line coloring
C. SU(n) coloring
D. 1/n expansions


LATTICE GAUGE THEORIES
A. Lattice action
B. Averaging color rotations
C. Link integrals
D. A few boring exercises
E. Plaquette integrals
F. Drouffe's bubbles


*ZERO-DIMENSIONAL QCD
	by D.J. Pritchard
A. World on one link


*INVARIANCE GROUPS

SU(n) FAMILY OF INVARIANCE GROUPS
A. Representations of SU(2)
B. Why do quarks come in three colors
C. Levi-Civita tensors and SU(n)
D. SU(4) - SO(6) isomorphism


QUARKS IN SEVEN COLORS
A. Two-quark states
B. Alternativity and reduction of G  tensors
C. Two-quark states reduction, completed
D. Primitivity implies alternativity
E. Hurwitz's theorem  
F. Representations of G2
*G. Quartic casimirs

*SYMMETRIC THREE-QUARK BARYONS
A. E6 family
B. E6 family evaluation rules
C. Higher representations
D. Quartic casimirs
E. Springer's construction of E6
F. Representations of E6 

*SYMMETRIC THREE-QUARK BARYONS
A. F4 family
B. Two-quark states
C. Quark-gluon states
D. Two-gluon states
E. Quartic casimirs
F. F4 family evaluation rules
G. Jordan algebra and F4(26)
H. Representations of F4

*SKEWSYMMETRIC FOUR-QUARK BARYONS
(based on P. Cvitanovic, Nucl. Phys. B188, 373 (1981))
A. E7 family
B. Self-duality for SO(4)
C. Quartic casimirs
D. Brown's construction of E7(56)
E. Representations of E7


*AN EXCEPTIONAL FAMILY
A. Two-gluon states for E8 family
B. Diophantine conditions
C. Representations of E8


*EXCEPTIONAL MAGIC
A. Magic triangle
B. Sub-algebra chains
C. D4 triality


*MAGIC NEGATIVE DIMENSIONS
A. E7 and SO(4)
B. E6 and SU(3)