finite-dimensional lie algebras and their representations ... · finite-dimensional lie algebras...
TRANSCRIPT
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Finite-Dimensional Lie Algebras and Their Representations for
Unified Model Building
Naoki Yamatsu ∗
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
November 30, 2015
Abstract
We give information about finite-dimensional Lie algebras and their representations formodel building in 4 and 5 dimensions; e.g., conjugacy classes, types of representations, Weyldimensional formulas, Dynkin indices, quadratic Casimir invariants, anomaly coefficients,projection matrices, and branching rules of Lie algebras and their subalgebras up to rank-15and D16. We show what kind of Lie algebras can be applied for grand unified theories in 4and 5 dimensions.
Contents
1 Introduction 16
2 Lie algebras and their subalgebras 17
2.1 (Extended) Dynkin diagramsand Cartan matrices . . . . . . . 21
2.2 Subalgebras . . . . . . . . . . . . 25
3 Representations of algebras 34
3.1 Conjugacy class . . . . . . . . . . 34
3.2 Complex, self-conjugate, real,and pseudo-real . . . . . . . . . . 36
3.3 Weyl dimension formula . . . . . 41
3.4 Dynkin index and Casimir in-variant . . . . . . . . . . . . . . . 43
3.5 Anomaly coefficient . . . . . . . . 47
3.6 Higher order Dynkin indices andCasimir invariants . . . . . . . . 49
4 Representations of subalgebras 49
4.1 Branching rules and projectionmatrices . . . . . . . . . . . . . . 50
4.2 Dynkin diagrams . . . . . . . . . 51
4.3 Recipe for calculating branchingrules . . . . . . . . . . . . . . . . 54
5 Tensor product 59
5.1 Dynkin’s theorem for secondhighest representation . . . . . . 59
5.2 Conjugacy class . . . . . . . . . . 59
5.3 Dynkin’s method of parts . . . . 60
5.4 Recipe for calculating tensorproduct . . . . . . . . . . . . . . 62
6 Summary for representations of
Lie algebras and their subalgebras 62
6.1 An = sun+1 . . . . . . . . . . . . 65
6.2 Bn = so2n+1 . . . . . . . . . . . . 70
6.3 Cn = usp2n . . . . . . . . . . . . 76
6.4 Dn = so2n . . . . . . . . . . . . . 81
6.5 E6 . . . . . . . . . . . . . . . . . 89
6.6 E7 . . . . . . . . . . . . . . . . . 91
6.7 E8 . . . . . . . . . . . . . . . . . 93
6.8 F4 . . . . . . . . . . . . . . . . . 95
6.9 G2 . . . . . . . . . . . . . . . . . 96
7 Application for model building 98
7.1 Projection matrices of GUTgauge groups . . . . . . . . . . . 99
7.1.1 Rank 4 . . . . . . . . . . 99
7.1.2 Rank 5 . . . . . . . . . . 100
7.2 Branching rules of GUT gaugegroups . . . . . . . . . . . . . . . 101
7.2.1 Rank 4 . . . . . . . . . . 101
7.2.2 Rank 5 . . . . . . . . . . 104
8 Summary and discussion 107
∗Electronic address: [email protected]
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http://arxiv.org/abs/1511.08771v1
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A Representations 113
A.1 An = sun+1 . . . . . . . . . . . . 113
A.2 Bn = so2n+1 . . . . . . . . . . . . 353
A.3 Cn = usp2n . . . . . . . . . . . . 457
A.4 Dn = so2n . . . . . . . . . . . . . 569
A.5 En . . . . . . . . . . . . . . . . . 681
A.6 F4 . . . . . . . . . . . . . . . . . 705
A.7 G2 . . . . . . . . . . . . . . . . . 713
B Positive roots 722
C Weight diagrams 729
D Projection matrices 756
D.1 An . . . . . . . . . . . . . . . . . 756
D.2 Bn . . . . . . . . . . . . . . . . . 776
D.3 Cn . . . . . . . . . . . . . . . . . 805
D.4 Dn . . . . . . . . . . . . . . . . . 823
D.5 En . . . . . . . . . . . . . . . . . 850
D.6 F4 . . . . . . . . . . . . . . . . . 853
D.7 G2 . . . . . . . . . . . . . . . . . 853
E Branching rules 854
E.1 An = sun+1 . . . . . . . . . . . . 854
E.1.1 Rank 2 . . . . . . . . . . 854
E.1.2 Rank 3 . . . . . . . . . . 857
E.1.3 Rank 4 . . . . . . . . . . 862
E.1.4 Rank 5 . . . . . . . . . . 867
E.1.5 Rank 6 . . . . . . . . . . 876
E.1.6 Rank 7 . . . . . . . . . . 882
E.1.7 Rank 8 . . . . . . . . . . 893
E.1.8 Rank 9 . . . . . . . . . . 902
E.1.9 Rank 10 . . . . . . . . . . 920
E.1.10 Rank 11 . . . . . . . . . . 930
E.1.11 Rank 12 . . . . . . . . . . 947
E.1.12 Rank 13 . . . . . . . . . . 958
E.1.13 Rank 14 . . . . . . . . . . 974
E.1.14 Rank 15 . . . . . . . . . . 996
E.2 Bn = so2n+1 . . . . . . . . . . . . 1018
E.2.1 Rank 3 . . . . . . . . . . 1018
E.2.2 Rank 4 . . . . . . . . . . 1026
E.2.3 Rank 5 . . . . . . . . . . 1043
E.2.4 Rank 6 . . . . . . . . . . 1058
E.2.5 Rank 7 . . . . . . . . . . 1071
E.2.6 Rank 8 . . . . . . . . . . 1085
E.2.7 Rank 9 . . . . . . . . . . 1092
E.2.8 Rank 10 . . . . . . . . . . 1099
E.2.9 Rank 11 . . . . . . . . . . 1107
E.2.10 Rank 12 . . . . . . . . . . 1114E.2.11 Rank 13 . . . . . . . . . . 1121E.2.12 Rank 14 . . . . . . . . . . 1128E.2.13 Rank 15 . . . . . . . . . . 1134
E.3 Cn = usp2n . . . . . . . . . . . . 1142E.3.1 Rank 2 . . . . . . . . . . 1142E.3.2 Rank 3 . . . . . . . . . . 1149E.3.3 Rank 4 . . . . . . . . . . 1159E.3.4 Rank 5 . . . . . . . . . . 1175E.3.5 Rank 6 . . . . . . . . . . 1190E.3.6 Rank 7 . . . . . . . . . . 1206E.3.7 Rank 8 . . . . . . . . . . 1213E.3.8 Rank 9 . . . . . . . . . . 1220E.3.9 Rank 10 . . . . . . . . . . 1227E.3.10 Rank 11 . . . . . . . . . . 1233E.3.11 Rank 12 . . . . . . . . . . 1237E.3.12 Rank 13 . . . . . . . . . . 1244E.3.13 Rank 14 . . . . . . . . . . 1247E.3.14 Rank 15 . . . . . . . . . . 1251
E.4 Dn = so2n . . . . . . . . . . . . . 1256E.4.1 Rank 4 . . . . . . . . . . 1256E.4.2 Rank 5 . . . . . . . . . . 1269E.4.3 Rank 6 . . . . . . . . . . 1282E.4.4 Rank 7 . . . . . . . . . . 1303E.4.5 Rank 8 . . . . . . . . . . 1325E.4.6 Rank 9 . . . . . . . . . . 1340E.4.7 Rank 10 . . . . . . . . . . 1352E.4.8 Rank 11 . . . . . . . . . . 1362E.4.9 Rank 12 . . . . . . . . . . 1369E.4.10 Rank 13 . . . . . . . . . . 1379E.4.11 Rank 14 . . . . . . . . . . 1386E.4.12 Rank 15 . . . . . . . . . . 1392E.4.13 Rank 16 . . . . . . . . . . 1399
E.5 En . . . . . . . . . . . . . . . . . 1407E.5.1 Rank 6 . . . . . . . . . . 1407E.5.2 Rank 7 . . . . . . . . . . 1423E.5.3 Rank 8 . . . . . . . . . . 1434
E.6 F4 . . . . . . . . . . . . . . . . . 1442E.7 G2 . . . . . . . . . . . . . . . . . 1449
F Tensor products 1461
F.1 An = sun+1 . . . . . . . . . . . . 1461F.2 Bn = so2n+1 . . . . . . . . . . . . 1680F.3 Cn = usp2n . . . . . . . . . . . . 1708F.4 Dn = so2n . . . . . . . . . . . . . 1743F.5 En . . . . . . . . . . . . . . . . . 1831F.6 F4 . . . . . . . . . . . . . . . . . 1843F.7 G2 . . . . . . . . . . . . . . . . . 1847
List of Tables
1 Extended Dynkin diagrams . . . 212 Cartan matrices . . . . . . . . . . 22
3 Inverse Cartan matrices . . . . . 24
4 Maximal S-subalgebras of clas-sical algebras (1) . . . . . . . . . 27
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5 Maximal S-subalgebras of clas-sical algebras (2) . . . . . . . . . 27
6 Maximal S-subalgebras of ex-ceptional algebras . . . . . . . . . 28
7 Maximal subalgebras . . . . . . . 28
8 Conjugacy classes . . . . . . . . . 34
9 Complex representations (1) . . . 37
10 Complex representations (2) . . . 37
11 Self-conjugate representations . . 37
12 Examples of complex represen-tations . . . . . . . . . . . . . . . 40
13 Examples of self-conjugate rep-resentations . . . . . . . . . . . . 40
14 Higher order Casimir invariants . 49
15 Lowest dimensional representa-tions . . . . . . . . . . . . . . . . 62
16 Adjoint representations . . . . . 63
17 Dimension of fundamental rep-resentations . . . . . . . . . . . . 63
18 Dynkin indices of fundamentalrepresentations . . . . . . . . . . 64
19 Types of representations of An . 65
20 Representations of An−1 = sun . 66
21 Maximal subalgebras of An . . . 67
22 Types of representations of Bn . 70
23 Representations of Bn = so2n+1 . 71
24 Maximal subalgebras of Bn . . . 72
25 Types of representations of Cn . 76
26 Representations of Cn = usp2n . 77
27 Maximal subalgebras of Cn . . . 77
28 Types of representations of Dn . 81
29 Representations of Dn = so2n . . 82
30 Maximal subalgebras of Dn . . . 83
31 Types of representations of E6 . 90
32 Representations of E6 . . . . . . 91
33 Maximal subalgebras of E6 . . . 91
34 Types of representations of E7 . 92
35 Representations of E7 . . . . . . 92
36 Maximal subalgebras of E7 . . . 93
37 Types of representations of E8 . 93
38 Representations of E8 . . . . . . 94
39 Maximal subalgebras of E8 . . . 94
40 Types of representations of F4 . . 95
41 Representations of F4 . . . . . . 95
42 Maximal subalgebras of F4 . . . 96
43 Types of representations of G2 . 96
44 Representations of G2 . . . . . . 97
45 Maximal subalgebras of G2 . . . 97
46 Candidates for 4D GUT gaugegroup . . . . . . . . . . . . . . . 98
47 Candidates for GUT gaugegroup in general . . . . . . . . . 98
48 Candidates for gauge-HiggsGUT gauge group . . . . . . . . 99
49 Branching rules of SU(5) ⊃SU(3)× SU(2)× U(1) . . . . . 101
50 Branching rules of SO(9) ⊃SU(3)× SU(2)× U(1) . . . . . 102
51 Branching rules of USp(8) ⊃SU(3)× SU(2)× U(1) . . . . . 102
52 Branching rules of F4 ⊃ SU(3)×SU(2)× U(1)(1) . . . . . . . . . 103
53 Branching rules of F4 ⊃ SU(3)×SU(2)× U(1)(2) . . . . . . . . . 103
54 Branching rules of SU(6) ⊃SU(3)× SU(2)× U(1) × U(1) . 104
55 Branching rules of SO(11) ⊃SU(3)× SU(2)× U(1) × U(1) . 105
56 Branching rules of USp(10) ⊃SU(3)× SU(2)× U(1) × U(1)(1) 105
57 Branching rules of USp(10) ⊃SU(3)× SU(2)× U(1) × U(1)(2) 106
58 Branching rules of SO(10) ⊃SU(3)× SU(2)× U(1) × U(1) . 107
59 Representations of A1 . . . . . . 113
60 Representations of A2 . . . . . . 129
61 Representations of A3 . . . . . . 145
62 Representations of A4 . . . . . . 161
63 Representations of A5 . . . . . . 177
64 Representations of A6 . . . . . . 193
65 Representations of A7 . . . . . . 209
66 Representations of A8 . . . . . . 225
67 Representations of A9 . . . . . . 241
68 Representations of A10 . . . . . . 257
69 Representations of A11 . . . . . . 273
70 Representations of A12 . . . . . . 289
71 Representations of A13 . . . . . . 305
72 Representations of A14 . . . . . . 321
73 Representations of A15 . . . . . . 337
74 Representations of B3 . . . . . . 353
75 Representations of B4 . . . . . . 361
76 Representations of B5 . . . . . . 369
77 Representations of B6 . . . . . . 377
78 Representations of B7 . . . . . . 385
79 Representations of B8 . . . . . . 393
80 Representations of B9 . . . . . . 401
81 Representations of B10 . . . . . . 409
82 Representations of B11 . . . . . . 417
83 Representations of B12 . . . . . . 425
84 Representations of B13 . . . . . . 433
85 Representations of B14 . . . . . . 441
86 Representations of B15 . . . . . . 449
87 Representations of C2 . . . . . . 457
88 Representations of C3 . . . . . . 465
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89 Representations of C4 . . . . . . 473
90 Representations of C5 . . . . . . 481
91 Representations of C6 . . . . . . 489
92 Representations of C7 . . . . . . 497
93 Representations of C8 . . . . . . 505
94 Representations of C9 . . . . . . 513
95 Representations of C10 . . . . . . 521
96 Representations of C11 . . . . . . 529
97 Representations of C12 . . . . . . 537
98 Representations of C13 . . . . . . 545
99 Representations of C14 . . . . . . 553
100 Representations of C15 . . . . . . 561
101 Representations of D4 . . . . . . 569
102 Representations of D5 . . . . . . 585
103 Representations of D6 . . . . . . 593
104 Representations of D7 . . . . . . 601
105 Representations of D8 . . . . . . 609
106 Representations of D9 . . . . . . 617
107 Representations of D10 . . . . . . 625
108 Representations of D11 . . . . . . 633
109 Representations of D12 . . . . . . 641
110 Representations of D13 . . . . . . 649
111 Representations of D14 . . . . . . 657
112 Representations of D15 . . . . . . 665
113 Representations of D16 . . . . . . 673
114 Representations of E6 . . . . . . 681
115 Representations of E7 . . . . . . 689
116 Representations of E8 . . . . . . 697
117 Representations of F4 . . . . . . 705
118 Representations of G2 . . . . . . 713
119 Positive roots of An . . . . . . . 722
120 Positive roots of Bn . . . . . . . 722
121 Positive roots of Cn . . . . . . . 723
122 Positive roots of Dn . . . . . . . 724
123 Positive roots of En . . . . . . . 725
124 Positive roots of F4 . . . . . . . . 727
125 Positive roots of G2 . . . . . . . 728
126 Weight diagrams of A1 . . . . . . 729
127 Weight diagrams of A2 . . . . . . 729
128 Weight diagrams of A3 . . . . . . 731
129 Weight diagrams of A4 . . . . . . 732
130 Weight diagrams of A5 . . . . . . 733
131 Weight diagrams of A6 . . . . . . 734
132 Weight diagrams of A7 . . . . . . 734
133 Weight diagrams of A8 . . . . . . 734
134 Weight diagrams of A9 . . . . . . 735
135 Weight diagrams of A10 . . . . . 736
136 Weight diagrams of B2 . . . . . . 736
137 Weight diagrams of B3 . . . . . . 737
138 Weight diagrams of B4 . . . . . . 737
139 Weight diagrams of B5 . . . . . . 738
140 Weight diagrams of B6 . . . . . . 739
141 Weight diagrams of B7 . . . . . . 739
142 Weight diagrams of B8 . . . . . . 740
143 Weight diagrams of B9 . . . . . . 740
144 Weight diagrams of B10 . . . . . 741
145 Weight diagrams of C2 . . . . . . 741
146 Weight diagrams of C3 . . . . . . 742
147 Weight diagrams of C4 . . . . . . 743
148 Weight diagrams of C5 . . . . . . 743
149 Weight diagrams of C6 . . . . . . 744
150 Weight diagrams of C7 . . . . . . 744
151 Weight diagrams of C8 . . . . . . 744
152 Weight diagrams of C9 . . . . . . 745
153 Weight diagrams of C10 . . . . . 745
154 Weight diagrams of D3 . . . . . . 746
155 Weight diagrams of D4 . . . . . . 746
156 Weight diagrams of D5 . . . . . . 747
157 Weight diagrams of D6 . . . . . . 748
158 Weight diagrams of D7 . . . . . . 748
159 Weight diagrams of D8 . . . . . . 748
160 Weight diagrams of D9 . . . . . . 749
161 Weight diagrams of D10 . . . . . 749
162 Weight diagrams of E6 . . . . . . 750
163 Weight diagrams of E7 . . . . . . 751
164 Weight diagrams of E8 . . . . . . 752
165 Weight diagrams of F4 . . . . . . 754
166 Weight diagrams of G2 . . . . . . 754
167 Projection matrices of A1 . . . . 756
168 Projection matrices of A2 . . . . 756
169 Projection matrices of A3 . . . . 756
170 Projection matrices of A4 . . . . 756
171 Projection matrices of A5 . . . . 757
172 Projection matrices of A6 . . . . 757
173 Projection matrices of A7 . . . . 758
174 Projection matrices of A8 . . . . 759
175 Projection matrices of A9 . . . . 760
176 Projection matrices of A10 . . . . 762
177 Projection matrices of A11 . . . . 763
178 Projection matrices of A12 . . . . 765
179 Projection matrices of A13 . . . . 767
180 Projection matrices of A14 . . . . 769
181 Projection matrices of A15 . . . . 772
182 Projection matrices of B3 . . . . 776
183 Projection matrices of B4 . . . . 776
184 Projection matrices of B5 . . . . 777
185 Projection matrices of B6 . . . . 777
186 Projection matrices of B7 . . . . 778
187 Projection matrices of B8 . . . . 780
188 Projection matrices of B9 . . . . 781
189 Projection matrices of B10 . . . . 783
190 Projection matrices of B11 . . . . 785
191 Projection matrices of B12 . . . . 788
192 Projection matrices of B13 . . . . 792
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193 Projection matrices of B14 . . . . 795
194 Projection matrices of B15 . . . . 799
195 Projection matrices of C2 . . . . 805
196 Projection matrices of C3 . . . . 805
197 Projection matrices of C4 . . . . 805
198 Projection matrices of C5 . . . . 806
199 Projection matrices of C6 . . . . 806
200 Projection matrices of C7 . . . . 807
201 Projection matrices of C8 . . . . 808
202 Projection matrices of C9 . . . . 809
203 Projection matrices of C10 . . . . 810
204 Projection matrices of C11 . . . . 812
205 Projection matrices of C12 . . . . 813
206 Projection matrices of C13 . . . . 816
207 Projection matrices of C14 . . . . 818
208 Projection matrices of C15 . . . . 820
209 Projection matrices of D4 . . . . 823
210 Projection matrices of D5 . . . . 823
211 Projection matrices of D6 . . . . 824
212 Projection matrices of D7 . . . . 825
213 Projection matrices of D8 . . . . 826
214 Projection matrices of D9 . . . . 828
215 Projection matrices of D10 . . . . 830
216 Projection matrices of D11 . . . . 833
217 Projection matrices of D12 . . . . 836
218 Projection matrices of D13 . . . . 839
219 Projection matrices of D14 . . . . 841
220 Projection matrices of D15 . . . . 844
221 Projection matrices of D16 . . . . 846
222 Projection matrices of E6 . . . . 850
223 Projection matrices of E7 . . . . 850
224 Projection matrices of E8 . . . . 852
225 Projection matrices of F4 . . . . 853
226 Projection matrices of G2 . . . . 853
227 Branching rules of su3 ⊃ su2 ⊕u1(R) . . . . . . . . . . . . . . . 854
228 Branching rules of su3 ⊃ su2(S) 855
229 Branching rules of su4 ⊃ su3 ⊕u1(R) . . . . . . . . . . . . . . . 857
230 Branching rules of su4 ⊃ su2 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 858
231 Branching rules of su4 ⊃ usp4(S) 860
232 Branching rules of su4 ⊃ su2 ⊕su2(S) . . . . . . . . . . . . . . . 861
233 Branching rules of su5 ⊃ su4 ⊕u1(R) . . . . . . . . . . . . . . . 862
234 Branching rules of su5 ⊃ su3 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 864
235 Branching rules of su5 ⊃ usp4(S) 866
236 Branching rules of su6 ⊃ su5 ⊕u1(R) . . . . . . . . . . . . . . . 867
237 Branching rules of su6 ⊃ su4 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 868
238 Branching rules of su6 ⊃ su3 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 870
239 Branching rules of su6 ⊃ su3(S) 872
240 Branching rules of su6 ⊃ su4(S) 873
241 Branching rules of su6 ⊃ usp6(S) 874
242 Branching rules of su6 ⊃ su3 ⊕su2(S) . . . . . . . . . . . . . . . 875
243 Branching rules of su7 ⊃ su6 ⊕u1(R) . . . . . . . . . . . . . . . 876
244 Branching rules of su7 ⊃ su5 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 877
245 Branching rules of su7 ⊃ su4 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 879
246 Branching rules of su7 ⊃ so7(S) 881
247 Branching rules of su8 ⊃ su7 ⊕u1(R) . . . . . . . . . . . . . . . 882
248 Branching rules of su8 ⊃ su6 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 883
249 Branching rules of su8 ⊃ su5 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 885
250 Branching rules of su8 ⊃ su4 ⊕su4 ⊕ u1(R) . . . . . . . . . . . . 887
251 Branching rules of su8 ⊃ so8(S) 889
252 Branching rules of su8 ⊃ usp8(S) 890
253 Branching rules of su8 ⊃ su4 ⊕su2(S) . . . . . . . . . . . . . . . 892
254 Branching rules of su9 ⊃ su8 ⊕u1(R) . . . . . . . . . . . . . . . 893
255 Branching rules of su9 ⊃ su7 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 894
256 Branching rules of su9 ⊃ su6 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 896
257 Branching rules of su9 ⊃ su5 ⊕su4 ⊕ u1(R) . . . . . . . . . . . . 898
258 Branching rules of su9 ⊃ so9(S) 900
259 Branching rules of su9 ⊃ su3 ⊕su3(S) . . . . . . . . . . . . . . . 901
260 Branching rules of su10 ⊃ su9 ⊕u1(R) . . . . . . . . . . . . . . . 902
261 Branching rules of su10 ⊃ su8 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 904
262 Branching rules of su10 ⊃ su7 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 905
263 Branching rules of su10 ⊃ su6 ⊕su4 ⊕ u1(R) . . . . . . . . . . . . 907
264 Branching rules of su10 ⊃ su5 ⊕su5 ⊕ u1(R) . . . . . . . . . . . . 909
265 Branching rules of su10 ⊃ su3(S) 911
266 Branching rules of su10 ⊃ su4(S) 913
267 Branching rules of su10 ⊃ su5(S) 914
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268 Branching rules of su10 ⊃usp4(S) . . . . . . . . . . . . . . 915
269 Branching rules of su10 ⊃so10(S) . . . . . . . . . . . . . . 916
270 Branching rules of su10 ⊃usp10(S) . . . . . . . . . . . . . . 918
271 Branching rules of su10 ⊃ su5 ⊕su2(S) . . . . . . . . . . . . . . . 919
272 Branching rules of su11 ⊃ su10⊕u1(R) . . . . . . . . . . . . . . . 920
273 Branching rules of su11 ⊃ su9 ⊕su2 ⊕ u1(R) . . . . . . . . . . . . 921
274 Branching rules of su11 ⊃ su8 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 923
275 Branching rules of su11 ⊃ su7 ⊕su4 ⊕ u1(R) . . . . . . . . . . . . 925
276 Branching rules of su11 ⊃ su6 ⊕su5 ⊕ u1(R) . . . . . . . . . . . . 927
277 Branching rules of su11 ⊃so11(S) . . . . . . . . . . . . . . 929
278 Branching rules of su12 ⊃ su11⊕u1(R) . . . . . . . . . . . . . . . 930
279 Branching rules of su12 ⊃ su10⊕su2 ⊕ u1(R) . . . . . . . . . . . . 931
280 Branching rules of su12 ⊃ su9 ⊕su3 ⊕ u1(R) . . . . . . . . . . . . 933
281 Branching rules of su12 ⊃ su8 ⊕su4 ⊕ u1(R) . . . . . . . . . . . . 935
282 Branching rules of su12 ⊃ su7 ⊕su5 ⊕ u1(R) . . . . . . . . . . . . 937
283 Branching rules of su12 ⊃ su6 ⊕su6 ⊕ u1(R) . . . . . . . . . . . . 939
284 Branching rules of su12 ⊃so12(S) . . . . . . . . . . . . . . 941
285 Branching rules of su12 ⊃usp12(S) . . . . . . . . . . . . . . 942
286 Branching rules of su12 ⊃ su6 ⊕su2(S) . . . . . . . . . . . . . . . 943
287 Branching rules of su12 ⊃ su4 ⊕su3(S) . . . . . . . . . . . . . . . 945
288 Branching rules of su13 ⊃ su12⊕u1(R) . . . . . . . . . . . . . . . 947
289 Branching rules of su13 ⊃ su11⊕su2 ⊕ u1(R) . . . . . . . . . . . . 948
290 Branching rules of su13 ⊃ su10⊕su3 ⊕ u1(R) . . . . . . . . . . . . 950
291 Branching rules of su13 ⊃ su9 ⊕su4 ⊕ u1(R) . . . . . . . . . . . . 951
292 Branching rules of su13 ⊃ su8 ⊕su5 ⊕ u1(R) . . . . . . . . . . . . 953
293 Branching rules of su13 ⊃ su7 ⊕su6 ⊕ u1(R) . . . . . . . . . . . . 955
294 Branching rules of su13 ⊃so13(S) . . . . . . . . . . . . . . 957
295 Branching rules of su14 ⊃ su13⊕u1(R) . . . . . . . . . . . . . . . 958
296 Branching rules of su14 ⊃ su12⊕su2 ⊕ u1(R) . . . . . . . . . . . . 960
297 Branching rules of su14 ⊃ su11⊕su3 ⊕ u1(R) . . . . . . . . . . . . 961
298 Branching rules of su14 ⊃ su10⊕su4 ⊕ u1(R) . . . . . . . . . . . . 963
299 Branching rules of su14 ⊃ su9 ⊕su5 ⊕ u1(R) . . . . . . . . . . . . 965
300 Branching rules of su14 ⊃ su8 ⊕su6 ⊕ u1(R) . . . . . . . . . . . . 967
301 Branching rules of su14 ⊃ su7 ⊕su7 ⊕ u1(R) . . . . . . . . . . . . 969
302 Branching rules of su14 ⊃so14(S) . . . . . . . . . . . . . . 971
303 Branching rules of su14 ⊃usp14(S) . . . . . . . . . . . . . . 972
304 Branching rules of su14 ⊃ su7 ⊕su2(S) . . . . . . . . . . . . . . . 973
305 Branching rules of su15 ⊃ su14⊕u1(R) . . . . . . . . . . . . . . . 975
306 Branching rules of su15 ⊃ su13⊕su2 ⊕ u1(R) . . . . . . . . . . . . 976
307 Branching rules of su15 ⊃ su12⊕su3 ⊕ u1(R) . . . . . . . . . . . . 977
308 Branching rules of su15 ⊃ su11⊕su4 ⊕ u1(R) . . . . . . . . . . . . 979
309 Branching rules of su15 ⊃ su10⊕su5 ⊕ u1(R) . . . . . . . . . . . . 981
310 Branching rules of su15 ⊃ su9 ⊕su6 ⊕ u1(R) . . . . . . . . . . . . 983
311 Branching rules of su15 ⊃ su8 ⊕su7 ⊕ u1(R) . . . . . . . . . . . . 985
312 Branching rules of su15 ⊃so15(S) . . . . . . . . . . . . . . 987
313 Branching rules of su15 ⊃ su5 ⊕su3(S) . . . . . . . . . . . . . . . 988
314 Branching rules of su15 ⊃ su3(S) 989
315 Branching rules of su15 ⊃ su3(S) 992
316 Branching rules of su15 ⊃ su5(S) 994
317 Branching rules of su15 ⊃ su6(S) 995
318 Branching rules of su16 ⊃ su15⊕u1(R) . . . . . . . . . . . . . . . 997
319 Branching rules of su16 ⊃ su14⊕su2 ⊕ u1(R) . . . . . . . . . . . . 998
320 Branching rules of su16 ⊃ su13⊕su3 ⊕ u1(R) . . . . . . . . . . . . 999
321 Branching rules of su16 ⊃ su12⊕su4 ⊕ u1(R) . . . . . . . . . . . . 1001
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322 Branching rules of su16 ⊃ su11⊕su5 ⊕ u1(R) . . . . . . . . . . . . 1003
323 Branching rules of su16 ⊃ su10⊕su6 ⊕ u1(R) . . . . . . . . . . . . 1005
324 Branching rules of su16 ⊃ su9 ⊕su7 ⊕ u1(R) . . . . . . . . . . . . 1007
325 Branching rules of su16 ⊃ su8 ⊕su8 ⊕ u1(R) . . . . . . . . . . . . 1009
326 Branching rules of su16 ⊃so16(S) . . . . . . . . . . . . . . 1011
327 Branching rules of su16 ⊃usp16(S) . . . . . . . . . . . . . . 1012
328 Branching rules of su16 ⊃so10(S) . . . . . . . . . . . . . . 1013
329 Branching rules of su16 ⊃ su8 ⊕su2(S) . . . . . . . . . . . . . . . 1014
330 Branching rules of su16 ⊃ su4 ⊕su4(S) . . . . . . . . . . . . . . . 1016
331 Branching rules of so7 ⊃ su4(R) 1018
332 Branching rules of so7 ⊃ su2 ⊕su2 ⊕ su2(R) . . . . . . . . . . . 1019
333 Branching rules of so7 ⊃ usp4 ⊕u1(R) . . . . . . . . . . . . . . . 1022
334 Branching rules of so7 ⊃ G2(S) . 1025
335 Branching rules of so9 ⊃ so8(R) 1026
336 Branching rules of so9 ⊃ su2 ⊕su2 ⊕ usp4(R) . . . . . . . . . . 1028
337 Branching rules of so9 ⊃ su4 ⊕su2(R) . . . . . . . . . . . . . . 1031
338 Branching rules of so9 ⊃ so7 ⊕u1(R) . . . . . . . . . . . . . . . 1034
339 Branching rules of so9 ⊃ su2(S) 1037
340 Branching rules of so9 ⊃ su2 ⊕su2(S) . . . . . . . . . . . . . . . 1039
341 Branching rules of so11 ⊃so10(R) . . . . . . . . . . . . . . 1043
342 Branching rules of so11 ⊃ so8 ⊕su2(R) . . . . . . . . . . . . . . 1044
343 Branching rules of so11 ⊃ su4 ⊕usp4(R) . . . . . . . . . . . . . . 1046
344 Branching rules of so11 ⊃ su2 ⊕su2 ⊕ so7(R) . . . . . . . . . . . 1049
345 Branching rules of so11 ⊃ so9 ⊕u1(R) . . . . . . . . . . . . . . . 1053
346 Branching rules of so11 ⊃ su2(S) 1056
347 Branching rules of so13 ⊃so12(R) . . . . . . . . . . . . . . 1058
348 Branching rules of so13 ⊃ so10⊕su2(R) . . . . . . . . . . . . . . 1059
349 Branching rules of so13 ⊃ so8 ⊕usp4(R) . . . . . . . . . . . . . . 1061
350 Branching rules of so13 ⊃ su4 ⊕so7(R) . . . . . . . . . . . . . . 1063
351 Branching rules of so13 ⊃ su2 ⊕su2 ⊕ so9(R) . . . . . . . . . . . 1065
352 Branching rules of so13 ⊃ so11⊕u1(R) . . . . . . . . . . . . . . . 1067
353 Branching rules of so13 ⊃ su2(S) 1069
354 Branching rules of so15 ⊃so14(R) . . . . . . . . . . . . . . 1071
355 Branching rules of so15 ⊃ so12⊕su2(R) . . . . . . . . . . . . . . 1072
356 Branching rules of so15 ⊃ so10⊕usp4(R) . . . . . . . . . . . . . . 1073
357 Branching rules of so15 ⊃ so8 ⊕so7(R) . . . . . . . . . . . . . . 1074
358 Branching rules of so15 ⊃ su4 ⊕so9(R) . . . . . . . . . . . . . . 1076
359 Branching rules of so15 ⊃ su2 ⊕su2 ⊕ so11(R) . . . . . . . . . . . 1077
360 Branching rules of so15 ⊃ so13⊕u1(R) . . . . . . . . . . . . . . . 1078
361 Branching rules of so15 ⊃ su2(S) 1079
362 Branching rules of so15 ⊃ su4(S) 1081
363 Branching rules of so15 ⊃ su2 ⊕usp4(S) . . . . . . . . . . . . . . 1082
364 Branching rules of so17 ⊃so16(R) . . . . . . . . . . . . . . 1085
365 Branching rules of so17 ⊃ so14⊕su2(R) . . . . . . . . . . . . . . 1085
366 Branching rules of so17 ⊃ so12⊕usp4(R) . . . . . . . . . . . . . . 1086
367 Branching rules of so17 ⊃ so10⊕so7(R) . . . . . . . . . . . . . . 1087
368 Branching rules of so17 ⊃ so8 ⊕so9(R) . . . . . . . . . . . . . . 1088
369 Branching rules of so17 ⊃ su4 ⊕so11(R) . . . . . . . . . . . . . . 1088
370 Branching rules of so17 ⊃ su2 ⊕su2 ⊕ so13(R) . . . . . . . . . . . 1089
371 Branching rules of so17 ⊃ so15⊕u1(R) . . . . . . . . . . . . . . . 1090
372 Branching rules of so17 ⊃ su2(S) 1091
373 Branching rules of so19 ⊃so18(R) . . . . . . . . . . . . . . 1092
374 Branching rules of so19 ⊃ so16⊕su2(R) . . . . . . . . . . . . . . 1093
375 Branching rules of so19 ⊃ so14⊕usp4(R) . . . . . . . . . . . . . . 1093
376 Branching rules of so19 ⊃ so12⊕so7(R) . . . . . . . . . . . . . . 1094
377 Branching rules of so19 ⊃ so10⊕so9(R) . . . . . . . . . . . . . . 1095
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378 Branching rules of so19 ⊃ so8 ⊕so11(R) . . . . . . . . . . . . . . 1095
379 Branching rules of so19 ⊃ su4 ⊕so13(R) . . . . . . . . . . . . . . 1096
380 Branching rules of so19 ⊃ su2 ⊕su2 ⊕ so15(R) . . . . . . . . . . . 1096
381 Branching rules of so19 ⊃ so17⊕u1(R) . . . . . . . . . . . . . . . 1097
382 Branching rules of so19 ⊃ su2(S) 1098
383 Branching rules of so21 ⊃so20(R) . . . . . . . . . . . . . . 1099
384 Branching rules of so21 ⊃ so18⊕su2(R) . . . . . . . . . . . . . . 1099
385 Branching rules of so21 ⊃ so16⊕usp4(R) . . . . . . . . . . . . . . 1100
386 Branching rules of so21 ⊃ so14⊕so7(R) . . . . . . . . . . . . . . 1100
387 Branching rules of so21 ⊃ so12⊕so9(R) . . . . . . . . . . . . . . 1101
388 Branching rules of so21 ⊃ so10⊕so11(R) . . . . . . . . . . . . . . 1101
389 Branching rules of so21 ⊃ so8 ⊕so13(R) . . . . . . . . . . . . . . 1102
390 Branching rules of so21 ⊃ su4 ⊕so15(R) . . . . . . . . . . . . . . 1102
391 Branching rules of so21 ⊃ su2 ⊕su2 ⊕ so17(R) . . . . . . . . . . . 1103
392 Branching rules of so21 ⊃ so19⊕u1(R) . . . . . . . . . . . . . . . 1103
393 Branching rules of so21 ⊃ su2(S) 1104
394 Branching rules of so21 ⊃ su2 ⊕so7(S) . . . . . . . . . . . . . . . 1105
395 Branching rules of so21 ⊃ so7(S) 1106
396 Branching rules of so21 ⊃usp6(S) . . . . . . . . . . . . . . 1107
397 Branching rules of so23 ⊃so22(R) . . . . . . . . . . . . . . 1107
398 Branching rules of so23 ⊃ so20⊕su2(R) . . . . . . . . . . . . . . 1108
399 Branching rules of so23 ⊃ so18⊕usp4(R) . . . . . . . . . . . . . . 1108
400 Branching rules of so23 ⊃ so16⊕so7(R) . . . . . . . . . . . . . . 1109
401 Branching rules of so23 ⊃ so14⊕so9(R) . . . . . . . . . . . . . . 1109
402 Branching rules of so23 ⊃ so12⊕so11(R) . . . . . . . . . . . . . . 1110
403 Branching rules of so23 ⊃ so10⊕so13(R) . . . . . . . . . . . . . . 1110
404 Branching rules of so23 ⊃ so8 ⊕so15(R) . . . . . . . . . . . . . . 1111
405 Branching rules of so23 ⊃ su4 ⊕so17(R) . . . . . . . . . . . . . . 1111
406 Branching rules of so23 ⊃ su2 ⊕su2 ⊕ so19(R) . . . . . . . . . . . 1112
407 Branching rules of so23 ⊃ so21⊕u1(R) . . . . . . . . . . . . . . . 1112
408 Branching rules of so23 ⊃ su2(S) 1113
409 Branching rules of so25 ⊃so24(R) . . . . . . . . . . . . . . 1114
410 Branching rules of so25 ⊃ so22⊕su2(R) . . . . . . . . . . . . . . 1114
411 Branching rules of so25 ⊃ so20⊕usp4(R) . . . . . . . . . . . . . . 1115
412 Branching rules of so25 ⊃ so18⊕so7(R) . . . . . . . . . . . . . . 1115
413 Branching rules of so25 ⊃ so16⊕so9(R) . . . . . . . . . . . . . . 1116
414 Branching rules of so25 ⊃ so14⊕so11(R) . . . . . . . . . . . . . . 1116
415 Branching rules of so25 ⊃ so12⊕so13(R) . . . . . . . . . . . . . . 1116
416 Branching rules of so25 ⊃ so10⊕so15(R) . . . . . . . . . . . . . . 1117
417 Branching rules of so25 ⊃ so8 ⊕so17(R) . . . . . . . . . . . . . . 1117
418 Branching rules of so25 ⊃ su4 ⊕so19(R) . . . . . . . . . . . . . . 1118
419 Branching rules of so25 ⊃ su2 ⊕su2 ⊕ so21(R) . . . . . . . . . . . 1118
420 Branching rules of so25 ⊃ so23⊕u1(R) . . . . . . . . . . . . . . . 1119
421 Branching rules of so25 ⊃ su2(S) 1119
422 Branching rules of so25 ⊃ usp4⊕usp4(S) . . . . . . . . . . . . . . 1120
423 Branching rules of so27 ⊃so26(R) . . . . . . . . . . . . . . 1121
424 Branching rules of so27 ⊃ so24⊕su2(R) . . . . . . . . . . . . . . 1121
425 Branching rules of so27 ⊃ so22⊕usp4(R) . . . . . . . . . . . . . . 1122
426 Branching rules of so27 ⊃ so20⊕so7(R) . . . . . . . . . . . . . . 1122
427 Branching rules of so27 ⊃ so18⊕so9(R) . . . . . . . . . . . . . . 1122
428 Branching rules of so27 ⊃ so16⊕so11(R) . . . . . . . . . . . . . . 1123
429 Branching rules of so27 ⊃ so14⊕so13(R) . . . . . . . . . . . . . . 1123
430 Branching rules of so27 ⊃ so12⊕so15(R) . . . . . . . . . . . . . . 1124
431 Branching rules of so27 ⊃ so10⊕so17(R) . . . . . . . . . . . . . . 1124
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432 Branching rules of so27 ⊃ so8 ⊕so19(R) . . . . . . . . . . . . . . 1124
433 Branching rules of so27 ⊃ su4 ⊕so21(R) . . . . . . . . . . . . . . 1125
434 Branching rules of so27 ⊃ su2 ⊕su2 ⊕ so23(R) . . . . . . . . . . . 1125
435 Branching rules of so27 ⊃ so25⊕u1(R) . . . . . . . . . . . . . . . 1126
436 Branching rules of so27 ⊃ su2(S) 1126
437 Branching rules of so27 ⊃ su3(S) 1127
438 Branching rules of so27 ⊃ so7(S) 1128
439 Branching rules of so27 ⊃ su2 ⊕so9(S) . . . . . . . . . . . . . . . 1128
440 Branching rules of so29 ⊃so28(R) . . . . . . . . . . . . . . 1129
441 Branching rules of so29 ⊃ so26⊕su2(R) . . . . . . . . . . . . . . 1129
442 Branching rules of so29 ⊃ so24⊕usp4(R) . . . . . . . . . . . . . . 1129
443 Branching rules of so29 ⊃ so22⊕so7(R) . . . . . . . . . . . . . . 1130
444 Branching rules of so29 ⊃ so20⊕so9(R) . . . . . . . . . . . . . . 1130
445 Branching rules of so29 ⊃ so18⊕so11(R) . . . . . . . . . . . . . . 1130
446 Branching rules of so29 ⊃ so16⊕so13(R) . . . . . . . . . . . . . . 1131
447 Branching rules of so29 ⊃ so14⊕so15(R) . . . . . . . . . . . . . . 1131
448 Branching rules of so29 ⊃ so12⊕so17(R) . . . . . . . . . . . . . . 1131
449 Branching rules of so29 ⊃ so10⊕so19(R) . . . . . . . . . . . . . . 1132
450 Branching rules of so29 ⊃ so8 ⊕so21(R) . . . . . . . . . . . . . . 1132
451 Branching rules of so29 ⊃ su4 ⊕so23(R) . . . . . . . . . . . . . . 1132
452 Branching rules of so29 ⊃ su2 ⊕su2 ⊕ so25(R) . . . . . . . . . . . 1133
453 Branching rules of so29 ⊃ so27⊕u1(R) . . . . . . . . . . . . . . . 1133
454 Branching rules of so29 ⊃ su2(S) 1133
455 Branching rules of so31 ⊃so30(R) . . . . . . . . . . . . . . 1134
456 Branching rules of so31 ⊃ so28⊕su2(R) . . . . . . . . . . . . . . 1135
457 Branching rules of so31 ⊃ so26⊕usp4(R) . . . . . . . . . . . . . . 1135
458 Branching rules of so31 ⊃ so24⊕so7(R) . . . . . . . . . . . . . . 1135
459 Branching rules of so31 ⊃ so22⊕so9(R) . . . . . . . . . . . . . . 1136
460 Branching rules of so31 ⊃ so20⊕so11(R) . . . . . . . . . . . . . . 1136
461 Branching rules of so31 ⊃ so18⊕so13(R) . . . . . . . . . . . . . . 1136
462 Branching rules of so31 ⊃ so16⊕so15(R) . . . . . . . . . . . . . . 1137
463 Branching rules of so31 ⊃ so14⊕so17(R) . . . . . . . . . . . . . . 1137
464 Branching rules of so31 ⊃ so12⊕so19(R) . . . . . . . . . . . . . . 1138
465 Branching rules of so31 ⊃ so10⊕so21(R) . . . . . . . . . . . . . . 1138
466 Branching rules of so31 ⊃ so8 ⊕so23(R) . . . . . . . . . . . . . . 1138
467 Branching rules of so31 ⊃ su4 ⊕so25(R) . . . . . . . . . . . . . . 1139
468 Branching rules of so31 ⊃ su2 ⊕su2 ⊕ so27(R) . . . . . . . . . . . 1139
469 Branching rules of so31 ⊃ so29⊕u1(R) . . . . . . . . . . . . . . . 1139
470 Branching rules of so31 ⊃ su2(S) 1140
471 Branching rules of usp4 ⊃ su2 ⊕su2(R) . . . . . . . . . . . . . . 1142
472 Branching rules of usp4 ⊃ su2 ⊕u1(R) . . . . . . . . . . . . . . . 1143
473 Branching rules of usp4 ⊃ su2(S) 1147
474 Branching rules of usp6 ⊃ su3 ⊕u1(R) . . . . . . . . . . . . . . . 1149
475 Branching rules of usp6 ⊃ su2 ⊕usp4(R) . . . . . . . . . . . . . . 1153
476 Branching rules of usp6 ⊃ su2(S) 1155
477 Branching rules of usp6 ⊃ su2 ⊕su2(S) . . . . . . . . . . . . . . . 1156
478 Branching rules of usp8 ⊃ su4 ⊕u1(R) . . . . . . . . . . . . . . . 1159
479 Branching rules of usp8 ⊃ su2 ⊕usp6(R) . . . . . . . . . . . . . . 1163
480 Branching rules of usp8 ⊃ usp4⊕usp4(R) . . . . . . . . . . . . . . 1165
481 Branching rules of usp8 ⊃ su2(S) 1167
482 Branching rules of usp8 ⊃ su2 ⊕su2 ⊕ su2(S) . . . . . . . . . . . 1170
483 Branching rules of usp10 ⊃ su5⊕u1(R) . . . . . . . . . . . . . . . 1175
484 Branching rules of usp10 ⊃ su2⊕usp8(R) . . . . . . . . . . . . . . 1179
485 Branching rules of usp10 ⊃usp4 ⊕ usp6(R) . . . . . . . . . . 1181
486 Branching rules of usp10 ⊃su2(S) . . . . . . . . . . . . . . . 1183
487 Branching rules of usp10 ⊃ su2⊕usp4(S) . . . . . . . . . . . . . . 1186
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488 Branching rules of usp12 ⊃ su6⊕u1(R) . . . . . . . . . . . . . . . 1190
489 Branching rules of usp12 ⊃ su2⊕usp10(R) . . . . . . . . . . . . . 1192
490 Branching rules of usp12 ⊃usp4 ⊕ usp8(R) . . . . . . . . . . 1193
491 Branching rules of usp12 ⊃usp6 ⊕ usp6(R) . . . . . . . . . . 1195
492 Branching rules of usp12 ⊃su2(S) . . . . . . . . . . . . . . . 1197
493 Branching rules of usp12 ⊃ su2⊕su4(S) . . . . . . . . . . . . . . . 1199
494 Branching rules of usp12 ⊃ su2⊕usp4(S) . . . . . . . . . . . . . . 1202
495 Branching rules of usp14 ⊃ su7⊕u1(R) . . . . . . . . . . . . . . . 1206
496 Branching rules of usp14 ⊃ su2⊕usp12(R) . . . . . . . . . . . . . 1207
497 Branching rules of usp14 ⊃usp4 ⊕ usp10(R) . . . . . . . . . 1208
498 Branching rules of usp14 ⊃usp6 ⊕ usp8(R) . . . . . . . . . . 1209
499 Branching rules of usp14 ⊃su2(S) . . . . . . . . . . . . . . . 1210
500 Branching rules of usp14 ⊃ su2⊕so7(S) . . . . . . . . . . . . . . . 1212
501 Branching rules of usp16 ⊃ su8⊕u1(R) . . . . . . . . . . . . . . . 1213
502 Branching rules of usp16 ⊃ su2⊕usp14(R) . . . . . . . . . . . . . 1214
503 Branching rules of usp16 ⊃usp4 ⊕ usp12(R) . . . . . . . . . 1214
504 Branching rules of usp16 ⊃usp6 ⊕ usp10(R) . . . . . . . . . 1215
505 Branching rules of usp16 ⊃usp8 ⊕ usp8(R) . . . . . . . . . . 1216
506 Branching rules of usp16 ⊃su2(S) . . . . . . . . . . . . . . . 1217
507 Branching rules of usp16 ⊃usp4(S) . . . . . . . . . . . . . . 1218
508 Branching rules of usp16 ⊃ su2⊕so8(S) . . . . . . . . . . . . . . . 1219
509 Branching rules of usp18 ⊃ su9⊕u1(R) . . . . . . . . . . . . . . . 1220
510 Branching rules of usp18 ⊃ su2⊕usp16(R) . . . . . . . . . . . . . 1221
511 Branching rules of usp18 ⊃usp4 ⊕ usp14(R) . . . . . . . . . 1221
512 Branching rules of usp18 ⊃usp6 ⊕ usp12(R) . . . . . . . . . 1222
513 Branching rules of usp18 ⊃usp8 ⊕ usp10(R) . . . . . . . . . 1223
514 Branching rules of usp18 ⊃su2(S) . . . . . . . . . . . . . . . 1223
515 Branching rules of usp18 ⊃ su2⊕so9(S) . . . . . . . . . . . . . . . 1225
516 Branching rules of usp18 ⊃ su2⊕usp6(S) . . . . . . . . . . . . . . 1225
517 Branching rules of usp20 ⊃su10 ⊕ u1(R) . . . . . . . . . . . 1227
518 Branching rules of usp20 ⊃ su2⊕usp18(R) . . . . . . . . . . . . . 1227
519 Branching rules of usp20 ⊃usp4 ⊕ usp16(R) . . . . . . . . . 1228
520 Branching rules of usp20 ⊃usp6 ⊕ usp14(R) . . . . . . . . . 1228
521 Branching rules of usp20 ⊃usp8 ⊕ usp12(R) . . . . . . . . . 1229
522 Branching rules of usp20 ⊃usp10 ⊕ usp10(R) . . . . . . . . . 1229
523 Branching rules of usp20 ⊃su2(S) . . . . . . . . . . . . . . . 1230
524 Branching rules of usp20 ⊃usp4 ⊕ usp4(S) . . . . . . . . . . 1231
525 Branching rules of usp20 ⊃ su2⊕so10(S) . . . . . . . . . . . . . . 1232
526 Branching rules of usp20 ⊃su6(S) . . . . . . . . . . . . . . . 1232
527 Branching rules of usp22 ⊃su11 ⊕ u1(R) . . . . . . . . . . . 1233
528 Branching rules of usp22 ⊃ su2⊕usp20(R) . . . . . . . . . . . . . 1233
529 Branching rules of usp22 ⊃usp4 ⊕ usp18(R) . . . . . . . . . 1234
530 Branching rules of usp22 ⊃usp6 ⊕ usp16(R) . . . . . . . . . 1234
531 Branching rules of usp22 ⊃usp8 ⊕ usp14(R) . . . . . . . . . 1235
532 Branching rules of usp22 ⊃usp10 ⊕ usp12(R) . . . . . . . . . 1235
533 Branching rules of usp22 ⊃su2(S) . . . . . . . . . . . . . . . 1236
534 Branching rules of usp24 ⊃su12 ⊕ u1(R) . . . . . . . . . . . 1237
535 Branching rules of usp24 ⊃ su2⊕usp22(R) . . . . . . . . . . . . . 1237
536 Branching rules of usp24 ⊃usp4 ⊕ usp20(R) . . . . . . . . . 1238
537 Branching rules of usp24 ⊃usp6 ⊕ usp18(R) . . . . . . . . . 1238
538 Branching rules of usp24 ⊃usp8 ⊕ usp16(R) . . . . . . . . . 1239
539 Branching rules of usp24 ⊃usp10 ⊕ usp14(R) . . . . . . . . . 1239
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540 Branching rules of usp24 ⊃usp12 ⊕ usp12(R) . . . . . . . . . 1240
541 Branching rules of usp24 ⊃su2(S) . . . . . . . . . . . . . . . 1240
542 Branching rules of usp24 ⊃ su2⊕su2 ⊕ usp6(S) . . . . . . . . . . . 1241
543 Branching rules of usp24 ⊃ su2⊕usp8(S) . . . . . . . . . . . . . . 1242
544 Branching rules of usp24 ⊃ su4⊕usp4(S) . . . . . . . . . . . . . . 1243
545 Branching rules of usp26 ⊃su13 ⊕ u1(R) . . . . . . . . . . . 1244
546 Branching rules of usp26 ⊃ su2⊕usp24(R) . . . . . . . . . . . . . 1244
547 Branching rules of usp26 ⊃usp4 ⊕ usp22(R) . . . . . . . . . 1245
548 Branching rules of usp26 ⊃usp6 ⊕ usp20(R) . . . . . . . . . 1245
549 Branching rules of usp26 ⊃usp8 ⊕ usp18(R) . . . . . . . . . 1245
550 Branching rules of usp26 ⊃usp10 ⊕ usp16(R) . . . . . . . . . 1246
551 Branching rules of usp26 ⊃usp12 ⊕ usp14(R) . . . . . . . . . 1246
552 Branching rules of usp26 ⊃su2(S) . . . . . . . . . . . . . . . 1247
553 Branching rules of usp28 ⊃su14 ⊕ u1(R) . . . . . . . . . . . 1247
554 Branching rules of usp28 ⊃ su2⊕usp26(R) . . . . . . . . . . . . . 1248
555 Branching rules of usp28 ⊃usp4 ⊕ usp24(R) . . . . . . . . . 1248
556 Branching rules of usp28 ⊃usp6 ⊕ usp22(R) . . . . . . . . . 1248
557 Branching rules of usp28 ⊃usp8 ⊕ usp20(R) . . . . . . . . . 1249
558 Branching rules of usp28 ⊃usp10 ⊕ usp18(R) . . . . . . . . . 1249
559 Branching rules of usp28 ⊃usp12 ⊕ usp16(R) . . . . . . . . . 1249
560 Branching rules of usp28 ⊃usp14 ⊕ usp14(R) . . . . . . . . . 1250
561 Branching rules of usp28 ⊃su2(S) . . . . . . . . . . . . . . . 1250
562 Branching rules of usp28 ⊃ so7⊕usp4(S) . . . . . . . . . . . . . . 1251
563 Branching rules of usp30 ⊃su15 ⊕ u1(R) . . . . . . . . . . . 1251
564 Branching rules of usp30 ⊃ su2⊕usp28(R) . . . . . . . . . . . . . 1251
565 Branching rules of usp30 ⊃usp4 ⊕ usp26(R) . . . . . . . . . 1252
566 Branching rules of usp30 ⊃usp6 ⊕ usp24(R) . . . . . . . . . 1252
567 Branching rules of usp30 ⊃usp8 ⊕ usp22(R) . . . . . . . . . 1252
568 Branching rules of usp30 ⊃usp10 ⊕ usp20(R) . . . . . . . . . 1253
569 Branching rules of usp30 ⊃usp12 ⊕ usp18(R) . . . . . . . . . 1253
570 Branching rules of usp30 ⊃usp14 ⊕ usp16(R) . . . . . . . . . 1253
571 Branching rules of usp30 ⊃su2(S) . . . . . . . . . . . . . . . 1254
572 Branching rules of usp30 ⊃ su2⊕usp10(S) . . . . . . . . . . . . . . 1254
573 Branching rules of usp30 ⊃usp4 ⊕ usp6(S) . . . . . . . . . . 1255
574 Branching rules of so8 ⊃ su2 ⊕su2 ⊕ su2 ⊕ su2(R) . . . . . . . . 1256
575 Branching rules of so8 ⊃ su4 ⊕u1(R) . . . . . . . . . . . . . . . 1260
576 Branching rules of so8 ⊃ su3(S) 1263
577 Branching rules of so8 ⊃ so7(S) . 1265
578 Branching rules of so8 ⊃ su2 ⊕usp4(S) . . . . . . . . . . . . . . 1267
579 Branching rules of so10 ⊃ su5 ⊕u1(R) . . . . . . . . . . . . . . . 1269
580 Branching rules of so10 ⊃ su2 ⊕su2 ⊕ su4(R) . . . . . . . . . . . 1271
581 Branching rules of so10 ⊃ so8 ⊕u1(R) . . . . . . . . . . . . . . . 1274
582 Branching rules of so10 ⊃usp4(S) . . . . . . . . . . . . . . 1276
583 Branching rules of so10 ⊃ so9(S) 1278
584 Branching rules of so10 ⊃ su2 ⊕so7(S) . . . . . . . . . . . . . . . 1279
585 Branching rules of so10 ⊃ usp4⊕usp4(S) . . . . . . . . . . . . . . 1280
586 Branching rules of so12 ⊃ su6 ⊕u1(R) . . . . . . . . . . . . . . . 1282
587 Branching rules of so12 ⊃ su2 ⊕su2 ⊕ so8(R) . . . . . . . . . . . 1285
588 Branching rules of so12 ⊃ su4 ⊕su4(R) . . . . . . . . . . . . . . 1287
589 Branching rules of so12 ⊃ so10⊕u1(R) . . . . . . . . . . . . . . . 1289
590 Branching rules of so12 ⊃ su2 ⊕usp6(S) . . . . . . . . . . . . . . 1292
591 Branching rules of so12 ⊃ su2 ⊕su2 ⊕ su2(S) . . . . . . . . . . . 1294
592 Branching rules of so12 ⊃so11(S) . . . . . . . . . . . . . . 1299
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593 Branching rules of so12 ⊃ su2 ⊕so9(S) . . . . . . . . . . . . . . . 1300
594 Branching rules of so12 ⊃ usp4⊕so7(R) . . . . . . . . . . . . . . 1302
595 Branching rules of so14 ⊃ su7 ⊕u1(R) . . . . . . . . . . . . . . . 1303
596 Branching rules of so14 ⊃ su2 ⊕su2 ⊕ so10(R) . . . . . . . . . . . 1306
597 Branching rules of so14 ⊃ su4 ⊕so8(R) . . . . . . . . . . . . . . 1309
598 Branching rules of so14 ⊃ so12⊕u1(R) . . . . . . . . . . . . . . . 1311
599 Branching rules of so14 ⊃usp4(S) . . . . . . . . . . . . . . 1313
600 Branching rules of so14 ⊃usp6(S) . . . . . . . . . . . . . . 1316
601 Branching rules of so14 ⊃ G2(S) 1317
602 Branching rules of so14 ⊃so13(S) . . . . . . . . . . . . . . 1319
603 Branching rules of so14 ⊃ su2 ⊕so11(S) . . . . . . . . . . . . . . 1320
604 Branching rules of so14 ⊃ usp4⊕so9(S) . . . . . . . . . . . . . . . 1322
605 Branching rules of so14 ⊃ so7 ⊕so7(S) . . . . . . . . . . . . . . . 1323
606 Branching rules of so16 ⊃ su8 ⊕u1(R) . . . . . . . . . . . . . . . 1325
607 Branching rules of so16 ⊃ su2 ⊕su2 ⊕ so12(R) . . . . . . . . . . . 1327
608 Branching rules of so16 ⊃ su4 ⊕so10(R) . . . . . . . . . . . . . . 1328
609 Branching rules of so16 ⊃ so8 ⊕so8(R) . . . . . . . . . . . . . . 1329
610 Branching rules of so16 ⊃ so14⊕u1(R) . . . . . . . . . . . . . . . 1331
611 Branching rules of so16 ⊃ so9(S) 1332
612 Branching rules of so16 ⊃ su2 ⊕usp8(S) . . . . . . . . . . . . . . 1333
613 Branching rules of so16 ⊃ usp4⊕usp4(S) . . . . . . . . . . . . . . 1334
614 Branching rules of so16 ⊃so15(S) . . . . . . . . . . . . . . 1336
615 Branching rules of so16 ⊃ su2 ⊕so13(S) . . . . . . . . . . . . . . 1337
616 Branching rules of so16 ⊃ usp4⊕so11(S) . . . . . . . . . . . . . . 1338
617 Branching rules of so16 ⊃ so7 ⊕so9(S) . . . . . . . . . . . . . . . 1339
618 Branching rules of so18 ⊃ su9 ⊕u1(R) . . . . . . . . . . . . . . . 1340
619 Branching rules of so18 ⊃ su2 ⊕su2 ⊕ so14(R) . . . . . . . . . . . 1341
620 Branching rules of so18 ⊃ su4 ⊕so12(R) . . . . . . . . . . . . . . 1342
621 Branching rules of so18 ⊃ so8 ⊕so10(R) . . . . . . . . . . . . . . 1343
622 Branching rules of so18 ⊃ so16⊕u1(R) . . . . . . . . . . . . . . . 1344
623 Branching rules of so18 ⊃ su2 ⊕su4(S) . . . . . . . . . . . . . . . 1345
624 Branching rules of so18 ⊃so17(S) . . . . . . . . . . . . . . 1348
625 Branching rules of so18 ⊃ su2 ⊕so15(S) . . . . . . . . . . . . . . 1348
626 Branching rules of so18 ⊃ usp4⊕so13(S) . . . . . . . . . . . . . . 1349
627 Branching rules of so18 ⊃ so7 ⊕so11(S) . . . . . . . . . . . . . . 1350
628 Branching rules of so18 ⊃ so9 ⊕so9(S) . . . . . . . . . . . . . . . 1351
629 Branching rules of so20 ⊃ su10⊕u1(R) . . . . . . . . . . . . . . . 1352
630 Branching rules of so20 ⊃ su2 ⊕su2 ⊕ so16(R) . . . . . . . . . . . 1353
631 Branching rules of so20 ⊃ su4 ⊕so14(R) . . . . . . . . . . . . . . 1353
632 Branching rules of so20 ⊃ so8 ⊕so12(R) . . . . . . . . . . . . . . 1354
633 Branching rules of so20 ⊃ so10⊕so10(R) . . . . . . . . . . . . . . 1355
634 Branching rules of so20 ⊃ so18⊕u1(R) . . . . . . . . . . . . . . . 1355
635 Branching rules of so20 ⊃ su2 ⊕usp10(S) . . . . . . . . . . . . . . 1356
636 Branching rules of so20 ⊃so19(S) . . . . . . . . . . . . . . 1357
637 Branching rules of so20 ⊃ su2 ⊕so17(S) . . . . . . . . . . . . . . 1357
638 Branching rules of so20 ⊃ usp4⊕so15(S) . . . . . . . . . . . . . . 1358
639 Branching rules of so20 ⊃ so7 ⊕so13(S) . . . . . . . . . . . . . . 1358
640 Branching rules of so20 ⊃ so9 ⊕so11(S) . . . . . . . . . . . . . . 1359
641 Branching rules of so20 ⊃ su2 ⊕su2 ⊕ usp4(S) . . . . . . . . . . . 1359
642 Branching rules of so20 ⊃ su4(S) 1361
643 Branching rules of so22 ⊃ su11⊕u1(R) . . . . . . . . . . . . . . . 1362
644 Branching rules of so22 ⊃ su2 ⊕su2 ⊕ so18(R) . . . . . . . . . . . 1363
645 Branching rules of so22 ⊃ su4 ⊕so16(R) . . . . . . . . . . . . . . 1364
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646 Branching rules of so22 ⊃ so8 ⊕so14(R) . . . . . . . . . . . . . . 1364
647 Branching rules of so22 ⊃ so10⊕so12(R) . . . . . . . . . . . . . . 1365
648 Branching rules of so22 ⊃ so20⊕u1(R) . . . . . . . . . . . . . . . 1365
649 Branching rules of so22 ⊃so21(S) . . . . . . . . . . . . . . 1366
650 Branching rules of so22 ⊃ su2 ⊕so19(S) . . . . . . . . . . . . . . 1367
651 Branching rules of so22 ⊃ usp4⊕so17(S) . . . . . . . . . . . . . . 1367
652 Branching rules of so22 ⊃ so7 ⊕so15(S) . . . . . . . . . . . . . . 1368
653 Branching rules of so22 ⊃ so9 ⊕so13(S) . . . . . . . . . . . . . . 1368
654 Branching rules of so22 ⊃ so11⊕so11(S) . . . . . . . . . . . . . . 1369
655 Branching rules of so24 ⊃ su12⊕u1(R) . . . . . . . . . . . . . . . 1369
656 Branching rules of so24 ⊃ su2 ⊕su2 ⊕ so20(R) . . . . . . . . . . . 1370
657 Branching rules of so24 ⊃ su4 ⊕so18(R) . . . . . . . . . . . . . . 1371
658 Branching rules of so24 ⊃ so8 ⊕so16(R) . . . . . . . . . . . . . . 1371
659 Branching rules of so24 ⊃ so10⊕so14(R) . . . . . . . . . . . . . . 1372
660 Branching rules of so24 ⊃ so12⊕so12(R) . . . . . . . . . . . . . . 1372
661 Branching rules of so24 ⊃ so22⊕u1(R) . . . . . . . . . . . . . . . 1373
662 Branching rules of so24 ⊃so23(S) . . . . . . . . . . . . . . 1373
663 Branching rules of so24 ⊃ su2 ⊕so21(S) . . . . . . . . . . . . . . 1374
664 Branching rules of so24 ⊃ usp4⊕so19(S) . . . . . . . . . . . . . . 1374
665 Branching rules of so24 ⊃ so7 ⊕so17(S) . . . . . . . . . . . . . . 1375
666 Branching rules of so24 ⊃ so9 ⊕so15(S) . . . . . . . . . . . . . . 1376
667 Branching rules of so24 ⊃ so11⊕so13(S) . . . . . . . . . . . . . . 1376
668 Branching rules of so24 ⊃ usp6⊕usp4(S) . . . . . . . . . . . . . . 1377
669 Branching rules of so24 ⊃ su2 ⊕so8(S) . . . . . . . . . . . . . . . 1378
670 Branching rules of so24 ⊃ su5(S) 1379
671 Branching rules of so26 ⊃ su13⊕u1(R) . . . . . . . . . . . . . . . 1379
672 Branching rules of so26 ⊃ su2 ⊕su2 ⊕ so22(R) . . . . . . . . . . . 1380
673 Branching rules of so26 ⊃ su4 ⊕so20(R) . . . . . . . . . . . . . . 1380
674 Branching rules of so26 ⊃ so8 ⊕so18(R) . . . . . . . . . . . . . . 1381
675 Branching rules of so26 ⊃ so10⊕so16(R) . . . . . . . . . . . . . . 1381
676 Branching rules of so26 ⊃ so12⊕so14(R) . . . . . . . . . . . . . . 1382
677 Branching rules of so26 ⊃ so24⊕u1(R) . . . . . . . . . . . . . . . 1382
678 Branching rules of so26 ⊃so25(S) . . . . . . . . . . . . . . 1382
679 Branching rules of so26 ⊃ su2 ⊕so23(S) . . . . . . . . . . . . . . 1383
680 Branching rules of so26 ⊃ usp4⊕so21(S) . . . . . . . . . . . . . . 1383
681 Branching rules of so26 ⊃ so7 ⊕so19(S) . . . . . . . . . . . . . . 1384
682 Branching rules of so26 ⊃ so9 ⊕so17(S) . . . . . . . . . . . . . . 1384
683 Branching rules of so26 ⊃ so11⊕so15(S) . . . . . . . . . . . . . . 1384
684 Branching rules of so26 ⊃ so13⊕so13(S) . . . . . . . . . . . . . . 1385
685 Branching rules of so26 ⊃ F4(S) 1385
686 Branching rules of so28 ⊃ su14⊕u1(R) . . . . . . . . . . . . . . . 1386
687 Branching rules of so28 ⊃ su2 ⊕su2 ⊕ so24(R) . . . . . . . . . . . 1386
688 Branching rules of so28 ⊃ su4 ⊕so22(R) . . . . . . . . . . . . . . 1386
689 Branching rules of so28 ⊃ so8 ⊕so20(R) . . . . . . . . . . . . . . 1387
690 Branching rules of so28 ⊃ so10⊕so18(R) . . . . . . . . . . . . . . 1387
691 Branching rules of so28 ⊃ so12⊕so16(R) . . . . . . . . . . . . . . 1388
692 Branching rules of so28 ⊃ so14⊕so14(R) . . . . . . . . . . . . . . 1388
693 Branching rules of so28 ⊃ so26⊕u1(R) . . . . . . . . . . . . . . . 1388
694 Branching rules of so28 ⊃so27(S) . . . . . . . . . . . . . . 1389
695 Branching rules of so28 ⊃ su2 ⊕so25(S) . . . . . . . . . . . . . . 1389
696 Branching rules of so28 ⊃ usp4⊕so23(S) . . . . . . . . . . . . . . 1389
697 Branching rules of so28 ⊃ so7 ⊕so21(S) . . . . . . . . . . . . . . 1390
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698 Branching rules of so28 ⊃ so9 ⊕so19(S) . . . . . . . . . . . . . . 1390
699 Branching rules of so28 ⊃ so11⊕so17(S) . . . . . . . . . . . . . . 1391
700 Branching rules of so28 ⊃ so13⊕so15(S) . . . . . . . . . . . . . . 1391
701 Branching rules of so28 ⊃ su2 ⊕su2 ⊕ so7(S) . . . . . . . . . . . 1391
702 Branching rules of so30 ⊃ su15⊕u1(R) . . . . . . . . . . . . . . . 1392
703 Branching rules of so30 ⊃ su2 ⊕su2 ⊕ so26(R) . . . . . . . . . . . 1392
704 Branching rules of so30 ⊃ su4 ⊕so24(R) . . . . . . . . . . . . . . 1393
705 Branching rules of so30 ⊃ so8 ⊕so22(R) . . . . . . . . . . . . . . 1393
706 Branching rules of so30 ⊃ so10⊕so20(R) . . . . . . . . . . . . . . 1394
707 Branching rules of so30 ⊃ so12⊕so18(R) . . . . . . . . . . . . . . 1394
708 Branching rules of so30 ⊃ so14⊕so16(R) . . . . . . . . . . . . . . 1394
709 Branching rules of so30 ⊃ so28⊕u1(R) . . . . . . . . . . . . . . . 1395
710 Branching rules of so30 ⊃so29(S) . . . . . . . . . . . . . . 1395
711 Branching rules of so30 ⊃ su2 ⊕so27(S) . . . . . . . . . . . . . . 1396
712 Branching rules of so30 ⊃ usp4⊕so25(S) . . . . . . . . . . . . . . 1396
713 Branching rules of so30 ⊃ so7 ⊕so23(S) . . . . . . . . . . . . . . 1396
714 Branching rules of so30 ⊃ so9 ⊕so21(S) . . . . . . . . . . . . . . 1397
715 Branching rules of so30 ⊃ so11⊕so19(S) . . . . . . . . . . . . . . 1397
716 Branching rules of so30 ⊃ so13⊕so17(S) . . . . . . . . . . . . . . 1397
717 Branching rules of so30 ⊃ so15⊕so15(S) . . . . . . . . . . . . . . 1398
718 Branching rules of so30 ⊃ su2 ⊕so10(S) . . . . . . . . . . . . . . 1398
719 Branching rules of so30 ⊃ usp4⊕su4(S) . . . . . . . . . . . . . . . 1399
720 Branching rules of so32 ⊃ su16⊕u1(R) . . . . . . . . . . . . . . . 1399
721 Branching rules of so32 ⊃ su2 ⊕su2 ⊕ so28(R) . . . . . . . . . . . 1400
722 Branching rules of so32 ⊃ su4 ⊕so26(R) . . . . . . . . . . . . . . 1400
723 Branching rules of so32 ⊃ so8 ⊕so24(R) . . . . . . . . . . . . . . 1400
724 Branching rules of so32 ⊃ so10⊕so22(R) . . . . . . . . . . . . . . 1401
725 Branching rules of so32 ⊃ so12⊕so20(R) . . . . . . . . . . . . . . 1401
726 Branching rules of so32 ⊃ so14⊕so18(R) . . . . . . . . . . . . . . 1401
727 Branching rules of so32 ⊃ so16⊕so16(R) . . . . . . . . . . . . . . 1402
728 Branching rules of so32 ⊃ so30⊕u1(R) . . . . . . . . . . . . . . . 1402
729 Branching rules of so32 ⊃so31(S) . . . . . . . . . . . . . . 1402
730 Branching rules of so32 ⊃ su2 ⊕so29(S) . . . . . . . . . . . . . . 1403
731 Branching rules of so32 ⊃ usp4⊕so27(S) . . . . . . . . . . . . . . 1403
732 Branching rules of so32 ⊃ so7 ⊕so25(S) . . . . . . . . . . . . . . 1403
733 Branching rules of so32 ⊃ so9 ⊕so23(S) . . . . . . . . . . . . . . 1404
734 Branching rules of so32 ⊃ so11⊕so21(S) . . . . . . . . . . . . . . 1404
735 Branching rules of so32 ⊃ so13⊕so19(S) . . . . . . . . . . . . . . 1404
736 Branching rules of so32 ⊃ so15⊕so17(S) . . . . . . . . . . . . . . 1405
737 Branching rules of so32 ⊃ su2 ⊕su2 ⊕ so8(S) . . . . . . . . . . . 1405
738 Branching rules of so32 ⊃ usp4⊕usp8(S) . . . . . . . . . . . . . . 1405
739 Branching rules of E6 ⊃ so10 ⊕u1(R) . . . . . . . . . . . . . . . 1407
740 Branching rules of E6 ⊃ su6 ⊕su2(R) . . . . . . . . . . . . . . 1409
741 Branching rules of E6 ⊃ su3 ⊕su3 ⊕ su3(R) . . . . . . . . . . . 1410
742 Branching rules of E6 ⊃ F4(S) . 1416
743 Branching rules of E6 ⊃ su3 ⊕G2(S) . . . . . . . . . . . . . . . 1416
744 Branching rules of E6 ⊃ usp8(S) 1419
745 Branching rules of E6 ⊃ G2(S) . 1420
746 Branching rules of E6 ⊃ su3(S) . 1421
747 Branching rules of E7 ⊃ E6 ⊕u1(R) . . . . . . . . . . . . . . . 1423
748 Branching rules of E7 ⊃ su8(R) . 1424
749 Branching rules of E7 ⊃ so12 ⊕su2(R) . . . . . . . . . . . . . . 1424
750 Branching rules of E7 ⊃ su6 ⊕su3(R) . . . . . . . . . . . . . . 1425
751 Branching rules of E7 ⊃ su2 ⊕F4(S) . . . . . . . . . . . . . . . 1426
14
-
752 Branching rules of E7 ⊃ G2 ⊕usp6(S) . . . . . . . . . . . . . . 1427
753 Branching rules of E7 ⊃ su2 ⊕G2(S) . . . . . . . . . . . . . . . 1428
754 Branching rules of E7 ⊃ su3(S) . 1429755 Branching rules of E7 ⊃ su2 ⊕
su2(S) . . . . . . . . . . . . . . . 1430756 Branching rules of E7 ⊃ su2(S) . 1432757 Branching rules of E7 ⊃ su2(S) . 1433758 Branching rules of E8 ⊃ so16(R) 1434759 Branching rules of E8 ⊃ su5 ⊕
su5(R) . . . . . . . . . . . . . . 1434760 Branching rules of E8 ⊃ E6 ⊕
su3(R) . . . . . . . . . . . . . . 1436761 Branching rules of E8 ⊃ E7 ⊕
su2(R) . . . . . . . . . . . . . . 1436762 Branching rules of E8 ⊃ su9(R) . 1437763 Branching rules of E8 ⊃ G2 ⊕
F4(S) . . . . . . . . . . . . . . . 1437764 Branching rules of E8 ⊃ su2 ⊕
su3(S) . . . . . . . . . . . . . . . 1438765 Branching rules of E8 ⊃ usp4(S) 1439766 Branching rules of E8 ⊃ su2(S) . 1440767 Branching rules of E8 ⊃ su2(S) . 1440768 Branching rules of E8 ⊃ su2(S) . 1441769 Branching rules of F4 ⊃ so9(R) . 1442770 Branching rules of F4 ⊃ su3 ⊕
su3(R) . . . . . . . . . . . . . . 1443771 Branching rules of F4 ⊃ su2 ⊕
usp6(R) . . . . . . . . . . . . . . 1445772 Branching rules of F4 ⊃ su2(S) . 1446773 Branching rules of F4 ⊃ su2 ⊕
G2(S) . . . . . . . . . . . . . . . 1447774 Branching rules of G2 ⊃ su3(R) 1449775 Branching rules of G2 ⊃ su2 ⊕
su2(R) . . . . . . . . . . . . . . 1452776 Branching rules of G2 ⊃ su2(S) . 1458777 Tensor products of A1 . . . . . . 1461778 Tensor products of A2 . . . . . . 1475779 Tensor products of A3 . . . . . . 1493780 Tensor products of A4 . . . . . . 1506781 Tensor products of A5 . . . . . . 1523782 Tensor products of A6 . . . . . . 1539783 Tensor products of A7 . . . . . . 1553784 Tensor products of A8 . . . . . . 1566785 Tensor products of A9 . . . . . . 1583786 Tensor products of A10 . . . . . . 1598787 Tensor products of A11 . . . . . . 1611788 Tensor products of A12 . . . . . . 1625
789 Tensor products of A13 . . . . . . 1637
790 Tensor products of A14 . . . . . . 1650
791 Tensor products of A15 . . . . . . 1664
792 Tensor products of B3 . . . . . . 1680
793 Tensor products of B4 . . . . . . 1681
794 Tensor products of B5 . . . . . . 1684
795 Tensor products of B6 . . . . . . 1686
796 Tensor products of B7 . . . . . . 1689
797 Tensor products of B8 . . . . . . 1691
798 Tensor products of B9 . . . . . . 1692
799 Tensor products of B10 . . . . . . 1694
800 Tensor products of B11 . . . . . . 1696
801 Tensor products of B12 . . . . . . 1698
802 Tensor products of B13 . . . . . . 1700
803 Tensor products of B14 . . . . . . 1702
804 Tensor products of B15 . . . . . . 1704
805 Tensor products of C2 . . . . . . 1708
806 Tensor products of C3 . . . . . . 1709
807 Tensor products of C4 . . . . . . 1712
808 Tensor products of C5 . . . . . . 1714
809 Tensor products of C6 . . . . . . 1716
810 Tensor products of C7 . . . . . . 1720
811 Tensor products of C8 . . . . . . 1723
812 Tensor products of C9 . . . . . . 1725
813 Tensor products of C10 . . . . . . 1727
814 Tensor products of C11 . . . . . . 1730
815 Tensor products of C12 . . . . . . 1733
816 Tensor products of C13 . . . . . . 1735
817 Tensor products of C14 . . . . . . 1737
818 Tensor products of C15 . . . . . . 1740
819 Tensor products of D4 . . . . . . 1743
820 Tensor products of D5 . . . . . . 1803
821 Tensor products of D6 . . . . . . 1806
822 Tensor products of D7 . . . . . . 1808
823 Tensor products of D8 . . . . . . 1810
824 Tensor products of D9 . . . . . . 1812
825 Tensor products of D10 . . . . . . 1814
826 Tensor products of D11 . . . . . . 1816
827 Tensor products of D12 . . . . . . 1820
828 Tensor products of D13 . . . . . . 1822
829 Tensor products of D14 . . . . . . 1824
830 Tensor products of D15 . . . . . . 1826
831 Tensor products of D16 . . . . . . 1828
832 Tensor products of E6 . . . . . . 1831
833 Tensor products of E7 . . . . . . 1834
834 Tensor products of E8 . . . . . . 1836
835 Tensor products of F4 . . . . . . 1843
836 Tensor products of G2 . . . . . . 1847
15
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1 Introduction
We will discuss finite-dimensional Lie algebras and their representations for unified model build-ing. There is already a good report Ref. [1] of Lie algebras and their representations for particlephysicists with the title “Group Theory for Unified Model Building” written by R. Slansky. Thepaper contains almost all knowledge for usual model building to construct grand unified modelsin four-dimensional spacetime with the finite degrees of freedom of internal space or mattercontent. However, in modern sense, several exceptional cases have emerged. In this paper, wefind missing information for further unified model building, but it does not contain definitions ofLie algebras, their theorems and lemmas and their proofs, completely. They can be confirmed inDynkin’s original papers Refs. [2–5], a Dynkin’s paper’s brief review “Semi-Simple Lie Algebrasand Their Representations” written by R. N. Cahn Ref. [6], or books about Lie algebras and Liegroups, e.g., Refs. [7–9]. Introductory-level knowledge about Lie algebras and groups is given inRef. [10].
At present, the irreducible representations of simple Lie algebras, whose rank is not exceeding8, are summarized in Ref. [11] written by W. G. McKay and J. Patera with the title “Tablesof Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras.” TheTable 1 of the book includes tables of dimensions [3, 7], second order and forth order Dynkinindices [12, 13], and type of representations, i.e. complex, self-conjugate, real, and pseudo-realrepresentations [11]. For each simple Lie algebra of rank 2 ≤ n ≤ 8 two pages are devoted,where two pages are used for E7 and E8.
It is known that branching rules of Lie algebras and their Lie subalgebras can be calculatedby using their corresponding projection matrices. The projection matrices of Lie algebras andtheir maximal regular and special Lie subalgebras are listed by W. Mckay et al. in Ref. [14] upto rank 8, where they do not contain u1 charges. Also, several generic projection matrices ofrank-n classical Lie algebras are derived by using Weyl group orbits in Refs. [15–18].
The Table 2 in Ref. [11] is devoted to the tables of the branching rules of up to 5,000-dimensional representations for the classical Lie algebras An, Bn, Cn, and Dn, and up to 10,000-dimensional representations for the five exceptional algebras E6, E7, E8, F4, and G2 by usingprojection matrices in Ref. [14]. Its tables do not contain u1 charges because u1 is not a semi-simple. R. Slansky give us its information in Ref. [1], but it is very limited.
There are useful public codes to calculate some features of irreducible representations ofLie algebras such as dimensions [3, 7], conjugacy classes [3, 19], (second order) Dynkin indices[3], quadratic Casimir invariants, and anomaly coefficients [20–22], and also tensor products,branching rules and some projection matrices [14]. For example, Susyno program [23] is aMathematica package, LieART [24] is also a Mathematica package, and LiE [25] is a C program.
Before we finish the introduction, we give some examples when we need Lie algebras, theirrepresentations, etc. in physics. For example, to construct a consistent chiral SU(n) gaugetheory [26] in four dimension, we must consider its SU(n) chiral gauge anomaly. (See e.g.,Ref. [27].) We can easily check whether a matter content is anomaly-free or not by using anomalycoefficients. To calculate the renormalization group equation for gauge coupling constant, weneed (second order) Dynkin indices of irreducible representations. (See e.g., Refs. [28–31].) Thenotion of types of representations (complex, self-conjugate, real, and pseudo-real representations)is important to find special Lie subalgebras. The notion of conjugacy classes is also useful toclassify irreducible representations and to calculate tensor products.
Tensor products of Lie algebras are also important for model building to write down invariantaction under certain symmetry transformation. There are several calculation techniques by usinge.g., Dynkin labels and Dynkin diagrams in Refs. [6,11] and Weyl group orbits in Refs. [32,33].
Branching rules are essential not only to construct grand unified theories but also to considermodels including explicit or spontaneous broken symmetries. For example, in an SU(5) grandunified theory discussed in Ref. [34], we have to know how to decompose representations ofSU(5) in its matter content into representations of GSM := SU(3) × SU(2) × U(1). E.g., the
16
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gauge bosons with the adjoint representation 24 of SU(5) are decomposed into the SM gaugebosons with (8,1)(0) ⊕ (1,3)(0) ⊕ (1,1)(0) of GSM and the so-called X and Y gauge bosonswith ⊕(3,2)(5) ⊕ (3,2)(−5) of GSM , where we take a normalization of U(1) charges as all theU(1) charges are integers.
A purpose of this paper is to provide basic and useful information about some propertiesof irreducible representations of Lie algebras, branching rules of Lie algebras and their subal-gebras, and tensor products, where the properties of irreducible representations of Lie algebrasinclude dimensions, quadratic Casimir invariants, Dynkin indices, anomaly coefficients, conju-gacy classes, and types of representations. The branching rules contain not only semi-simplesubalgebra but also u1 charges because the information is important for model building. An-other purpose is to inform one of calculation methods to obtain the above things. By using theabove information, we will check what kind of Lie algebras can be applied for grand unificationin 4 and 5 dimensions.
This paper is organized as follows. In Sec. 2, we check basics of Lie algebras and theirsubalgebras such as Dynkin diagrams, Cartan matrices, and notion of regular, special, and max-imal subalgebras. In Sec. 3, we see several features of representations of Lie algebras such asconjugacy classes, types of representations, Weyl dimension formulas, Dynkin indices, Casimirinvariants, and anomaly coefficients. In Sec. 4, we consider how to decompose irreducible repre-sentations of Lie algebras into irreducible representations of their Lie subalgebras. In Sec. 5, wediscuss a method to calculate tensor products of two irreducible representations of a Lie algebramainly by using its Dynkin diagram. In Sec. 6, several features of rank-n classical Lie algebrasand the five exceptional Lie algebras are summarized. This section includes several new resultsabout generic projection matrices of rank-n Lie algebras and their Lie subalgebras. In Sec. 7,we show which Lie algebras can be applied for grand unification in general by using the abovediscussion. Section 8 is devoted to a summary and discussion. Appendix A contains tables ofrepresentations of classical Lie algebras An, Bn, Cn, Dn (n = 1, 2, · · · , 15), and D16 and theexceptional Lie algebras E6, E7, E8, F4, and G2, which include dimensions, conjugacy classes,Dynkin indices, quadratic Casimir invariants, anomaly coefficients, and types of representa-tions. The tables are partially calculated by Susyno program [23]. Appendix B contains tablesof positive roots of some classical Lie algebras and the exceptional Lie algebras. Appendix Ccontains tables of weight diagrams of several representations of some classical Lie algebras andthe exceptional Lie algebras. Appendix D contains tables of all projection matrices of classicalLie algebras An, Bn, Cn, Dn (n = 1, 2, · · · , 15), and D16 and the exceptional Lie algebras E6,E7, E8, F4, and G2 and their maximal regular and special subalgebras. Appendix E containstables of branching rules of classical Lie algebras An, Bn, Cn, Dn (n = 1, 2, · · · , 15), and D16and the exceptional Lie algebras E6, E7, E8, F4, and G2 and their maximal regular and specialsubalgebras. The tables are also obtained by Susyno program [23] by using projection matricesshown in Appendix D. Appendix F contains tables of tensor products of classical Lie algebrasAn, Bn, Cn, Dn (n = 1, 2, · · · , 15), and D16 and the exceptional Lie algebras E6, E7, E8, F4,and G2. The tables are also calculated by Susyno program [23].
2 Lie algebras and their subalgebras
First, we list up some technical terms about Lie algebras and subalgebras. (For more detail, seee.g., Refs. [6, 35].)
• A Lie algebra g is an algebra such that its map [, ]: g ⊗ g → g satisfies the followingproperties:
(1) [x, y] = −[y, x] for ∀x, y ∈ g (antisymmetry);
(2) [x, [y, z]] + [y, [z, x]] + [z, [x, y]]] = 0 for ∀x, y, z ∈ g (Jacobi identity).
• A Lie subalgebra h (h ⊆ g ) of the Lie algebra g itself is a Lie algebra.
17
-
• A proper Lie subalgebra h is a Lie subalgebra if h 6= g; i.e., h ⊂ g;
• An ideal h of the Lie algebra g is a subalgebra that satisfies the property [g, h] ⊆ h.
• An Abelian Lie algebra is a Lie algebra that satisfies [g, g] = 0.
• A simple Lie algebra is a Lie algebra that does not contain proper ideals and that is notAbelian algebra u1.
• A semi-simple Lie algebra is an algebra that is the direct sum of simple Lie algebras.
• A non-semi-simple Lie algebra is an algebra that is the direct sum of a semi-simple andan Abelian Lie algebra.
Let us classify a finite-dimensional simple Lie algebra g by using a Cartan matrix Aij(g),
Aij(g) := 2(αi, αj)
(αj , αj), (2.1)
where αi(i = 1, 2, · · · , n) are the simple roots of rank-n algebra g, (∗, ∗) is a scalar product onthe root space. Here we define a Cartan matrix of a simple Lie algebra g as it satisfying thefollowing conditions:
(C0) Aij(g) ∈ Z,
(C1) Aii(g) = 2,
(C2) Aij(g) = 0 ⇔ Aji(g) = 0,
(C3) Aij(g) ∈ Z≤0 for i 6= j,
(C4) detAij(g) > 0,
(C5) irreducible. (2.2)
(See e.g., Refs. [6, 35].)We check what kind of Cartan matrices Aij(g) satisfy the above conditions. For rank-2, we
can write a Cartan matrix Aij(g) as
A(g) =
(
2 a12a21 2
)
. (2.3)
Its determinant detAij(g) must satisfy the following condition:
detAij(g) = 4− a12a21 > 0. (2.4)
Thus, the possible sets of (a12, a21) are
(a12, a21) = (0, 0), (−1,−1), (−1,−2), (−1,−3), (−2,−1), (−3,−1), (2.5)
where (0, 0) does not give us a simple Lie algebra, but D2 ≃ A1 ⊕A1. For usual notations, theycorrespond to D2, A2, C2(≃ B2), G2, B2, G2, respectively. (Note that for rank-2 Lie algebras, aCartan matrix is the same as its transpose one, where it can be understood e.g., by using theirDynkin diagrams discussed later; thus, the matrix with (a12, a21) = (−1,−2) is the same asthat with (a12, a21) = (−2,−1); the matrix with (a12, a21) = (−1,−3) is the same as that with(a12, a21) = (−3,−1).) The explicit matrices of A2, B2(≃ C2), C2, D2(≃ A1 ⊕A1), and G2 are
A(A2) =
(
2 −1−1 2
)
, A(B2) =
(
2 −2−1 2
)
, A(C2) =
(
2 −1−2 2
)
,
A(D2) =
(
2 00 2
)
, A(G2) =
(
2 −3−1 2
)
. (2.6)
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-
Notice that for (a12, a21) = (−2,−2), (−1,−4), the Cartan matrices are
A(A(1)1 ) =
(
2 −2−2 2
)
, A(A(2)1 ) =
(
2 −4−1 2
)
. (2.7)
They lead to detA(g) = 0, where the superscript of the algebra (r), e.g., (1) of A(1)1 , is cor-
responding to so-called Coxeter label. Thus, they are not finite-dimensional Lie algebras, butAffine Lie algebras. The class of the former matrix gives us important information about aso-called extended Dynkin diagram. (See e.g., [35, 36] for Affine Lie algebras and Kac-Moodyalgebras.)
For rank-3, we can write a Cartan matrix as
Aij(g) =
2 a12 a13a21 2 a23a31 a32 2
. (2.8)
Its determinant detAij(g) must satisfy the following condition:
detAij(g) = 8 + a12a23a31 + a21a32a13 − 2 (a12a21 + a13a31 + a23a32) > 0. (2.9)
Thus, the possible sets of (a12, a21; a13, a31; a23, a32) are
(a12, a21; a13, a31; a23, a32) =(−1,−1; 0, 0;−1,−1), (−1,−1; 0, 0;−2,−1),
(−1,−1; 0, 0;−1,−2), (−1,−1;−1,−1; 0, 0). (2.10)
For usual notations, they correspond to A3, B3, C3, and D3(≃ A3), respectively. (Note thatfor rank-3 Lie algebras, the Cartan matrix of A3 is the same as that of D3, where it can beunderstood e.g., by using their Dynkin diagrams discussed later.) The explicit matrices of A3,B3, C3, D3(≃ A3), and G2 are
A(A3) =
2 −1 0−1 2 −10 −1 2
, A(B3) =
2 −1 0−1 2 −20 −1 2
,
A(C3) =
2 −1 0−1 2 −10 −2 2
, A(D3) =
2 −1 −1−1 2 0−1 0 2
. (2.11)
Notice that for appropriate sets (a12, a21; a13, a31; a23, a32), the Cartan matrices with detA(g) = 0are
A(A(1)2 ) =
2 −1 −1−1 2 −1−1 −1 2
, A(D(2)5 ) =
2 −1 0−2 2 −20 −1 2
,
A(A(2)4 ) =
2 −2 0−1 2 −20 −1 2
, A(C(1)2 ) =
2 −2 0−1 2 −10 −2 2
,
A(G(1)2 ) =
2 −1 0−1 2 −30 −1 2
, A(D(3)4 ) =
2 −3 0−1 2 −10 −1 2
. (2.12)
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For rank-4, algebras A4, B4, C4, D4, and F4 satisfy the condition detA(g) > 0. The explicitmatrices of A4, B4, C4, D4, and F4 are
A(A4) =
2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2
, A(B4) =
2 −1 0 0−1 2 −1 00 −1 2 −20 0 −1 2
,
A(C4) =
2 −1 0 0−1 2 −1 00 −1 2 −10 0 −2 2
, A(D4) =
2 −1 0 0−1 2 −1 −10 −1 2 00 −1 0 2
,
A(F4) =
2 −1 0 0−1 2 −2 00 −1 2 −10 0 −1 2
. (2.13)
The algebras A(1)3 , B
(1)3 , C
(1)3 , A
(2)5 , A
(2)6 , and D
(2)6 satisfy the condition detA(g) = 0. The
explicit matrices of them are
A(A(1)3 ) =
2 −1 0 −1−1 2 −1 00 −1 2 −1−1 0 −1 2
, A(B(1)3 ) =
2 0 −1 00 2 −1 0−1 −1 2 −20 0 −1 2
,
A(A(2)5 ) =
2 0 −1 00 2 −1 0−1 −1 2 −20 0 −1 2
, A(C(1)3 ) =
2 −2 0 0−1 2 −1 00 −1 2 −10 0 −2 2
,
A(D(2)6 ) =
2 0 −1 00 2 −1 0−1 −1 2 −10 0 −2 2
, A(A(2)6 ) =
2 −2 0 0−1 2 −1 00 −1 2 −20 0 −1 2
. (2.14)
For rank-5, algebras A5, B5, C5, and D5 satisfy the condition detA(g) > 0. The explicitmatrices of A5, B5, C5, and D5 are
A(A5) =
2 −1 0 0 0−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −10 0 0 −1 2
, A(B5) =
2 −1 0 0 0−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −20 0 0 −1 2
,
A(C5) =
2 −1 0 0 0−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −10 0 0 −2 2
, A(D5) =
2 −1 0 0 0−1 2 −1 0 00 −1 2 −1 −10 0 −1 2 00 0 −1 0 0
. (2.15)
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The algebras A(1)4 , A
(1)3 , A
(1)3 , F
(1)4 , A
(2)7 , A
(2)8 , D
(2)7 , F
(1)4 , and E
(2)6 satisfy the condition
detA(g) = 0. The explicit matrices of them are
A(A(1)4 ) =
2 −1 0 0 −1−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −1−1 0 0 −1 2
, A(B(1)4 ) =
2 0 −1 0 00 2 −1 0 0−1 −1 2 −1 00 0 −1 2 −20 0 0 −1 2
,
A(C(1)4 ) =
2 −2 0 0 0−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −10 0 0 −2 2
, A(D(1)4 ) =
2 −1 0 0 0−1 2 −1 −1 −10 −1 2 0 00 −1 0 2 00 −1 0 0 2
,
A(F(1)4 ) =
2 −1 0 0 0−1 2 −2 0 00 −1 2 −1 00 0 −1 2 −10 0 0 −1 2
. (2.16)
We can also check rank-n (n ≥ 4) algebras by using the same technique. The condition (d)in Eq. (2.2) detAij > 0 does not constrain the rank of the classical algebras An, Bn, Cn, andDn, but it strongly constrains the rank of the exceptional algebras En, Fn, and Gn. In otherwords, the rank of the classical algebras is unlimited for large n, but the rank of the exceptionalalgebras En, Fn, and Gn is limited by n = 8, n = 4, and n = 2, respectively. By using words of
the Affine Lie algebras, E9 = E(1)8 , F5 = F
(1)4 , and G3 = G
(1)2 .
Note that detA(An) = n + 1 (n ≥ 1), detA(Bn) = 2 (n ≥ 1), detA(Cn) = 2 (n ≥ 1),detA(Dn) = 4 (n ≥ 3), detA(En) = 9− n (n ≥ 6), detA(F4) = 1, and detA(G2) = 1.
2.1 (Extended) Dynkin diagrams and Cartan matrices
Finite-dimensional Lie algebras are classified into An, Bn, Cn, Dn (n ≥ 1), En (n = 6, 7, 8), F4,and G2. For their associate groups, we only consider compact groups. We sometimes denotethe Lie algebras An = sun+1, Bn = so2n+1, Cn = usp2n, and Dn = so2n. Also, their associatecompact groups of An, Bn, Cn, Dn, En, F4, and G2 are SU(n + 1), SO(2n + 1), USp(2n),SO(2n), En, F4, and G2, respectively.
The simple Lie algebras, their associate compact groups, and their extended Dynkin diagramsare summarized in the following table.
Table 1: Extended Dynkin diagrams
Algebra Group Rank Extended Dynkin diagram
An = sun+1 SU(n+ 1)∀n ◦
1
◦❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧
x
◦❘❘
❘❘❘❘
❘❘❘❘
❘❘❘❘
❘❘❘
◦2
◦3
· · · ◦n−1
◦n
Bn = so2n+1 SO(2n+ 1)∀n ◦
2
◦ ❖❖❖❖❖❖1
◦ ♦♦♦♦♦♦
x
◦3
· · · ◦n−1
•n
Cn = usp2n USp(2n)∀n ◦
x
•1
•2
· · · •n−1
◦n
21
-
Table 1 (continued)
Algebra Group Rank Extended Dynkin diagram
Dn = so2n SO(2n)∀n ◦
2
◦ ❖❖❖❖❖❖1
◦ ♦♦♦♦♦♦
x
◦3
· · · ◦n−2
◦♦♦♦♦♦♦ n−1
◦❖❖
❖❖❖❖
n
E6 E6 6 ◦1
◦2
◦3
◦ 6
◦ x
◦4
◦5
E7 E7 7 ◦x
◦1
◦2
◦3
◦ 7
◦4
◦5
◦6
E8 E8 8 ◦1
◦2
◦3
◦ 8
◦4
◦5
◦6
◦7
◦x
F4 F4 4 ◦x
◦1
◦2
•3
•4
G2 G2 2 ◦x
◦1
•2
where Group stands for a compact group. Since D2 ≃ A1 ⊕ A1 (so4 ≃ su2 ⊕ su2), D2 = so4is not a simple Lie algebra. Note that we took the same notation in Refs. [1, 10] to write the(extended) Dynkin diagrams. The Dynkin diagram of a simple algebra is connected, and theDynkin diagram of a non-simple algebra is disconnected. Each simple algebra has simple rootsof one or two different lengths. The black circles denote the shorter roots.
The Dynkin diagrams have the same information of their corresponding Cartan matricesA(G) of Lie algebras g. (See, e.g., Ref. [1, 3, 8] in detail.) The Cartan matrices of simple Liealgebras are listed in the following table.
Table 2: Cartan matrices
Algebra Group Rank Cartan matrix
An = sun+1 SU(n+ 1)∀n A(An) =
2 −1 0 · · · 0 0 0−1 2 −1 · · · 0 0 00 −1 2 · · · 0 0 0...
......
. . ....
......
0 0 0 · · · 2 −1 00 0 0 · · · −1 2 −10 0 0 · · · 0 −1 2
22
-
Table 2 (continued)
Algebra Group Rank Cartan matrix
Bn = so2n+1 SO(2n+ 1)∀n A(Bn) =
2 −1 0 · · · 0 0 0−1 2 −1 · · · 0 0 00 −1 2 · · · 0 0 0...
......
. . ....
......
0 0 0 · · · 2 −1 00 0 0 · · · −1 2 −20 0 0 · · · 0 −1 2
Cn = usp2n USp(2n)∀n A(Cn) =
2 −1 0 · · · 0 0 0−1 2 −1 · · · 0 0 00 −1 2 · · · 0 0 0...
......
. . ....
......
0 0 0 · · · 2 −1 00 0 0 · · · −1 2 −10 0 0 · · · 0 −2 2
Dn = so2n SO(2n)∀n A(Dn) =
2 −1 0 · · · 0 0 0−1 2 −1 · · · 0 0 00 −1 2 · · · 0 0 0...
......
. . ....
......
0 0 0 · · · 2 −1 −10 0 0 · · · −1 2 00 0 0 · · · −1 0 2
E6 E6 6 A(E6) =
2 −1 0 0 0 0−1 2 −1 0 0 00 −1 2 −1 0 −10 0 −1 2 −1 00 0 0 −1 2 00 0 −1 0 0 2
E7 E7 7 A(E7) =
2 −1 0 0 0 0 0−1 2 −1 0 0 0 00 −1 2 −1 0 0 −10 0 −1 2 −1 0 00 0 0 −1 2 −1 00 0 0 0 −1 2 00 0 −1 0 0 0 2
E8 E8 8 A(E8) =
2 −1 0 0 0 0 0 0−1 2 −1 0 0 0 0 00 −1 2 −1 0 0 0 −10 0 −1 2 −1 0 0 00 0 0 −1 2 −1 0 00 0 0 0 −1 2 −1 00 0 0 0 0 −1 2 00 0 −1 0 0 0 0 2
F4 F4 4 A(F4) =
2 −1 0 0−1 2 −2 00 −1 2 −10 0 −1 2
G2 G2 2 A(G2) =
(
2 −3−1 2
)
23
-
The inverse Cartan matrices of simple Lie algebras g in the following table are defined by
G(g)ij :=(
A(g)−1)
ij
(αj , αj)
2, (2.17)
The matrices are useful when we calculate the Weyl dimension formulas, second order Casimirinvariants, etc.
Table 3: Inverse Cartan matrices
Algebra Inverse Cartan matrix
An G(An) =1
n+1
1 · n 1 · (n− 1) 1 · (n− 2) · · · 1 · 2 1 · 11 · (n− 1) 2 · (n− 1) 2 · (n− 2) · · · 2 · 2 2 · 11 · (n− 2) 2 · (n− 2) 3 · (n− 2) · · · 3 · 2 3 · 1
......
.... . .
......
1 · 2 2 · 2 3 · 2 · · · (n− 1)