Symmetry, Representations, and Invariants

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Springer Science & Business Media, Jul 30, 2009 - Mathematics - 716 pages
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Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications.

The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case.

Key Features of Symmetry, Representations, and Invariants:

• Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus

• Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux)

• Self-contained chapters, appendices, comprehensive bibliography

• More than 350 exercises (most with detailed hints for solutions) further explore main concepts

• Serves as an excellent main text for a one-year course in Lie group theory

• Benefits physicists as well as mathematicians as a reference work

 

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Contents

III
1
IV
3
V
7
VI
8
VII
11
VIII
13
IX
14
X
15
CLVI
314
CLVII
315
CLVIII
316
CLX
321
CLXI
322
CLXII
323
CLXIV
325
CLXV
329

XI
16
XII
17
XIII
18
XV
19
XVI
23
XVII
28
XVIII
30
XIX
31
XX
34
XXI
35
XXIII
39
XXIV
43
XXV
45
XXVI
47
XXVIII
49
XXIX
53
XXX
54
XXXI
55
XXXIII
57
XXXIV
58
XXXV
61
XXXVI
62
XXXVIII
65
XXXIX
67
XL
68
XLI
69
XLII
70
XLIII
72
XLIV
76
XLV
77
XLVII
79
XLVIII
81
XLIX
83
LI
84
LII
86
LIII
88
LIV
90
LV
91
LVII
95
LVIII
99
LIX
104
LX
106
LXI
108
LXIII
114
LXIV
118
LXV
122
LXVI
125
LXVIII
127
LXIX
128
LXX
132
LXXI
138
LXXII
141
LXXIII
147
LXXIV
148
LXXV
153
LXXVI
157
LXXVII
158
LXXVIII
161
LXXIX
163
LXXXI
166
LXXXII
169
LXXXIII
173
LXXXIV
174
LXXXV
175
LXXXVI
180
LXXXVII
181
LXXXVIII
182
LXXXIX
184
XCI
187
XCII
191
XCIII
195
XCV
197
XCVI
199
XCVII
200
XCVIII
203
XCIX
204
C
206
CIII
208
CIV
209
CV
211
CVI
214
CVII
215
CVIII
216
CIX
217
CX
218
CXI
221
CXII
222
CXIII
224
CXIV
225
CXV
226
CXVII
228
CXVIII
234
CXIX
237
CXX
238
CXXI
246
CXXIII
247
CXXIV
248
CXXV
256
CXXVII
258
CXXIX
259
CXXXI
268
CXXXII
272
CXXXIII
273
CXXXIV
275
CXXXV
278
CXXXVII
282
CXXXVIII
284
CXXXIX
287
CXL
290
CXLI
292
CXLII
293
CXLIV
295
CXLV
296
CXLVI
298
CXLVIII
301
CLI
303
CLII
307
CLIII
310
CLIV
311
CLV
312
CLXVI
334
CLXVII
337
CLXVIII
339
CLXIX
342
CLXXI
344
CLXXII
346
CLXXIII
347
CLXXIV
352
CLXXV
353
CLXXVI
354
CLXXVIII
356
CLXXIX
358
CLXXX
360
CLXXXI
361
CLXXXII
362
CLXXXIII
363
CLXXXV
366
CLXXXVI
370
CLXXXVIII
371
CLXXXIX
372
CXC
373
CXCII
375
CXCIII
378
CXCIV
379
CXCV
384
CXCVI
386
CXCVII
388
CXCVIII
391
CXCIX
394
CC
396
CCI
399
CCIII
401
CCIV
405
CCV
406
CCVI
407
CCVIII
412
CCIX
414
CCX
416
CCXI
418
CCXII
421
CCXIII
425
CCXIV
426
CCXV
435
CCXVII
436
CCXVIII
440
CCXIX
442
CCXX
446
CCXXI
451
CCXXIII
453
CCXXIV
455
CCXXV
458
CCXXVI
460
CCXXVII
461
CCXXIX
463
CCXXX
464
CCXXXI
469
CCXXXII
475
CCXXXIII
476
CCXXXIV
479
CCXXXV
481
CCXXXVI
484
CCXXXVII
485
CCXXXVIII
490
CCXXXIX
491
CCXLI
493
CCXLII
496
CCXLIII
497
CCXLIV
500
CCXLVII
501
CCXLVIII
506
CCXLIX
507
CCL
510
CCLI
516
CCLII
519
CCLIV
520
CCLV
522
CCLVI
524
CCLVII
525
CCLVIII
526
CCLIX
530
CCLX
531
CCLXII
536
CCLXIII
538
CCLXV
540
CCLXVI
541
CCLXVII
543
CCLXIX
545
CCLXX
546
CCLXXI
548
CCLXXII
549
CCLXXIII
550
CCLXXIV
551
CCLXXV
552
CCLXXVI
553
CCLXXVII
554
CCLXXVIII
557
CCLXXIX
566
CCLXXXI
575
CCLXXXII
585
CCLXXXIII
587
CCLXXXIV
588
CCLXXXV
589
CCLXXXVI
590
CCLXXXVII
593
CCLXXXVIII
597
CCLXXXIX
601
CCXC
605
CCXCI
607
CCXCII
609
CCXCIII
611
CCXCVI
615
CCXCVII
616
CCXCVIII
617
CCC
619
CCCI
621
CCCII
697
CCCIII
705
CCCIV
709
Copyright

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About the author (2009)

Dr. Roe Goodman has been a professor for 45 years, and is currently a professor at Rutgers University. He as travelled internationally as a visiting professor to numerous prestigious universities. He has authored two books, and co-authored the previous highly successful version of this book. He has edited two books, and has published over 30 articles in refereed journals.

Dr. Nolan R. Wallach has been a professor since 1966, and is currently a professor at the University of California, San Diego. He has authored or co-authored over 100 publications. In 1992, he was the Chair of the Editorial Boards Committee of the American Mathematical Society. He has been an editor of Birkhäuser's series, Mathematics: Theory and Applications, since 2001. In addition to numerous other prizes, recognitions and professional organization affiliations, in 2004 he became and Elected Fellow of the American Academy of Arts and Sciences.

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