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: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.
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 Birkhauser'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.
Lie Groups and Algebraic Groups.- Structure of Classical Groups.- Highest-Weight Theory.- Algebras and Representations.- Classical Invariant Theory.- Spinors.- Character Formulas.- Branching Laws.- Tensor Representations of GL(V).- Tensor Representations of O(V) and Sp(V).- Algebraic Groups and Homogeneous Spaces.- Representations on Spaces of Regular Functions.