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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: (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.
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.
From the reviews:
"This is a terrific book, succeeding in its considerable ambitions, and doing so in the remarkable style ... . Symmetry, Representations, and Invariants is indeed capable of instructing the reader in the three themes given in its title, taking him from elementary and foundational notions to very advanced material ... . Symmetry, Representations, and Invariants promises to be a graduate text of major importance." (Michael Berg, The Mathematical Association of America, August, 2009)
"The book under review is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups. ... can be used as a source for various kinds of courses. ... supported by the rich collections of exercises (mostly with detailed hints for solutions) accompanying each section. Local reading is well supported by the structure of the book. The book can be recommended for a ... wide audience of readers: for graduate and postgraduate students as well as for researchers as a reference work." (Sergei Platonov, Zentralblatt MATH, Vol. 1173, 2009)
"The book under review is a substantial rewriting and extension of the authors' previous work 'Representations and Invariants of the Classical Groups'. It is a thorough and comprehensive treatment of the geometry of Lie and algebraic group actions. ... More than 350 exercises are presented. This monograph is written with great care and provides an excellent basis for a variety of courses from this field." (M. Kunzinger, Monatshefte für Mathematik, Vol. 163 (1), May, 2011)
"The book will serve as a useful reference for a broad range of mathematicians, as well as physicists seeking a rigorous, systematic development of the representation theory underlying much of modern quantum mechanics. Detailed and careful statements of definitions, theorems, and proofs are provided, as well as over 350 exercises, many of them substantial. ... can be used as thebasis for graduate-level courses in Lie groups and algebraic groups, representation theory, invariant theory, and a variety of applications." (Peter J. Olver, SIAM Review, Vol. 53 (3), 2011)
"This volume is clearly a labour of love on the part of the authors, who have obviously thought very carefully about the best way to explain and motivate this often sophisticated material. ... The authors have taken great pains to make this book a textbook as well as a useful reference. ... It should be in the library of every university with a graduate mathematics program, as well as on the shelf of every lecturer teaching, or with research interests in, the material covered here." (Mark Hunacek, The Mathematical Gazette, Vol. 96 (536), July, 2012)
"This is a terrific book, succeeding in its considerable ambitions, and doing so in the remarkable style ... . Symmetry, Representations, and Invariants is indeed capable of instructing the reader in the three themes given in its title, taking him from elementary and foundational notions to very advanced material ... . Symmetry, Representations, and Invariants promises to be a graduate text of major importance." (Michael Berg, The Mathematical Association of America, August, 2009)
"The book under review is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups. ... can be used as a source for various kinds of courses. ... supported by the rich collections of exercises (mostly with detailed hints for solutions) accompanying each section. Local reading is well supported by the structure of the book. The book can be recommended for a ... wide audience of readers: for graduate and postgraduate students as well as for researchers as a reference work." (Sergei Platonov, Zentralblatt MATH, Vol. 1173, 2009)
"The book under review is a substantial rewriting and extension of the authors' previous work 'Representations and Invariants of the Classical Groups'. It is a thorough and comprehensive treatment of the geometry of Lie and algebraic group actions. ... More than 350 exercises are presented. This monograph is written with great care and provides an excellent basis for a variety of courses from this field." (M. Kunzinger, Monatshefte für Mathematik, Vol. 163 (1), May, 2011)
"The book will serve as a useful reference for a broad range of mathematicians, as well as physicists seeking a rigorous, systematic development of the representation theory underlying much of modern quantum mechanics. Detailed and careful statements of definitions, theorems, and proofs are provided, as well as over 350 exercises, many of them substantial. ... can be used as thebasis for graduate-level courses in Lie groups and algebraic groups, representation theory, invariant theory, and a variety of applications." (Peter J. Olver, SIAM Review, Vol. 53 (3), 2011)
"This volume is clearly a labour of love on the part of the authors, who have obviously thought very carefully about the best way to explain and motivate this often sophisticated material. ... The authors have taken great pains to make this book a textbook as well as a useful reference. ... It should be in the library of every university with a graduate mathematics program, as well as on the shelf of every lecturer teaching, or with research interests in, the material covered here." (Mark Hunacek, The Mathematical Gazette, Vol. 96 (536), July, 2012)