Symmetry, representations, and invariants

Roe Goodman, Nolan R. Wallach (auth.)038779851X, 9780387798516

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

Table of contents :
Front Matter….Pages i-xix
Lie Groups and Algebraic Groups….Pages 1-68
Structure of Classical Groups….Pages 69-126
Highest-Weight Theory….Pages 127-174
Algebras and Representations….Pages 175-224
Classical Invariant Theory….Pages 225-300
Spinors….Pages 301-328
Character Formulas….Pages 329-362
Branching Laws….Pages 363-385
Tensor Representations of GL(V)….Pages 387-424
Tensor Representations of O(V) and Sp(V)….Pages 425-477
Algebraic Groups and Homogeneous Spaces….Pages 479-544
Representations on Spaces of Regular Functions….Pages 545-610
Back Matter….Pages 1-103