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.
