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.
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