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Characters of Groups and Lattices over Orders - From Ordinary to Integral Representation Theory (Paperback)
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Characters of Groups and Lattices over Orders - From Ordinary to Integral Representation Theory (Paperback)
Series: De Gruyter Textbook
Expected to ship within 10 - 15 working days
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This is the fi rst textbook leading coherently from classical
character theory to the theory of lattices over orders and integral
representations of fi nite groups. Character theory is developed in
a highly pedagogical way including many examples and exercises
covering at once the fi rst defi nitions up to Clifford theory,
Brauer's induction theorem and the splitting fi eld theorem, as
well as self-dual simple modules allowing bilinear forms. This
latter part is done step by step using the approach given by Sin
and Willems. Dirichlet characters and Dirichlet's result on primes
in arithmetic progressions are given as an application. Examples of
integral representations of fi nite groups are already detailed at
a quite early stage where appropriate, so that the more systematic
treatment of lattices over orders is natural. After that, the
necessary number theory and homological algebra is developed as far
as needed. Maximal as well as hereditary orders are introduced and
the Auslander- Buchsbaum theorem is proved. The Jordan-Zassenhaus
theorem on the fi niteness of lattices in a given vector space is
fully proved. Then the development and properties of class groups
of orders is a fi rst focus. As a further highlight Swan's example
of a stably free but not free ideal over the integral group ring of
the generalised quaternion group of order 32 is developed in great
detail. A student friendly introduction to ordinary representation
theory Many examples and exercises, including solutions for some of
them, make the book well suited for self-study Leads coherently
from ordinary character theory to the integral representation
theory of lattices over orders Several topics appear for the fi rst
time in a textbook, such as Sin-Willems' approach to self-dual
simple modules and Swan's example of a stably free non free ideal.
General
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