Introduction to Group Characters
CUP Archive, 30 juil. 1987 - 227 pages
To an algebraist the theory of group characters presents one of those fascinating situations, where the structure of an abstract system is elucidated by a unique set of numbers inherent in the system. But the subject also has a practical aspect, since group characters have gained importance in several branches of science, in which considerations of symmetry play a decisive part. This is an introductory text, suitable for final-year undergraduates or postgraduate students. The only prerequisites are a standard knowledge of linear algebra and a modest acquaintance with group theory. Especial care has been taken to explain how group characters are computed. The character tables of most of the familiar accessible groups are either constructed in the text or included amongst the exercise, all of which are supplied with solutions. The chapter on permutation groups contains a detailed account of the characters of the symmetric group based on the generating function of Frobenius and on the Schur functions. The exposition has been made self-sufficient by the inclusion of auxiliary material on skew-symmetric polynomials, determinants and symmetric functions.
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affords algebraic alternating applying arbitrary associated assume basis called centre Chapter character of G character table coefficients columns commute complete conjugacy classes conjugate consider consists construct correspondence coset cycle deduce defined Definition denoted described determinant distinct element equal equation equivalent established example Exercise exists expressed fact field finite group fixed follows formula function G-module given group G Hence holds implies induced integers invariant invertible irreducible representation Lemma linear linear characters matrix representation minimal multiplication obtain orthogonal particular partition permutation polynomial positive Proof Proposition Prove rational reducible refer regarded relations remains representation of G respectively result root satisfies Schur Show side simple character Substituting Suppose symmetric symmetric matrix Table taking tensor Theorem theory trivial unity values vector space virtue whence write zero
The Symmetric Group: Representations, Combinatorial Algorithms, and ...
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