Representation Theory(D)
REPRESENTATION THEORY (D)
Linear Algebra, and Groups, Rings and Modules are esssential.
Representations of finite groups
- Representations of groups on vector spaces, matrix representations. Equivalence of representations. Invariant subspaces and submodules. Irreducibility and Schur’s Lemma. Complete reducibility for finite groups. Irreducible representations of Abelian groups.
Character theory
- Determination of a representation by its character. The group algebra, conjugacy classes, and orthogonality relations. Regular representation. Permutation representations and their characters. Induced representations and the Frobenius reciprocity theorem. Mackey’s theorem. Frobenius’s Theorem. [12]
Arithmetic properties of characters
- Divisibility of the order of the group by the degrees of its irreducible characters. Burnside’s p^aq^b theorem. [2]
Tensor products
- Tensor products of representations and products of characters. The character ring. Tensor, symmetric and exterior algebras. [3]
Representations of S1 and SU2
- The groups S1, SU2 and SO(3), their irreducible representations, complete reducibility. The ClebschGordan formula. *Compact groups.* [4]
Further worked examples
- The characters of one of GL2(Fq), Sn or the Heisenberg group. [3]
Appropriate books
- J.L. Alperin and R.B. Bell Groups and representations. Springer 1995
- I.M. Isaacs Character theory of finite groups. Dover Publications 1994
- G.D. James and M.W. Liebeck Representations and characters of groups. Second Edition, CUP 2001
- J-P. Serre Linear representations of finite groups. Springer-Verlag 1977
- M. Artin Algebra. Prentice Hall 1999
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