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I'm about to finish Rotman's "Introduction to the Theory of Groups" and I would like to continue my study of group theory with a book on representation theory. The book should give a broad overview over the different areas of representation theory. It may assume the reader to be familiar with the basic theory of groups. I like Rotman's system of offering a large amount of excersices some of which are used in latter sections of the book.

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    $\begingroup$ you may follow this link > mathoverflow.net/questions/2755/… $\endgroup$ – srijan Apr 16 '13 at 11:40
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    $\begingroup$ I found Serre's book rather useful. It is called "Linear Representations of Finite Groups". The first part was originally written for quantum chemists, so you know that it is going to be manageable. The previous sentence has sarcastic undertones. $\endgroup$ – user1729 May 7 '13 at 8:33
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I would recommend the books of I.M. Isaacs Character Theory of Finite Groups (1976) and J.P. Serre Linear Representations of Finite Groups (1977). Both excellent books written by the real masters and with plenty of exercises!
Also the five (!) volumes of Gregory Karpilovsky on Group Representations are fantastic sources of knowledge. If you want to go into modular representation theory and projective representation theory, look for the books of Bertram Huppert or B. M. Puttaswamaiah, John D. Dixon, and again Karpilovsky respectively.

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The subject "Representation theory" can be learned interestingly through examples of representations. The most interesting examples are "Representations of cyclic group, dihedral group, $A_4$, $S_4$ and $A_5$. The representations of these groups can actually be understood visually in 3-dimensional Euclidean geometry. Later, it would be best, if one has collection of representations of groups of small order, and analyze the examples frequently. The book by James and Liebeck is an interesting book, who describes the characters of many small groups.

After understanding the examples of the small groups, one can move to "Theory". The book "Representation Theory of Finite Groups: Algebra and Arithmatic" by S. Weintraub describes the theory with interesting proofs, and also with "necessary hypothesis" (whereas, many books on the subject describe the theory, over algebraically closed field, and we miss here arithmatic of the field).

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