By Peter Webb
This graduate-level textual content presents an intensive grounding within the illustration thought of finite teams over fields and jewelry. The e-book offers a balanced and accomplished account of the topic, detailing the tools had to learn representations that come up in lots of components of arithmetic. Key issues comprise the development and use of personality tables, the function of induction and restrict, projective and easy modules for crew algebras, indecomposable representations, Brauer characters, and block idea. This classroom-tested textual content offers motivation via a great number of labored examples, with routines on the finish of every bankruptcy that try out the reader's wisdom, supply additional examples and perform, and contain effects no longer confirmed within the textual content. necessities contain a graduate path in summary algebra, and familiarity with the houses of teams, earrings, box extensions, and linear algebra.
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Extra info for A Course in Finite Group Representation Theory
Given that there is exactly one error, determine where it is, and what the correct entry should be. 9. A finite group has seven conjugacy classes with representatives c1 , . . , c7 (where c1 = 1), and the values of five of its irreducible characters are given by the following table: c1 c2 c3 c4 c5 c6 c7 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 4 1 −1 0 2 −1 0 4 1 −1 0 −2 1 0 5 −1 0 1 1 1 −1 Calculate the numbers of elements in the various conjugacy classes and the remaining simple characters. 10. Let g ∈ G.
This computes the character table as far as it was presented at the start of this example. In Exercise 16 at the end of this chapter you are asked to show that this G is the only group that has this character table. CHAPTER 3. 5 38 Algebraic integers and divisibility of character degrees So far we have established that the degrees of the irreducible complex characters of G have the properties that their number equals the number of conjugacy classes of G, and the sum of their squares is |G|. We now establish a further important property, which is that the character degrees all divide |G| – a big restriction on the possible degrees that may occur.
The Ai are indecomposable as rings if and only if the ei are primitive central idempotent elements. If every Ai is indecomposable as a ring then the subsets Ai and also the primitive central idempotents ei are uniquely determined; furthermore, every central idempotent can be written as a sum of certain of the ei . Proof. Given any ring decomposition A = A1 ⊕ · · · ⊕ Ar we may write 1 = e1 + · · · + er where ei ∈ Ai and it is clear that the ei are orthogonal central idempotent elements. Conversely, given an expression 1 = e1 + · · · + er where the ei are orthogonal central idempotent elements we have A = Ae1 ⊕ · · · ⊕ Aer as rings.
A Course in Finite Group Representation Theory by Peter Webb