By Peter Webb

This graduate-level textual content presents a radical grounding within the illustration concept of finite teams over fields and earrings. The publication presents a balanced and accomplished account of the topic, detailing the tools had to study representations that come up in lots of parts of arithmetic. Key subject matters comprise the development and use of personality tables, the function of induction and restrict, projective and straightforward modules for workforce algebras, indecomposable representations, Brauer characters, and block conception. This classroom-tested textual content offers motivation via quite a few labored examples, with routines on the finish of every bankruptcy that try the reader's wisdom, offer extra examples and perform, and contain effects no longer confirmed within the textual content. necessities contain a graduate direction in summary algebra, and familiarity with the houses of teams, earrings, box extensions, and linear algebra.

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**Sample text**

Step 6. We show that |χ(g)| = χ(1) and put ζ = χ(g)/χ(1). We consider the algebraic conjugates of ζ, which are the roots of the minimal polynomial of ζ over Q. They are all algebraic integers, since ζ and its algebraic conjugates are all roots of the same polynomials over Q. Thus the product N (ζ) of the algebraic conjugates is an algebraic integer. Since it is also ± the constant term of the minimal polynomial of ζ, it is rational and non-zero. 1. Now χ(g) is the sum of the eigenvalues of χ(g), of which there are χ(1), each of which is a root of unity.

Let V1 , . . , Vm and W1 , . . , Wn be complete lists of the simple complex representations of groups G1 and G2 . Then the representations Vi ⊗ Wj with 1 ≤ i ≤ m and 1 ≤ j ≤ n form a complete list of the simple complex G1 × G2 representations. 2 is false in general when the field over which we are working is not algebraically closed (see Exercise 10 at the end of this chapter). The theorem is an instance of a more general fact to do with representations of finite dimensional algebras A and B over an algebraically closed field k: the simple representations of A ⊗k B are precisely the modules S ⊗k T , where S is a simple A-module and T is a simple Bmodule.

Any morphism f : M → N equals α( ni=1 pi f ⊗ vi ), so α is surjective. If α( ni=1 gi ⊗ vi ) = 0 for certain gi ∈ M ∗ then for all u ∈ M , ni=1 gi (u)vi = 0, which implies that gi (u) = 0 for all i and u. This means that gi is the zero map for all i, so that ni=1 gi ⊗ vi = 0 and α is injective. We now observe that α is a map of RG-modules, since for g ∈ G, α(g(f ⊗ w)) = α(gf ⊗ gw) = (v → (gf )(v) · gw) = (v → g(f (g −1 v)w)) = g(v → f (v)w) = gα(f ⊗ w). 3 then the argument can be simplified. Since M and N are always free as R-modules in this situation, only one of the two arguments given is needed.