# Download PDF by Peter Webb: A Course in Finite Group Representation Theory

By Peter Webb

This graduate-level textual content presents a radical grounding within the illustration concept of finite teams over fields and jewelry. The ebook offers a balanced and complete account of the topic, detailing the tools had to study representations that come up in lots of components of arithmetic. Key themes comprise the development and use of personality tables, the position of induction and restrict, projective and straightforward modules for team algebras, indecomposable representations, Brauer characters, and block idea. This classroom-tested textual content presents motivation via various labored examples, with routines on the finish of every bankruptcy that try out the reader's wisdom, offer extra 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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Example text

Hence if x1 , . . , xr are representatives of the conjugacy classes of G, xi ∈ ZG is the sum of the elements conjugate to xi , and λ1 , . . , λr ∈ C are algebraic integers then the element ri=1 λi xi ∈ Z(CG) is integral over Z. Proof. 1, since such a subring is in particular a subgroup of the finitely-generated free abelian group ZG, and hence is finitely generated as a Z-module. 2 that the elements x1 , . . 1 the linear combination ri=1 λi xi is integral also. ) Let ρ1 , . . , ρr be the simple representations of G over C with degrees d1 , .

It is a central idempotent element if it lies in the center Z(A). Two idempotent elements e and f are orthogonal if ef = f e = 0. An idempotent element e is called primitive if whenever e = e1 + e2 where e1 and e2 are orthogonal idempotent elements then either e1 = 0 or e2 = 0. We say that e is a primitive central idempotent element if it is primitive as an idempotent element in Z(A), that is, e is central and has no proper decomposition as a sum of orthogonal central idempotent elements. We comment that the term ‘idempotent element’ is very often abbreviated to ‘idempotent’, thereby elevating the adjective to the status of a noun.

Conversely, show that if every character of G is real-valued, then every element of G is conjugate to its inverse. [The quaternion group of order 8 in its action on the algebra of quaternions provides an example of a complex representation that is not equivalent to a real representation, but whose character is real-valued (see Chapter 2 Exercise 12). In this example, the representation has complex dimension 2, but there is no basis over C for the representation space such that the group acts by matrices with real entries.