By Peter Webb

This graduate-level textual content offers an intensive grounding within the illustration concept of finite teams over fields and earrings. The e-book offers a balanced and entire account of the topic, detailing the tools had to examine representations that come up in lots of components of arithmetic. Key subject matters contain the development and use of personality tables, the function of induction and limit, projective and straightforward modules for workforce algebras, indecomposable representations, Brauer characters, and block concept. This classroom-tested textual content presents motivation via a good number of labored examples, with workouts on the finish of every bankruptcy that attempt the reader's wisdom, offer additional examples and perform, and comprise effects no longer confirmed within the textual content. necessities comprise a graduate path in summary algebra, and familiarity with the houses of teams, earrings, box extensions, and linear algebra.

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An element e of a ring A is said to be idempotent if e2 = e. 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.

Without this exponent we would get a right RG-module. If R happens to be a field and we have bases v1 , . . , vm for V and w1 , . . , wn for W then V ⊗ W has a basis {vi ⊗ wj 1 ≤ i ≤ m, 1 ≤ j ≤ n} and V ∗ has a dual basis CHAPTER 3. CHARACTERS 26 vˆ1 , . . , vˆm . With respect to these bases an element g ∈ G acts on V ⊗ W with the matrix that is the tensor product of the two matrices giving its action on V and W , and on V ∗ it acts with the transpose of the inverse of the matrix of its action on V .

In particular ni is determined by V independently of the choice of decomposition. 2. Let G = S3 and denote by C the trivial representation, the sign representation and V the 2-dimensional simple representation over C. We decompose the 4-dimensional representation V ⊗V as a direct sum of simple representations. Since the values of the character χV give the row of the character table χV 2 0 −1 CHAPTER 3. CHARACTERS 31 we see that V ⊗ V has character values χV ⊗V Thus 4 0 1 1 χV ⊗V , χC = (4 · 1 + 0 + 2 · 1 · 1) = 1 6 1 χV ⊗V , χ = (4 · 1 + 0 + 2 · 1 · 1) = 1 6 1 χV ⊗V , χV = (4 · 2 + 0 − 2 · 1 · 1) = 1 6 and we deduce that V ⊗V ∼ = C ⊕ ⊕ V.

### A Course in Finite Group Representation Theory by Peter Webb

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