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

By Peter Webb

This graduate-level textual content offers an intensive grounding within the illustration concept of finite teams over fields and jewelry. The booklet presents a balanced and entire account of the topic, detailing the tools had to learn representations that come up in lots of parts of arithmetic. Key themes comprise the development and use of personality tables, the position of induction and restrict, projective and easy modules for team algebras, indecomposable representations, Brauer characters, and block concept. This classroom-tested textual content presents motivation via loads of labored examples, with workouts on the finish of every bankruptcy that try out the reader's wisdom, supply extra examples and perform, and contain effects now not confirmed within the textual content. must haves contain a graduate direction in summary algebra, and familiarity with the homes of teams, earrings, box extensions, and linear algebra.

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3 Consequences of the orthogonality relations We will describe many consequences of the orthogonality relations, and the first is that they provide a way of determining the decomposition of a given representation as a direct sum of simple representations. This procedure is similar to the way of finding the coefficients in the Fourier expansion of a function using orthogonality of the functions sin(mx) and cos(nx). 1. Let V be a CG-module. In any expression V = S1n1 ⊕ · · · ⊕ Srnr in which S1 , .

From the fact that the character table is square we get orthogonality relations between its columns. 4 (Column orthogonality relations). Let X be the character table of G, regarded as a matrix, and let   |CG (x1 )| 0 ··· 0   0 |CG (x2 )|   C=  .. ..   . . 0 · · · |CG (xr )| where x1 , . . , xr are representatives of the conjugacy classes of elements of G. Then X T X = C, where the bar denotes complex conjugation. Proof. The orthogonality relations between the rows may be written XC −1 X T = I.

Vi (g1 )χVi (g1 )χVj (g2 )χVj (g2 ) (g1 ,g2 )∈G1 ×G2 χVi (g1 )χVi (g1 ) · g1 ∈G1 1 |G2 | χVj (g2 )χVj (g2 ) g2 ∈G2 CHAPTER 4. CONSTRUCTION OF CHARACTERS 52 The characters of these representations are distinct, since by a similar calculation if (i, j) = (r, s) then χVi ⊗Wj , χVr ⊗Ws = 0. To show that we have the complete list, we observe that if dim Vi = di and dim Wj = ej then Vi ⊗ Wj is a representation of degree di ej and m n m n (di ej )2 = i=1 j=1 d2i · i=1 e2j = |G1 ||G2 |. 13. 3. Putting the last two results together enables us to compute the character table of any finite abelian group.

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A Course in Finite Group Representation Theory by Peter Webb

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