A Course in Finite Group Representation Theory by Peter Webb PDF

By Peter Webb

This graduate-level textual content presents an intensive grounding within the illustration idea of finite teams over fields and earrings. The publication offers a balanced and finished account of the topic, detailing the equipment had to examine representations that come up in lots of components of arithmetic. Key issues contain 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 offers motivation via a lot of labored examples, with workouts on the finish of every bankruptcy that attempt the reader's wisdom, supply additional examples and perform, and comprise effects now not confirmed within the textual content. must haves comprise a graduate path in summary algebra, and familiarity with the homes of teams, jewelry, box extensions, and linear algebra.

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Extra resources for A Course in Finite Group Representation Theory

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Xt . Proof. We first show that xi ∈ Z(RG). Write xi = y∼xi y, where ∼ denotes conjugacy. Then gxi = gy = ( gyg −1 )g = xi g y∼xi y∼xi since as y runs through the elements of G conjugate to xi , so does gyg −1 , and from this it follows that xi is central. Next suppose g∈G ag g ∈ Z(RG). We show that if g1 ∼ g2 then ag1 = ag2 . Suppose that g2 = hg1 h−1 . The coefficient of g2 in h( g∈G ag g)h−1 is ag1 and in ( g∈G ag g) is ag2 . Since elements of G are independent in RG, these coefficients must be equal.

If (apq ), (brs ) are an m × m matrix and an n × n matrix their tensor product is the mn × mn matrix (cij ) where if i = (p − 1)n + r and j = (q − 1)n + s with 1 ≤ p, q ≤ m and 1 ≤ r, s ≤ n then cij = apq brs . For example,   ae af be bf ag ah bg bh  a b e f  ⊗ =  ce cf de df  . c d g h cg ch dg dh If α : V → V and β : W → W are endomorphisms, then the matrix of α⊗β :V ⊗W →V ⊗W is the tensor product of the matrices that represent α and β (provided the basis elements vi ⊗wj are taken in an appropriate order).

Proof. Direct evaluation gives χCG , χi = 1 |G|χi (1) = di |G| and hence χCG = d1 χ1 + · · · + dr χr . Evaluating at 1 gives (1), and at g = 1 gives (2). 4 The number of simple characters It is an immediate deduction from the fact that the rows of the character table are orthogonal that the number of simple complex characters of a group is at most the number of conjugacy classes of elements in the group. We shall now prove that there is always equality here. The proof follows a surprising approach in which examine the center of the group algebra.

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A Course in Finite Group Representation Theory by Peter Webb

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