By Hahl H., Salzmann H.

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**Additional resources for 16-dimensional compact projective planes with a large group fixing two points and two lines**

**Example text**

Let w=Nlx1 +fl2x2+ ... + flaxa be another element of G. Then y = w if and only if a ; = /3 (i = 1, 2, ... 20) 1v=v. Furthermore, G c possesses a multiplicative structure. 21) where j (i, j) is a well-defined integer lying between 1 and g. i)• This multiplication is associative by virtue of the associative law for G. Thus, if u, y, w E G e , then (uv)w = u(vw). As we have already remarked (p. 27), a vector space which is endowed with an associative multiplication is called an algebra. Accordingly, we call G c the group algebra of G over C .

O+ U,, where UJ (i = 1, 2, ... , 1) are irreducible G-modules over K. Of course, the case 1= 1 refers to an irreducible representation A(x) (an irreducible G-module V). 3. Let G be a finite group of order g, and let K be a field whose characteristic is zero or else prime to g. Then (i) every matrix representation of G over K is completely reducible, or equivalently, (ii) every G-module over K is completely reducible. In subsequent chapters of this book we shall be exclusively concerned with finite groups and with ground fields of characteristic zero.

Prove that A (x) is irreducible. 5, dim ((A) = 1. 1. Orthogonality relations From now on we assume that G is a finite group of order g and that K is the field of complex numbers. Hence, by Maschke's Theorem, all representations of G are completely reducible. Furthermore, irreducibility henceforth means absolute irreducibility. Let A(x) and $(x) be inequivalent irreducible representations of degrees f and f' respectively, and write (i, j =1, 2, . 2) A (x ) = ( a (x)) ;; For brevity, we adopt the convention in this section that i and j always run from 1 to f, and that p and q run from 1 to f'.