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By Kondratev A.S.

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Ht„x1 is the group „ n t 1Ht . ; ; ^-^ 2. Show that the map a^C(a)—^ -0 1 -1/ 1 defines a representation of the cyclic group gp{a I =1 }. Prove that this representation is irreducible over the field of real numbers. 3. Let E be a linear map of the m-dimensional vector space V into itself such that E2 = E, e # i (the identity map). E # 0, Define W = {wiwe = 0}. U= {ulue = u}, Show that U and W are subspaces of V which are invariant under UEGU, WEcW. Prove that E, that is V= UO+W. Deduce that if E is an m x m matrix such that E 2 =E, E#0, E #I, there exists an integer r satisfying 1 r < m and a non-singular matrix T such that T- 'ET = (0 34 0 ) = i.

E m (X). 37) - where the right-hand side is an abbreviation for 4)(x). This enables us to recast the expression for the inner product of two characters 4(x) and 0(x). When x lies in Ca , then 4(x) = O a , i (x} _ 111a , {x ') = t1ra . 38) In particular, if x and x' are simple characters, the character relations (of the first kind, p. 39) state that 1 g ^ haXaX►a--{0 ifX#X' ►. 39) (i, j = 1, 2, ... , k). 1(11„1 g)). 39) states that the rows of U form a system of unitary-orthogonal k-tuples.

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 .

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