Ledermann, Walter. Introduction to Group Characters. Cambridge: Cambridge University Press, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....when G = H and f is in the automorphism group Aut(G) of G. In this situation, the assignment f 7 f T gives a representation of Aut(G) acting on CT(G) and the assignment f 7 fK gives a representation of Aut(G) acting on CK(G) All the representation theory we need is in Chapter 1 of Ledermann [7]. Commutativity of the diagram means that G is an Aut(G) equivariant isomorphism, so these two representations are linearly equivalent. A representation of a nite group is determined up to linear equivalence by its group character, so G exists if and only if the characters T and K of these ....

W. Ledermann, \Introduction to Group Characters," Cambridge University Press, Cambridge, 1977.

....supply of completely reducible modules. Theorem 3.1.1 If G is a finite group then every G module V is completely reducible. 2 Although we do not have room here to prove the results that we will need from representation theory, the reader is encouraged to consult the excellent text of Ledermann [Led 77] or the up coming book of Sagan [Sag ip] The next question to ask is: given G, how many irreducible G modules are there First, however, we must know when two modules are the same. We say that G modules V and W are equivalent, written V = W , if there is a vector space isomorphism OE : V W ....

W. Ledermann, Introduction to group characters, Cambridge University Press, 1977.

....then TT (C) F OE (C) 3.3. Group Characters Now that we have established a nice theorem about orthogonal transforms and F(D;W 0 ) for finite groups W , it would be nice to demonstrate the existence of interesting transforms that meet the criteria of the theorem. Group representation theory [12] [13] provides such a family of tranforms. The simple characters of a group W are complex functions on W that are pairwise orthogonal. The inner product of a simple character with itself is k. Suppose W 1 ; W r are the conjugacy classes of W : then there are r simple characters, each of ....

....be its own conjugacy class would make life much nicer; this happens exactly when W is an abelian group. A result from algebra tells us that any finite abelian group isomorphic to the direct product of cyclic groups. A standard result from group representation theory (eg, Theorem 2. 4 in Ledermann [12]) tells us exactly what the simple characters of W are. Lemma 36 Suppose W = ZZ=k 1 Theta ZZ=k 2 Theta : Theta ZZ=k s Then the simple characters of W are the f w : w 2 Wg, where w (x) e 2 i P s j=1 x j w j k j So we can define OE(a; b) to be a (b) and obtain an orthogonal transform ....

Ledermann, Walter. Introduction to Group Characters. Cambridge: Cambridge University Press, 1977.

....of Frobenius in 1896. One of the great achievements of Frobenius is to find all the characters of the symmetric group S n for all n [3] The celebrated degree formula of Frobenius gives the degree of an arbitrary character of S n in a closed form formula, corresponding to a partition p of n [5, 2]. One particular step in the derivation of Frobenius s degree formula is the evaluation of a certain determinant. We will make use of this determinant and some easy generalizations in a new construction and proof of a family of polynomials that generalizes those polynomials discovered by Toda in ....

W. Ledermann, Introduction to Group Characters, Cambridge University Press, Cambridge, Second Edition, 1987.

....of an ffl biased distribution; the properties of this construction are studied in Section 5; finally, in Section 6 our construction is applied to linear codes. 2 Preliminaries 2.1 Characters of Finite Abelian Groups. Our discussion here follows the exposition of Babai [Ba] and Ledermann [Le]. Let C denote the multiplicative group of complex numbers with unit modulus. A character of a finite abelian group G is a homomorphism : G C . That is j (x)j = 1, for all x 2 G , and (xy) x) y) for all x; y 2 G . The number of different characters for an abelian group is jG j. ....

....r is of order n=gcd(r; n) We note that in the case of a multiplicative group modulo some prime p, viz. Z p , the notion of a multiplicative character can be extended to the finite field GF [p] by setting (0) 0. For the general case of a finite abelian group G , we invoke the Basis Theorem [Le] which states that G is the direct product of cyclic groups, say, Z n1 ; Z nk . Let V G denote the set of all k tuples (t 1 ; t k ) for 0 t i n i , for 1 i k; note that jV G j = jG j. Each element x 2 G can be uniquely expressed as x = z a1 1 z a2 2 Delta Delta ....

W. Ledermann. Introduction to Group Characters. Cambridge University Press (1987, 2nd edition).

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Ledermann, Walter. Introduction to Group Characters. Cambridge: Cambridge University Press, 1977.

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W. Ledermann, "Introduction to Group Characters," CUP, 1987.

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W. Ledermann. Introduction to Group Characters. Cambridge University Press (1987, 2nd edition).

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W. Ledermann, Introduction to Group Characters, Cambridge University Press, New York/London, 1977.

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