DOCSLIB.ORG

Some Applications of the Fourier Transform in Algebraic Coding Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Applications of the Fourier Transform in Algebraic Coding Theory Contemporary Mathematics Some Applications of the Fourier Transform in Algebraic Coding Theory Jay A. Wood In memory of Shoshichi Kobayashi, 1932{2012 Abstract. This expository article describes two uses of the Fourier transform that are of interest in algebraic coding theory: the MacWilliams identities and the decomposition of a semi-simple group algebra of a finite group into a direct sum of matrix rings. Introduction Because much of the material covered in my lectures at the ASReCoM CIMPA research school has already appeared in print ([18], an earlier research school spon- sored by CIMPA), the organizers requested that I discuss more broadly the role of the Fourier transform in coding theory. I have chosen to discuss two aspects of the Fourier transform of particular interest to me: the well-known use of the complex Fourier transform in proving the MacWilliams identities, and the less-well-known use of the Fourier transform as a way of understanding the decomposition of a semi-simple group algebra into a sum of matrix rings. In the latter, the Fourier transform essentially acts as a change of basis matrix. I make no claims of originality; the content is discussed in various forms in such sources as [1,4, 15, 16 ], to whose authors I express my thanks. The material is organized into three parts: the theory over the complex numbers needed to prove the MacWilliams identities, the theory over general fields relevant to decomposing group algebras, and examples over finite fields. Each part is further divided into numbered sections. Personal Remark. This article is dedicated to the memory of Professor Shoshichi Kobayashi, my doctoral advisor, who died 29 August 2012. I admired Professor 2010 Mathematics Subject Classification. Primary 94B05, Secondary 16P10, 20C15, 43A40. I thank the organizers of the CIMPA school, Edgar Mart´ınezMoro and Mustapha Lahyane, for their support and the invitation to write this article. I thank Steve Szabo for suggesting the topic of the Fourier transform as a change of basis in a group algebra. c 2013 American Mathematical Society 1 2 JAY A. WOOD Kobayashi's ability to write such wonderful research monographs, and I cherish the memory of his patience and kindness towards me. I. Fourier transform over C for finite abelian groups 1. Linear codes and weight enumerators We begin with a short summary of terminology from coding theory. Let R be a finite associative ring with 1; we do not assume that R is commu- tative. Any module over R will be assumed to be unital, i.e., 1 2 R acts as the identity. Fix a finite left R-module A; A will be the alphabet for linear codes. A (left) R-linear code of length n over A is a left R-submodule C ⊂ An. An important special case is when A = R, the ring itself. (One could also study right linear codes, of course.) Of interest in coding theory are the weights of codewords. The weights are often studied by means of weight enumerators. There are a number of different (but related) weight enumerators, of two main types. Our primary interest is in the first type. Composition-type. We begin with the most general form of weight enumerator, 1 n the complete weight enumerator. For any a 2 A and x = (x1; : : : ; xn) 2 A , define the counting function ca(x) = jfi : xi = agj; i.e., the number of entries in the vector x 2 An that equal a particular a 2 A. Note P n n that a2A ca(x) = n for all x 2 A . For a fixed x 2 A , the numbers ca(x); a 2 A, form the \composition" of x. For every a 2 A, let Za be an indeterminate. Form the polynomial ring C[Za : a 2 A] over the complex numbers. Given an R-linear code C ⊂ An, the complete weight enumerator of C is n X Y X Y ca(x) cweC (Z) = Zxi = Za : x2C i=1 x2C a2A The complete weight enumerator counts the number of codewords in C of any given composition. Experience has shown it useful to cluster certain compositions together; i.e., certain compositions are to be regarded as equivalent. To that end, let ∼ be an equivalence relation on the alphabet A. Denote the equivalence class of a 2 A by [a]. For x 2 An and equivalence class [a], a 2 A, define a symmetrized weight composition swc by X swc[a](x) = jfi : xi ∼ agj = cb(x): b2[a] Then define the symmetrized weight enumerator of a linear code C ⊂ An by n X Y X Y swc[a](x) sweC (S) = S[xi] = S[a] ; x2C i=1 x2C [a]2A=∼ 1There is the even-more-general full (or exact) weight enumerator, which is a copy of the code itself. See [14, 18]. FOURIER TRANSFORM 3 where A=∼ denotes the set of all ∼-equivalence classes and the S[a],[a] 2 A= ∼, are indeterminates. Note that the specialization of variables Za 7! S[a] induces a homomorphism of polynomial rings C[Za : a 2 A] ! C[S[a] :[a] 2 A=∼] that maps cweC (Z) to sweC (S). Suppose there are two equivalence relations on A and one is finer than the other. That is, a ∼1 b implies a ∼2 b; or every equivalence class of ∼1 is contained in an equivalence class of ∼2. Then the specialization of variables S[a]1 7! T[a]2 is well-defined and induces a ring homomorphism C[S[a]1 :[a]1 2 A= ∼1] ! C[T[a]2 : (1) (2) [a]2 2 A=∼2] taking sweC (S) to sweC (T ). There are two examples of equivalence relations of special note. The Hamming equivalence relation has just two equivalence classes: f0g and fa 2 A : a 6= 0g. It is n traditional to write wt(x) = jfi : xi 6= 0gj, the Hamming weight of x 2 A . Define the Hamming weight enumerator by X n−wt(x) wt(x) (1.1) hweC (X; Y ) = X Y : x2C The specialization of variables Z0 7! X and, for a 6= 0, Za 7! Y , induces a homo- morphism C[Za : a 2 A] ! C[X; Y ] carrying cweC (Z) to hweC (X; Y ). It is very common to see hweC (X; Y ) denoted WC (X; Y ). The other special case is an equivalence relation arising from a group action. Let Aut(A) be the group of all invertible R-linear transformations from A to itself. Because A is a left R-module, we will write mappings on the right, so that R- linearity is expressed as (ra)φ = r(aφ), for r 2 R, a 2 A, and φ 2 Aut(A). Fix a subgroup G ⊂ Aut(A) and define an equivalence relation via the action of G on A: a ∼G b if there exists φ 2 G with a = bφ. We will see more of this example later, in Remark 1.3. When A = R, the ring itself, Aut(A) is just the group of units of R. Fixing a subgroup G of the group of units, the equivalence relation is then a ∼G b when a = bu for some u 2 G. As a concrete example, let R = Z=4Z and let G = {±1g, the full group of units. Then 1 ∼G 3 in Z=4Z, and there are three equivalence classes: f0g, f1; 3g, and f2g. This case arises when studying the Lee weight on Z=4Z. Example 1.1. Let R = A = Z=4Z, and let C ⊂ R3 be the submodule generated by 123 and 022. Then C = f000; 123; 202; 321; 022; 101; 220; 303g: Let G = {±1g determine an equivalence relation. The three weight enumerators described above are: 3 2 2 2 cweC (Z) = Z0 + Z0Z1 + 3Z0Z2 + Z0Z3 + 2Z1Z2Z3 3 2 2 2 sweC (S) = S[0] + 2S[0]S[1] + 3S[0]S[2] + 2S[1]S[2] 3 2 3 hweC (X; Y ) = X + 5XY + 2Y : Weight-type. Although we have defined weight enumerators, we have not de- fined weights, except for the Hamming weight. A very general definition is: a weight on an alphabet A is any function w : A ! C with w(0) = 0. (One may wish to impose additional hypotheses on w, such as the triangle inequality, but we will not n Pn do so here.) The weight is then extended to A by setting w(x) = i=1 w(xi), for n x = (x1; : : : ; xn) 2 A . The Hamming weight satisfies w(a) = 1 for nonzero a 2 A. 4 JAY A. WOOD Suppose a weight w on an alphabet A has values in the nonnegative integers, and suppose that M is the largest value of w on A. Define the w-weight enumerator of a linear code C ⊂ An by n X Y M−w(xi) w(xi) X nM−w(x) w(x) wweC (X; Y ) = X Y = X Y : x2C i=1 x2C This weight enumerator counts the number of codewords of any given weight. For the Hamming weight, this definition agrees with that of hweC given in (1.1). Example 1.2. The Lee weight wL on R = A = Z=4Z is defined by wL(0) = 0, wL(1) = wL(3) = 1, and wL(2) = 2. Note that M = 2. For the code given in Example 1.1, we have 6 4 2 2 4 wweC (X; Y ) = X + 2X Y + 5X Y : Remark 1.3. Given a weight w on A, there is a symmetrized weight enumerator naturally determined by w. Define the right symmetry group G of w by G = fφ 2 Aut(A): w(aφ) = w(a); for all a 2 Ag.
Load more

Recommended publications
  • Notes on the Finite Fourier Transform
    Notes on the Finite Fourier Transform Peter Woit Department of Mathematics, Columbia University [email protected] April 28, 2020 1 Introduction In this final section of the course we'll discuss a topic which is in some sense much simpler than the cases of Fourier series for functions on S1 and the Fourier transform for functions on R, since it will involve functions on a finite set, and thus just algebra and no need for the complexities of analysis. This topic has important applications in the approximate computation of Fourier series (which we won't cover), and in number theory (which we'll say a little bit about). 2 The group Z(N) What we'll be doing is simplifying the topic of Fourier series by replacing the multiplicative group S1 = fz 2 C : jzj = 1g by the finite subgroup Z(N) = fz 2 C : zN = 1g If we write z = reiθ, then zN = 1 =) rN eiθN = 1 =) r = 1; θN = k2π (k 2 Z) k =) θ = 2π k = 0; 1; 2; ··· ;N − 1 N mod 2π The group Z(N) is thus explicitly given by the set i 2π i2 2π i(N−1) 2π Z(N) = f1; e N ; e N ; ··· ; e N g Geometrically, these are the points on the unit circle one gets by starting at 1 2π and dividing it into N sectors with equal angles N . (Draw a picture, N = 6) The set Z(N) is a group, with 1 identity element 1 (k = 0) inverse 2πi k −1 −2πi k 2πi (N−k) (e N ) = e N = e N multiplication law ik 2π il 2π i(k+l) 2π e N e N = e N One can equally well write this group as the additive group (Z=NZ; +) of inte- gers mod N, with isomorphism Z(N) $ Z=NZ ik 2π e N $ [k]N 1 $ [0]N −ik 2π e N $ −[k]N = [−k]N = [N − k]N 3 Fourier analysis on Z(N) An abstract point of view on the theory of Fourier series is that it is based on exploiting the existence of a particular othonormal basis of functions on the group S1 that are eigenfunctions of the linear transformations given by rotations.
    [Show full text]
  • An Introduction to the Trace Formula
    Clay Mathematics Proceedings Volume 4, 2005 An Introduction to the Trace Formula James Arthur Contents Foreword 3 Part I. The Unrefined Trace Formula 7 1. The Selberg trace formula for compact quotient 7 2. Algebraic groups and adeles 11 3. Simple examples 15 4. Noncompact quotient and parabolic subgroups 20 5. Roots and weights 24 6. Statement and discussion of a theorem 29 7. Eisenstein series 31 8. On the proof of the theorem 37 9. Qualitative behaviour of J T (f) 46 10. The coarse geometric expansion 53 11. Weighted orbital integrals 56 12. Cuspidal automorphic data 64 13. A truncation operator 68 14. The coarse spectral expansion 74 15. Weighted characters 81 Part II. Refinements and Applications 89 16. The first problem of refinement 89 17. (G, M)-families 93 18. Localbehaviourofweightedorbitalintegrals 102 19. The fine geometric expansion 109 20. Application of a Paley-Wiener theorem 116 21. The fine spectral expansion 126 22. The problem of invariance 139 23. The invariant trace formula 145 24. AclosedformulaforthetracesofHeckeoperators 157 25. Inner forms of GL(n) 166 Supported in part by NSERC Discovery Grant A3483. c 2005 Clay Mathematics Institute 1 2 JAMES ARTHUR 26. Functoriality and base change for GL(n) 180 27. The problem of stability 192 28. Localspectraltransferandnormalization 204 29. The stable trace formula 216 30. Representationsofclassicalgroups 234 Afterword: beyond endoscopy 251 References 258 Foreword These notes are an attempt to provide an entry into a subject that has not been very accessible. The problems of exposition are twofold. It is important to present motivation and background for the kind of problems that the trace formula is designed to solve.
    [Show full text]
  • Adams Operations and Symmetries of Representation Categories Arxiv
    Adams operations and symmetries of representation categories Ehud Meir and Markus Szymik May 2019 Abstract: Adams operations are the natural transformations of the representation ring func- tor on the category of finite groups, and they are one way to describe the usual λ–ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, deter- mines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and to show that monoidal equivalences that preserve the second Adams operations do not have to be symmetric. Along the way, we classify all possible symmetries and all monoidal auto- equivalences of representation categories of finite groups. MSC: 18D10, 19A22, 20C15 Keywords: Representation rings, Adams operations, λ–rings, symmetric monoidal cate- gories 1 Introduction Every finite group G can be reconstructed from the category Rep(G) of its finite-dimensional representations if one considers this category as a symmetric monoidal category. This follows from more general results of Deligne [DM82, Prop. 2.8], [Del90]. If one considers the repre- sentation category Rep(G) as a monoidal category alone, without its canonical symmetry, then it does not determine the group G. See Davydov [Dav01] and Etingof–Gelaki [EG01] for such arXiv:1704.03389v3 [math.RT] 3 Jun 2019 isocategorical groups. Examples go back to Fischer [Fis88]. The representation ring R(G) of a finite group G is a λ–ring.
    [Show full text]
  • On the Characters of ¿-Solvable Groups
    ON THE CHARACTERSOF ¿-SOLVABLEGROUPS BY P. FONGO) Introduction. In the theory of group characters, modular representation theory has explained some of the regularities in the behavior of the irreducible characters of a finite group; not unexpectedly, the theory in turn poses new problems of its own. These problems, which are asked by Brauer in [4], seem to lie deep. In this paper we look at the situation for solvable groups. We can answer some of the questions in [4] for these groups, and in doing so, obtain new properties for their characters. Finite solvable groups have recently been the object of much investigation by group theorists, especially with the end of relating the structure of such groups to their Sylow /»-subgroups. Our work does not lie quite in this direction, although we have one result tying up arith- metic properties of the characters to the structure of certain /»-subgroups. Since the prime number p is always fixed, we can actually work in the more general class of /»-solvable groups, and shall do so. Let © be a finite group of order g = pago, where p is a fixed prime number, a is an integer ^0, and ip, go) = 1. In the modular theory, the main results of which are in [2; 3; 5; 9], the characters of the irreducible complex-valued representations of ©, or as we shall say, the irreducible characters of ®, are partitioned into disjoint sets, these sets being the so-called blocks of © for the prime p. Each block B has attached to it a /»-subgroup 35 of © determined up to conjugates in @, the defect group of the block B.
    [Show full text]
  • Introduction to Fourier Methods
    Appendix L Introduction to Fourier Methods L.1. The Circle Group and its Characters We will denote by S the unit circle in the complex plane: S = {(x, y) ∈ R2 | x2 + y2 =1} = {(cos θ, sin θ) ∈ R2 | 0 ≤ θ< 2π} = {z ∈ C | |z| =1} = {eiθ | 0 ≤ θ< 2π}. We note that S is a group under multiplication, i.e., if z1 and z2 are in S then so is z1z2. and if z = x + yi ∈ S thenz ¯ = x − yi also is in S with zz¯ =zz ¯ =1, soz ¯ = z−1 = 1/z. In particular then, given any function f defined on S, and any z0 in S, we can define another function fz0 on S (called f translated by z0), by fz0 (z) = f(zz0). The set of piecewise continuous complex-valued functions on S will be denoted by H(S). The map κ : t 7→ eit is a continuous group homomorphism of the additive group of real numbers, R, onto S, and so it induces a homeomorphism and group isomorphism [t] 7→ eit of the compact quotient group K := R/2πZ with S. Here [t] ∈ K of course denotes the coset of the real number t, i.e., the set of all real numbers s that differ from t by an integer multiple of 2π, and we note that we can always choose a unique representative of [t] in [0, 2π). K is an additive model for S, and this makes it more convenient for some purposes. A function on K is clearly the “same thing” as a 2π-periodic function on R.
    [Show full text]
  • Beyond Endoscopy for the Relative Trace Formula II: Global Theory
    BEYOND ENDOSCOPY FOR THE RELATIVE TRACE FORMULA II: GLOBAL THEORY YIANNIS SAKELLARIDIS Abstract. For the group G = PGL2 we perform a comparison between two relative trace formulas: on one hand, the relative trace formula of Jacquet for the quotient T \G/T , where T is a non-trivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to non-standard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ “Beyond Endoscopy” program. Contents 1. Introduction. 1 Part 1. Poisson summation 18 2. Generalities and the baby case. 18 3. Direct proof in the baby case 31 4. Poisson summation for the relative trace formula 41 Part 2. Spectral analysis. 54 5. Main theorems of spectral decomposition 54 6. Completion of proofs 67 7. The formula of Waldspurger 85 Appendix A. Families of locally multiplicative functions 96 Appendix B. F-representations 104 arXiv:1402.3524v3 [math.NT] 9 Mar 2017 References 107 1. Introduction. 1.1. The result of Waldspurger. The celebrated result of Waldspurger [Wal85], relating periods of cusp forms on GL2 over a nonsplit torus (against a character of the torus, but here we will restrict ourselves to the trivial character) with the central special value of the corresponding quadratic base 2010 Mathematics Subject Classification. 11F70. Key words and phrases. relative trace formula, beyond endoscopy, periods, L-functions, Waldspurger’s formula. 1 2 YIANNIS SAKELLARIDIS change L-function, was reproven by Jacquet [Jac86] using the relative trace formula.
    [Show full text]
  • Chapter 4 Characters and Gauss Sums
    Chapter 4 Characters and Gauss sums 4.1 Characters on finite abelian groups In what follows, abelian groups are multiplicatively written, and the unit element of an abelian group A is denoted by 1. We denote the order (number of elements) of A by jAj. Let A be a finite abelian group. A character on A is a group homomorphism χ : A ! C∗ (i.e., C n f0g with multiplication). If jAj = n then an = 1, hence χ(a)n = 1 for each a 2 A and each character χ on A. Therefore, a character on A maps A to the roots of unity. The product χ1χ2 of two characters χ1; χ2 on A is defined by (χ1χ2)(a) = χ1(a)χ2(a) for a 2 A. With this product, the characters on A form an abelian group, the so-called character group of A, which we denote by Ab (or Hom(A; C∗)). (A) The unit element of Ab is the trivial character χ0 that maps A to 1. Since any character on A maps A to the roots of unity, the inverse χ−1 : a 7! χ(a)−1 of a character χ is equal to its complex conjugate χ : a 7! χ(a). It would have been possible to develop the theory of characters using the fact that every finite abelian groups is the direct sum of cyclic groups, but we prefer to start from scratch. Let B be a subgroup of A and χ a character on B. By an extension of χ to A 0 0 0 we mean a character χ on A such that χ jB = χ, i.e., χ (b) = χ(b) for b 2 B.
    [Show full text]
  • The Origin of Representation Theory
    THE ORIGIN OF REPRESENTATION THEORY KEITH CONRAD Abstract. Representation theory was created by Frobenius about 100 years ago. We describe the background that led to the problem which motivated Frobenius to define characters of a finite group and show how representation theory solves the problem. The first results about representation theory in characteristic p are also discussed. 1. Introduction Characters of finite abelian groups have been used since Gauss in the beginning of the 19th century, but it was only near the end of that century, in 1896, that Frobenius extended this concept to finite nonabelian groups [21]. This approach of Frobenius to group characters is not the one in common use today, although some of his ideas that were overlooked for a long period have recently been revived [30]. Here we trace the development of the problem whose solution led Frobenius to introduce characters of finite groups, show how this problem can be solved using classical representa- tion theory of finite groups, and indicate some relations between this problem and modular representations. Other surveys on the origins of representation theory are by Curtis [7], Hawkins [24, 25, 26, 27], Lam [32], and Ledermann [35]. While Curtis describes the development of modular representation theory focusing on the work of Brauer, we examine the earlier work in characteristic p of Dickson. 2. Circulants For a positive integer n, consider an n × n matrix where each row is obtained from the previous one by a cyclic shift one step to the right. That is, we look at a matrix of the form 0 1 X0 X1 X2 :::Xn−1 B Xn−1 X0 X1 :::Xn−2 C B C B .
    [Show full text]
  • Brauer Groups of Linear Algebraic Groups
    Volume 71, Number 2, September 1978 BRAUERGROUPS OF LINEARALGEBRAIC GROUPS WITH CHARACTERS ANDY R. MAGID Abstract. Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d - l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimen- sional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F.
    [Show full text]
  • 3 Representations of Finite Groups: Basic Results
    3 Representations of finite groups: basic results Recall that a representation of a group G over a field k is a k-vector space V together with a group homomorphism δ : G GL(V ). As we have explained above, a representation of a group G ! over k is the same thing as a representation of its group algebra k[G]. In this section, we begin a systematic development of representation theory of finite groups. 3.1 Maschke's Theorem Theorem 3.1. (Maschke) Let G be a finite group and k a field whose characteristic does not divide G . Then: j j (i) The algebra k[G] is semisimple. (ii) There is an isomorphism of algebras : k[G] EndV defined by g g , where V ! �i i 7! �i jVi i are the irreducible representations of G. In particular, this is an isomorphism of representations of G (where G acts on both sides by left multiplication). Hence, the regular representation k[G] decomposes into irreducibles as dim(V )V , and one has �i i i G = dim(V )2 : j j i Xi (the “sum of squares formula”). Proof. By Proposition 2.16, (i) implies (ii), and to prove (i), it is sufficient to show that if V is a finite-dimensional representation of G and W V is any subrepresentation, then there exists a → subrepresentation W V such that V = W W as representations. 0 → � 0 Choose any complement Wˆ of W in V . (Thus V = W Wˆ as vector spaces, but not necessarily � as representations.) Let P be the projection along Wˆ onto W , i.e., the operator on V defined by P W = Id and P W^ = 0.
    [Show full text]
  • Characters and the Macwilliams Identities
    Characters and the MacWilliams identities Jay A. Wood Department of Mathematics Western Michigan University http://homepages.wmich.edu/∼jwood/ Algebra for Secure and Reliable Communications Modeling Morelia, Michoac´an,Mexico October 8, 2012 Characters I Let G be a finite abelian group (usually additive). I A character of G is a group homomorphism π : G ! C×, where C× is the multiplicative group of nonzero complex numbers. I Example. Let G = Z=mZ and let a 2 Z=mZ. Then × πa : Z=mZ ! C , πa(g) = exp(2πiag=m) is a character. Every character of Z=mZ is of this form. JW (WMU) MacWilliams identities October 8, 2012 2 / 21 Character group I Let Gb be the set of all characters of G. I Then Gb is itself a finite abelian group under pointwise multiplication of characters. That is, (π1π2)(g) := π1(g)π2(g) for g 2 G. I Gb is the character group of G. ∼ I jGbj = jGj; in fact, Gb = G (but not naturally). ∼ I Gb = G, naturally: g 7! [π 7! π(g)]. JW (WMU) MacWilliams identities October 8, 2012 3 / 21 Products I If G1 and G2 are two finite abelian groups, then ∼ (G1 × G2)b = Gc1 × Gc2. I (π1; π2)(g1; g2) := π1(g1)π2(g2). JW (WMU) MacWilliams identities October 8, 2012 4 / 21 Vanishing formulas × I The character sending every g 2 G to 1 2 C is written 1, the principal character. ( X jGj; π = 1; π(g) = 0; π 6= 1: g2G ( X jGj; g = 0; π(g) = 0; g 6= 0: π2Gb JW (WMU) MacWilliams identities October 8, 2012 5 / 21 Linear independence of characters I Let F (G; C) = ff : G ! Cg be the set of all functions from G to C.
    [Show full text]
  • REPRESENTATION THEORY Tammo Tom Dieck
    REPRESENTATION THEORY Tammo tom Dieck Mathematisches Institut Georg-August-Universit¨at G¨ottingen Preliminary Version of February 9, 2009 Contents 1 Representations 4 1.1 Basic Definitions . 4 1.2 Group Actions and Permutation Representations . 9 1.3 The Orbit Category . 13 1.4 M¨obius Inversion . 17 1.5 The M¨obius Function . 19 1.6 One-dimensional Representations . 21 1.7 Representations as Modules . 24 1.8 Linear Algebra of Representations . 26 1.9 Semi-simple Representations . 28 1.10 The Regular Representation . 30 2 Characters 34 2.1 Characters . 34 2.2 Orthogonality . 36 2.3 Complex Representations . 39 2.4 Examples . 42 2.5 Real and Complex Representations . 45 3 The Group Algebra 46 3.1 The Theorem of Wedderburn . 46 3.2 The Structure of the Group Algebra . 47 4 Induced Representations 50 4.1 Basic Definitions and Properties . 50 4.2 Restriction to Normal Subgroups . 54 4.3 Monomial Groups . 58 4.4 The Character Ring and the Representation Ring . 60 4.5 Cyclic Induction . 62 4.6 Induction Theorems . 64 4.7 Elementary Abelian Groups . 67 5 The Burnside Ring 69 5.1 The Burnside ring . 69 5.2 Congruences . 73 5.3 Idempotents . 76 5.4 The Mark Homomorphism . 79 5.5 Prime Ideals . 81 Contents 3 5.6 Exterior and Symmetric Powers . 83 5.7 Burnside Ring and Euler Characteristic . 87 5.8 Units and Representations . 88 5.9 Generalized Burnside Groups . 90 6 Groups of Prime Power Order 94 6.1 Permutation Representations . 94 6.2 Basic Examples . 95 6.3 An Induction Theorem for p-Groups .
    [Show full text]