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Cocyclic Hadamard Matrices Kshitija Vaidya

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Cocyclic Hadamard Matrices Kshitija Vaidya Cocyclic Hadamard Matrices Kshitija Vaidya Supervised by Dr Santiago Barrera Acevedo Monash University Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute. T Abstract: A square {±1g-matrix H is Hadamard if HH = nIn. The existence of a Hadamard matrix of order 4n for each n 2 N is conjectured. A recent algebraic approach to constructing Hadamard matrices connects cohomology with combinatorics, giving rise to cocyclic Hadamard matrices. Flannery's paper 'Calculation of Cocyclic Matrices' [4], which is the focus of this report addresses the problem of computing cocyclic martrices. 1 Introduction T A Hadamard matrix H is an n × n {±1g-array such that HH = nIn, where In is the n × n identity matrix and HT is the transpose of H. Hadamard proved that all Hadamard matrices have order 1, 2 or 4n for some n 2 N [4]. The following conjecture, known as the Hadamard conjecture, remains an open problem: Conjecture 1.1. There exists a Hadamard matrix of order 4n for each n 2 N. Central to the study of Hadamard matrices is the notion of Hadamard equivalence. Definition 1.2. Two {±1g-matrices M1 and M2 are said to be Hadamard equivalent (written M1 ∼ M2) if M1 = PM2Q where P and Q are signed permutation matrices. This defines an equivalence relation on the set of Hadamard matrices of order n. Some classical constructions of Hadamard matrices include the Sylvester, Williamson and Ito constructions [7]. Cocyclic Hadamard matrices constitute another family of Hadamard matrices. The connection between 2-cocycles in cohomology and Hadamard matrices was established by Horadam and de Launey [6]. We now know that many classical constructions of Hadamard matrices are cocyclic. As such, cocyclic Hadamard matrices may provide a uniform approach to Conjecture 1.1. This makes the calculation of cocyclic matrices an important problem. In response to this, Flannery developed a new method of calculating cocyclic matrices [4]. Section 2 of this report covers background on cohomology, cocycles and cocyclic matrices. Sections 3, 4 and 5 detail the results of Flannery's paper [4]. In Section 5 we apply Flannery's theory to construct the cocycles from the Klein four-group to C2. The list we obtain matches earlier computations. The ideas presented in this report are Flannery's; we simply expand on their explanations [4]. 2 Preliminaries Knowledge of basic group theory is assumed. The following concepts are defined in Appendix A: the centre of a group, torsion subgroups, group actions, G-modules, exact sequences of groups, group 1 extensions, lifts, free groups, commutators and commutator subgroups. In the following two definitions G is a group and U a G-module under some action ?. Definition 2.1. [7, p.114 Definition 6.1] For n ≥ 0, the abelian group of n-cochains is the set n n C (G; U) = ff : G ! Ujf(x1; :::; xn) = 0 whenever xi = 1 for some ig with group action given by pointwise addition. By definition, jG0j = 1, and so C0(G; U) =∼ f0g. The n n+1 degree n differential is the homomorphism @n : C (G; U) ! C (G; U) defined by n X i @n(f)(x1; :::; xn+1) = x1 ? f(x2; :::; xn+1) + (−1) f(x1; :::; xi−1; xixi+1; :::; xn+1) i=1 n+1 + (−1) f(x1; :::; xn); n which satisfies @n ◦ @n−1 = 0. The group of n-cocycles from G to U is denoted Z (G; U) and de- n fined as ker(@n). The group of n-coboundaries from G to U is denoted B (G; U) and defined as 2 Im(@n−1). The nth-cohomology group of G with coefficients in U is denoted H (G; U) and defined as Zn(G; U)=Bn(G; U). Each coset f + B2(G; U) is called a cohomology class and is denoted [f]. The following is an equivalent definition of 2-cocycles and 2-coboundaries, or simply (normalised) cocycles and coboundaries. Multiplicative notation is from now on unless otherwise stated. Definition 2.2. [7, p.115, Definition 6.2] A (normalised) 2-cocycle (or simply cocycle) is a map : G × G ! U such that (g; h) (gh; k) = (g ? (h; k)) (g; hk), (g; 1) = 1 = (1; g), for all g; h; k 2 G.A coboundary @φ is a cocycle of the form @φ(g; h) = φ(g)−1(g ? φ(h))−1φ(gh) for some 1-cochain φ : G ! U. Definition 2.3. Let G = fg1 = 1; g2; :::; gng be a finite group and U a G-module. A v × v matrix M is G-cocylic over U if M ∼ [ (gi; gj)]1≤i;j≤v. A cocyclic Hadamard matrix is a Hadamard matrix that is G-cocyclic over C2. 3 Inflation, transgression and a five-term exact sequence Here, we introduce the inflation and transgression homomorphisms which will be used in Section 4 to construct embeddings into the second cohomology group. These homomorphisms appear in an exact sequence related to the Lyndon-Hoschild-Serre spectral sequence. 2 3.1 Inflation Let G be a finite group and U a G-module under the group action (g; u) 7! g ? u. Let N E G. Define U N := fu 2 U such that n ? u = u for all n 2 Ng. The action of G on U is used to define an induced action of G=N on U N where (gN; u) 7! g ? u for all u 2 U N and gN 2 G=N. Now define a map inf : Zn(G=N; U N ) ! Zn(G; U N ); where 7! inf ; and inf (g1; g2; ::::; gn) := (g1N; g2N; ::::; gnN). This is the inflation map on cocycles. We define an inflation map on cohomology classes inf : Hn(G=N; U N ) ! Hn(G; U N ); where [ ] 7! [inf ]: Inflation is well-defined (see Appendix B). It is straightforward to prove that inf is a homomorphism. 3.2 Transgression For the rest of Section 3 suppose N ≤ Z(G) and G acts trivially on U. This induces a trivial action of G=N on U. Let proj. 1 ! N −−!inc. G −−−! G=N ! 1 be a central extension of N by G=N (i.e. an extension such that N ≤ Z(G)). For convenience we identify i(N) with N, treating N as a subgroup of G. Let σ : G=N ! G be a normalised lift. Define µσ to be the cocycle associated with this extension, constructed in the usual way: −1 µσ(gN; hN) := σ(gN)σ(hN)σ(gN · hN) ; −1 where gN; hN 2 G=N. Clearly µσ is well-defined. Since π(σ(gN)σ(hN)σ(gN · hN) ) = 1, µσ maps elements from G=N into N. The construction of µσ is independent of the choice of normalised lift, 2 and µσ 2 Z (G=N; N). A more general proof of this is in Appendix E (see the proof of Theorem E.1). We define transgresssion: 2 τ : Hom(N; U) ! H (G=N; U); where φ 7! [φ ◦ µσ]: Again, it is straightforward to show that τ is a homomorphism. 3.3 A five-term exact sequence The inflation and transgression homomorphisms arise in the following five-term exact sequence which comes from the Lyndon-Hoschild-Serre spectral sequence (for a proof of exactness see Appendix C): inf 0 ! Hom(G=N; U) −−!infa Hom(G; U) −−!res Hom(N; U) −!τ H2(G=N; U) −−−!b H2(G; U): (1) 3 The homomorphism res is induced by restricting the domain of elements in Hom(G; U) to N. The exactness of (1) is used to show that the trangsression map is injective. Lemma 3.1. [4, p.183, Lemma 2.1] Suppose that G; N; U are defined as above, and N ≤ G0. Then τ : Hom(N; U) ! H2(G=N; U) is injective. Proof. Let 2 Hom(G; U) and n 2 N, then n = [g; h] := g−1h−1gh for some g; h 2 G. So, (n) = (g−1h−1gh) = (g−1) (h−1) (gh) = (h−1) (g−1) (gh) = (h−1g−1gh) = (1) = 1 The above implies that Im(res) = f1g where res : Hom(G; U) ! Hom(N; U). Since (1) is exact, we have Ker(τ) = f1g and thus τ is injective. 4 The Universal Coefficient Theorem and a decomposition of H2(G; U) In this section we introduce the Universal Coefficient Theorem which expresses H2(G; U) as an external direct sum. Via embeddings, we obtain this decomposition as an internal direct sum. This ultimately provides a method of calculating cocyclic matrices associated with the elements of Z2(G; U). 4.1 The Universal Coefficient Theorem From now on, let G be finite and U a finitely generated, trivial G-module. A cocylce 2 Z2(G; U) is symmetric if (g; h) = (h; g) for all g; h 2 G. If A is abelian then Ext(A; U) is defined to be the set of all 2-cocycle classes in H2(A; U) containing at least one symmetric cocycle. Theorem 4.1. [8, p.179, Theorem 3.3] (Universal Coefficient Theorem). For a finite group G and a trivial G module, U, 2 ∼ 0 H (G; U) = Ext(G=G ;U) × Hom(H2(G);U) (2) 2 ∗ where H2(G) denotes the Schur multiplier of G which we define in the usual way as H (G; C ). Proof. See [8, Theorem 3.3]. Note that G0 is the commutator subgroup of G which is defined in Appendix A. 0 0 Lemma 4.2.
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