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Fourier Analysis Using Representation Theory

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Fourier Analysis Using Representation Theory FOURIER ANALYSIS USING REPRESENTATION THEORY NEEL PATEL Abstract. In this paper, it is our goal to develop the concept of Fourier transforms by using Representation Theory. We begin by laying basic def- initions that will help us understand the definition of a representation of a group. Then, we define representations and provide useful definitions and the- orems. We study representations and key theorems about representations such as Schur's Lemma. In addition, we develop the notions of irreducible repre- senations, *-representations, and equivalence classes of representations. After doing so, we develop the concept of matrix realizations of irreducible represen- ations. This concept will help us come up with a few theorems that lead up to our study of the Fourier transform. We will develop the definition of a Fourier transform and provide a few observations about the Fourier transform. Contents 1. Essential Definitions 1 2. Group Representations 3 3. Tensor Products 7 4. Matrix Realizations of Representations 8 5. Fourier Analysis 11 Acknowledgments 12 References 12 1. Essential Definitions Definition 1.1. An internal law of composition on a set R is a product map P : R × R ! R Definition 1.2. A group G, is a set with an internal law of composition such that: (i) P is associative. i.e. P (x; P (y; z)) = P (P (x; y); z) (ii) 9 an identity; e; 3 if x 2 G; then P (x; e) = P (e; x) = x (iii) 9 inverses 8x 2 G, denoted by x−1, 3 P (x; x−1) = P (x−1; x) = e: Let it be noted that we shorthand P (x; y) as xy. Definition 1.3. Given a group G, we call the group abelian if xy = yx 8x; y 2 G. (In other words, the map is commutative.) Date: DEADLINE AUGUST 21, 2009. 1 2 NEEL PATEL Definition 1.4. If #(G) is finite, then we call #(G) the order of G and denote it by o(G). Definition 1.5. A is a subgroup of G if 8x; y 2 A, both xy and x−1 are in A. Definition 1.6. A bijection is a map φ : A ! B such that the map is: (i) surjective: 8b 2 B 9 a 2 A 3 φ(a) = b: (ii) injective: 8b; y 2 B and a; x 2 A 3 φ(a) = b and φ(x) = y, b = y implies a = x. Definition 1.7. A map φ : G ! H, where G and H are groups, is called a homomorphism if φ(xy) = φ(x)φ(y)8x; y 2 G. If a homomorphism is a bijection, then we call it an isomorphism. If an isomorphism is from G to itself, then it is called an automorphism. Note that this definition of a homomorphism implies that φ(x−1) = (φ(x))−1 and that φ(e) = e. Also, Hom(G,H) will be the family of homomorphisms from G to H. Similarly, Aut(G) will be the automorphisms of G. Definition 1.8. Given a group G and an element g 2 G, we define an inner automorphism −1 generated by g as ig(x) = gxg . Definition 1.9. If N is a subgroup of G and ig(N) ⊂ N 8g 2 G, then we call N a normal subgroup of G. Definition 1.10. A direct product of two groups G and H is denoted by G × H and is a Cartesian product, f(g; h) jg 2 G; h 2 Hg, equipped with the operations: 0 0 0 0 • (g; h)(g ; h ) = (gg ; hh ) • eGeH = eG×H • (g; h)−1 = (g−1; h−1): The following theorem is a simple theorem that illuminates several key points about the direct product. Theorem 1.11. Given groups G; K, let Ge = f(g; e)jg 2 Gg ; Ke = f(e; k)jk 2 Kg. Both Ge and Ke are contained in G × K. The following are true: (i) Ge and Ke are normal subgroups of G × K. (ii) Ge \ Ke = feG×K g. (iii)Ge and Ke generate G × K, i.e. if x 2 G × K then x = geek for some ge 2 Ge and ek 2 Ke. (iv) If H is a group and Ge and Ke obey (i)-(iii), where G × K is replaced by H, we can conclude that H is isomorphic to Ge × Ke. 0 0 0 0−1 Proof. (i) Let x 2 G × K, where x = (g ; k ). Then ix((g; e)) = (g gg ; e) and we 0 0−1 0 know that g gg 2 G since g 2 G; g 2 G. So ix(Ge) ⊂ Ge 8x 2 G × K: The proof is similar for Ke. (ii) In one direction, it is clear that Ge \ Ke ⊃ feG×K g To show the other direction, suppose Ge \ Ke ⊂ feG×K g is not true. Then 9 some (g; e) or (e; k) 3 it is in both Ge and Ke. Then, this implies that there exists more than one identity in G or K, which is impossible. (iii) Let x = (g; k) for some g 2 G and k 2 K. Then (g; e)(e; k) = (g; k) for FOURIER ANALYSIS USING REPRESENTATION THEORY 3 (g; e) 2 G;e (e; k) 2 Ke. −1 −1 (iv) First note that if geek = gk, where g;e g 2 Ge and ek; k 2 Ke, then g ge = kek 2 Ge \ Ke ) g = g;e k = ek. Furthermore, since H obeys (iii), we see that τ : Ge × Ke ! H, where (g;e ek) 7! geek, is a bijection. To show that τ preserves products, we first see that given g 2 Ge and k 2 Ke, gkg−1k−1 = (gkg−1)k−1 2 Ke. Similarly, g(kg−1k−1) 2 Ge. Both of the last two statement are true due to (i) and imply that gk = kg, i.e. we have commutativity by (ii). τ((g; k); (g;e ek)) = τ(gg;e kek) = (gge)(kek) = (gk)(geek) = τ((g; k))τ((g;e ek)). Thus τ preserves products and is an isomorphism. In the next definition, we assume the reader knows the definition of a vector space. Definition 1.12. A linear map T : V ! V , where V is a vector space, is linear if it preserves vector addition and scalar multiplication. That is, T (v+w) = T (v)+T (w) and T (a · v) = aT (v), 8v; w 2 V and 8 scalars a. In the next definition, the reader should know that a basis for a vector space is a set of linearly independent vectors such that any vector in the vector space can be written as a linear combination of vectors in the basis. Definition 1.13. The dimension of V , a vector space, is the number of elements of a basis of the vector space. Just for the reader's knowledge, any two bases of a vector space have the same cardinality, or number of vector elements. Also, if a linear map T has an inverse, then it is called invertible. The set of all invertible linear maps is denoted by GL(V ). 2. Group Representations Definition 2.1. Given a group G, a group representation of that group is an element of Hom(G, GL(V )) for some vector space V . The degree of a representation is simply the dimension of V . Definition 2.2. An inner product is a map h·; ·i: V × V ! C 3: (i) hv; ·i is a linear map if v is fixed. (ii) hv; wi = hw; vi (iii)hv; vi ≥ 0 and hv; vi = 0 iff v = 0. Definition 2.3. A unitary representation preserves inner product. In notation, this means that hU(g)v; U(g)wi = hv; wi 8g 2 G; v; w 2 V: Theorem 2.4. Given U : G ! GL(V ), a representation of a finite group G, 9 inner product h·; ·i 3 each U(g) is unitary. 0 Proof. Suppose we have inner product h ; i on our vector space V . Let us con- 1 P 0 struct another inner product hv; wi = o(G) g2G hU(g)v; U(g)wi Using our new inner product on V , 4 NEEL PATEL 1 X 0 hU(i)v; U(i)wi = hU(g)U(i)v; U(g)U(i)wi o(G) g2G 1 X 0 = hU(gi)v; U(gi)wi o(G) g2G 1 X 0 = hU(g)v; U(g)wi o(G) g2G since fgijg 2 Gg = G. 1 P 0 And, o(G) g2G hU(g)v; U(g)wi = hv; wi Note that we need our group G to be finite. Definition 2.5. A unitary representation is a homomorphism from a finite group G to U(V), the group of unitary operators on V . From here onwards, when we use the term representation, we mean unitary representation. This is not such a big deal because, by the theorem above, any representation can be made unitary by the appropriate choice of inner product. Definition 2.6. Given U : G ! U(V);T : G ! U(W), where G is finite, U is unitarily equivalent to T iff 9 unitary S : V ! W 3 T (g) = SU(g)S−1 8g 2 G: Definition 2.7. Given vector spaces V and W , we define the direct sum V ⊕W to be the Cartesian product of the two vector spaces with operations carried out coordinate-wise. (i.e. (v1; w1) + (v2; w2) = (v1 + v2; w1 + w2) and inner product is defined by h(v1; w1); (v2; w2)i = hv1; v2i + hw1; w2i).
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