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Note on Character Theory A NOTE ON CHARACTER THEORY FAN HUANG In this note, we explore two examples. We try to glimpse both the analogy between character theory of finite groups and representation theory of the infinite compact group, and the power of character theory for nonabelian groups. 1. Basic Facts of Character Theory of Finite Groups Let G be a finite group and ρ : G ! GL(V ) be a linear representation of G in the complex finite dimensional vector space V . For each g 2 G, put χρ(g) = T r(ρ(g)). The complex valued function χρ on G is called the character of the representation ρ and it is a class function{ those are any functions from G into F which are constant on the conjugacy classes of G, i.e., f : G ! F such that f(g−1xg) = f(x) for all g; x 2 G. It characterizes the representation ρ in the sense that representations are determined by their characters, up to isomorphisms. Let ρ : G ! GL(V ) τ : G ! GL(W ) be two representations of a group G on vector spaces V and W . An isomorphism from V and W is a G-linear map f : V ! W such that τ(g) ◦ f(v) = f ◦ ρ(g)(v); 8g 2 G; 8v 2 V i.e., for all g 2 G, the below square commutes. ρ(g) V V f f W W τ(g) In that case, we say that two representations ρ and τ are isomorphic. In terms of matrices, two matrix representations are isomorphic if and only if they describe the same representation in different bases. Specifically, choose a basis A = fa1; : : : ; ang for V and a basis B = fb1; : : : ; bng for W , and let ρA : G ! GLn(C) τB : G ! GLn(C) be the two matrix representations obtained by writing ρ and τ in these bases. Then ρ and τ are isomorphic if and only if ρA and τB are isomorphic. Proof. It can be illustrated via the commutative graphs: Date: August 2013. 1 A NOTE ON CHARACTER THEORY 2 τB(g) Cn Cn B B τ(g) W W S f f S−1 ρ(g) V V A A Cn Cn ρA(g) There is an inner product on the vector space of class functions given by 1 X hf; hi = f(g)h(g); jGj g2G where jGj is the order of the group G. The characters of irreducible representations form an orthonormal basis for the space of class functions on G, denoted as Cl(G). Every represen- tation is a direct sum of irreducible representations. Let ρ be any representation of G and ρi be an isomorphism class of irreducible representations of G. Let χ and χi be the characters of ρ and ρi respectively. Then the multiplicity of ρi in ρ is the inner product hχi; χi. 2. The representation theory of the circle group S1 (the simplest infinite compact group) Consider the unit circle S1 = fz 2 C× j jzj= 1g = f(x; y) 2 R2 j x2 + y2 = 1g in the complex plane. If we view the circle as the set of points feiθ : θ 2 Rg, the natural group operation is multiplication of complex numbers. We treat S1 as an abelian topological group under multiplication with its topology as a subset of C. (A topological group is a group G equipped with a topology such that the product map G×G ! G and the inverse map G ! G are both continuous. More generally, a representation of a topological group on a vector space V is a continuous group homomorphism ρ : G ! GL(V ) with the topology of GL(V ) inherited from the space End(V ) of continuous linear operators). However, if we identify points on S1 with their angle θ, then S1 becomes R modulo 2π, denote it R=2πZ, where the operation is addition.The group isomorphism R=2πZ ! S1 is given by t 7! eit = cos t+i sin t. Math 597 Fan Huang FAN HUANG 3 From [JM74], since it is a continuous bijection between compact topological spaces, it is a homeomorphism. In this note we will focus on unitary representations of S1, i.e., representations on a Hilbert space which preserve the inner product. Unitary representations are those for which the operators ρg are unitary, i.e. preserve the inner product. On the complex plane, this t means hρgv; ρgwi = hv; wi = wv, 8v; w 2 V; g 2 G. which means equivalently that ρg ρg = I. 1 1 We denote the unitary representation of S as ρn : S ! U(V ). n It is not hard to see that for each integer n, the formula ρn(z) = z gives a 1-dimensional irreducible representation of S1. We shall show that (1) Every irreducible unitary representation of S1 has dimension 1. (2) Every continuous irreducible representation of S1 is of this form. (3) For n 6= k, ρn and ρk are not isomorphic. 1 That is, Z is isomorphic to irreducible representations of S with bijection given by n 7! ρn. First, we note that if W is an irreducible unitary representation of S1, then dim W = 1. To see this, we take as a fact that ρ is determined by ρ(g) where g 2 S1 is a topological generator. Then by density, we know where every other element goes. The spectral theorem from linear algebra asserts that any unitary transformation is diagonalizable. Put another way, ρ0(g) = Sρ(g)S−1, where S 2 U(V ) and ρ0(g) is a diagonal matrix. The eigenvalues of unitary matrices all have absolute value 1. It is obvious that for each i the diagonal term 0 Ln ρi(g) is a 1-dimensional unitary irreducible representation. This follows from ρ = i=1 ρi, Ln −1 ρ = i=1 SρiS and thus dim W = 1 as desired. Proposition 2.1. Every irreducible unitary representations of S1 are the continuous homo- morphisms of the form z 7! zn for some n 2 Z. To prove this proposition, we need two Lemmas. φ R R 1 1 S ρ S Lemma 2.1. Consider (R; +). If φ : R ! R is a continuous homomorphism, then φ is a multiplication by a scalar. 1 1 c Proof. Set c = φ(1). Then φ(n) = nc; n 2 Z. Also mφ( m ) = c and then φ( m ) = m ; m 2 Z. n n Thus φ( m ) = c m and so φ(x) = cx, for x 2 Q. Since Q is dense in R and φ is continuous, we have φ(x) = cx for all x 2 R. Lemma 2.2. If : R ! S1 is a continuous homomorphism, then there exists c 2 R such that (x) = eicx for all x 2 R. Proof. We claim that there is a unique continuous homomorphism l : R ! R such that (x) = ei·l(x). The exponential map " : R ! S1 given by "(x) = eix maps the real line around the unit circle with period 2π and it is a universal cover. For any continuous : R ! S1 such that (0) = 1, there exists a unique continuous lift l of this function to the real line such that l(0) = 0, In other words, there exists a unique continuous function l : R ! R such that l(0) = 0 and (x) = "(l(x)) for all x, so the following diagram commutes: Math 597 Fan Huang A NOTE ON CHARACTER THEORY 4 R l " S1 R We also claim that if is a homomorphism, then its lift l is also a homomorphism and by Lemma 2.1 l(x) = cx for some c. Note that (s + t) = (s) (t), thus "(l(s + t) − l(s) − l(t)) = 1 = e0. It follows that l(s + t) − l(s) − l(t) = 2πn for some n 2 Z which depends only on s; t. Since s and t varies continuously, we find n is a constant. We set s = t = 0 to conclude that n = 0. Thus l is a homomorphism as promised, and so l(x) = cx for some c 2 R by Lemma 2.2. Proof of proposition 2.1. Going back to the irreducible unitary representations of S1, given a representation ρ : S1 ! C∗, it has a compact and thus bounded image. It follows that the image lies on S1. Thus ρ : S1 ! S1 is a continuous homomorphism. Precompose ρ with the exponential map ", then we have the following diagram commutes: ρ S1 S1 " ρε R By Lemma 2.2 there exists c 2 R with ρε(x) = eicx for all x. Thus we have ρ(eix) = eicx. 2πi 2πic Note that 1 = ρ(1) = ρ(e ) = ρε(2π) = e and thus c 2 Z. So we have ρ(z) = ρn(z) = n z . Finally, to show for n 6= k, that ρn and ρk are not isomorphic, because let f : C ! C be the G-linear map. Note that znf(v) = f(znv) = zkf(v) for all z 2 S1, then we have zn−k = 1, i.e., n = k. Because of the isomorphism from S1 to R=2πZ, another way to view the characters on S1 inx is as a map R=2πZ ! C. Then the characters are given by en(x) = e with n 2 Z. We shall show all characters have this form by following an exercise from [ST03]. First we denote by G^ the class of all characters of G, which is called the dual group of G. We observe that the trivial character e(g) = 1 plays the role of the unit, so G^ inherits the structure of an abelian group under the multiplication χ1χ2(g) = χ1(g)χ2(g) for all g 2 G and characters χ1; χ2.
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