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REPRESENTATIONS of FINITE GROUPS Contents 1. Introduction 1 REPRESENTATIONS OF FINITE GROUPS SANG HOON KIM Abstract. This paper provides the definition of a representation of a finite group and ways to study it with several concepts and remarkable theorems such as an irreducible representation, the character, and Maschke's Theorem. Contents 1. Introduction 1 2. Group representations 1 3. Important theorems regarding G-invariant Hermitian form and unitary representations 3 4. Irreducible representations and Maschke's Theorem 7 5. Characters 8 6. Permutation representations 12 7. Regular representations 13 8. Schur's Lemma and proof of Orthogonality relations 14 Acknowledgments 18 References 18 1. Introduction Representation theory studies linear operators that are given by group elements acting on a vector space. Thus, representation theory is very useful in that it makes it possible to delve into geometric symmetries effectively with using coordinates, which is not possible if abstract group elements are only considered. This paper assumes that readers prior knowledge of linear algebra, including familiarity with Hermitian form and Spectral Theorem. Moreover, we deal only with finite groups. 2. Group representations We will start with a simple example of a representation of the group T of rotations of a tetrahedron. Suppose that the group T acts on a three-dimensional vector space V . Choose a basis (v1; v2; v3) so that each element of the basis passes through the midpoints A, B, C of three edges as follows. 1 2 SANG HOON KIM v3 C D v B 2 A v1 Figure 1. Tetrahedron on a 3-dimensional space Let x 2 T be the counter clockwise rotation around the vertex D by 2π=3 and y1; y2; y3 2 T be the counter clockwise rotations by π around the vertices A, B, C, respectively. The matrices that correspond to the group elements are 2 13 21 3 2−1 3 (2.1) Mx = 41 5 ;My1 = 4 −1 5 ;My2 = 4 1 5 ; 1 −1 −1 2−1 3 My3 = 4 −1 5 : 1 The group T is generated by fx; y1; y2; y3g. Therefore, the group of matrices isomorphic to the group T is generated by fRx;Ry1 ;Ry2 ;Ry3 g. As in the example above, we define a matrix representation of a group G as follows. Definition 2.2. A matrix representation of a group G is a homomorphism M : G ! GLn(F) where GLn(F) is the general linear group of degree n over a field F, the set of n × n invertible matrices. We will denote each image of g, an element of G, by Mg; the matrix repre- sentation M carries g to the invertible matrix Mg. Since M is a homomorphism, −1 −1 Mg−1 ,the image of g , is the inverse of Mg, which is thereby Mg . Also, the identity 1 2 G is carried to the n-dimensional identity matrix In = M1. Since a vector space does not have a unique basis, it is possible to deal with a representation without fixing a basis. Therefore, we introduce the definition of a group representation on a finite-dimensional vector space V . Definition 2.3. A representation of G on V is a homomorphism ρ : G ! GL(V ) where GL(V ) is the group of invertible linear operators on V . Definition 2.4. The dimension of ρ is defined to be the dimension of the vector space V . Choosing a basis, we can construct an isomorphism Φ : GL(V ) ! GLn(F) as follows: let ρg be the image of g via ρ. ρg defines a matrix Mg for a given basis B1 = (v1; v2; :::; vn). REPRESENTATIONS OF FINITE GROUPS 3 M : G / GLn(F) O isomorphic ρ : G / GL(V ) If we treat the representation ρg with a different basis B2 = (w1; w2; : : : ; wn), we get a different matrix with respect to the basis B2 via a change of basis matrix P in V as follows: 0 −1 (2.5) Mg = PMgP for every g 2 G: A group representation can also be thought of as a group action on a vector space V . As long as a group element acts on a vector space, its action should be consistent with the vector space, which means it should act as a linear operator on V . Therefore, it should satisfy the two axioms of a group action, which are 1v = v and (gh)v = g(hv) for all g; h 2 G and all v 2 V; and the two conditions of linear operator, 0 0 g(v + v ) = gv + gv and g(cv) = cgv for c 2 F: No matter what is given in the first place between a group representation and a group action, with the rule gv = ρg(v), we can define one by another. From now on, we will only deal with a complex representation that operates on a complex vector space. 3. Important theorems regarding G-invariant Hermitian form and unitary representations After defining a unitary representation, we will delve into several representations. Definition 3.1. A unitary representation is a homomorphism M : G ! Un from the group G to the unitary group Un. Let V be a Hermitian vector space. Then, a linear operator T is unitary if hv; wi = hT (v);T (w)i: In the same way, we can say a representation ρ : G ! GL(V ) is unitary if ρg is a unitary operator for all g 2 G, i.e. if (3.2) hv; wi = hρgv; ρgwi; which is also (3.3) hv; wi = hgv; gwi for all v; w 2 V . Given an orthonormal basis, a matrix representation, which is isomorphic to the unitary representation ρ, is unitary. Definition 3.4. Given a representation ρ of G on a vector space V , a form h·; ·i on V is called G-invariant if (3.2), or (3.3), is satisfied. Theorem 3.5. Let ρ be a representation of a finite group G on a complex vector space V . There exists a G-invariant, positive definite Hermitian form h·; ·i on V . 4 SANG HOON KIM Proof. Choose an arbitrary positive definite Hermitian form on V , which is denoted by {·; ·}. We can then define a new form as follows: 1 X (3.6) hv; wi = fgv; gwg N g2G where N is the order of G. We will show by the next lemma that the new Hermitian form (3.6) is G-invariant and positive definite. Lemma 3.7. The form (3.6) is a G-invariant, positive definite Hermitian form. Proof. For the form to be Hermitian, it should be linear in the second variable, conjugate-linear in the first variable, and Hermitian symmetric. (1) (Linearity in the second variable) Since the form {·; ·} is Hermitian and g acts as a linear operator on V , fgv; g(w + w0)g = fgv; gw + gw0g = fgv; gwg + fgv; gw0g: Thus, 1 X hv; w + w0i = fgv; g(w + w0)g N g2G 1 X 1 X = fgv; gwg + fgv; gw0g N N g2G g2G = hv; wi + hv; w0i: Also, fgv; g(cv0)g = fgv; c(gv0)g = cfgv; gv0g: Thus, 1 X hv; cv0i = fgv; g(cv0)g N g2G 1 X = c fgv; gv0g N g2G = chv; v0i: (2) (Conjugate linearity in the first variable) Similarly, we have that fg(v + v0); gwg = fgv + gv0; gwg = fgv; gwg + fgv0; gwg: REPRESENTATIONS OF FINITE GROUPS 5 Thus, 1 X hv + v0; wi = fg(v + v0); gwg N g2G 1 X 1 X = fgv; gwg + fgv0; gwg N N g2G g2G = hv; wi + hv0; wi: Also, fg(cv); gv0g = fc(gv); gv0g =c ¯fgv; gv0g: Therefore, 1 X hcv; v0i = fc(gv); gv0g N g2G 1 X =c ¯ fgv; gv0g N g2G =c ¯hv; v0i: (3) (Hermitian symmetry) Note that fgv; gv0g = fgv0; gvg. 1 X hv; v0i = fgv; gv0g N g2G 1 X = fgv0; gvg N g2G = hv0; vi: 1 Since h·; ·i is the sum of positive definite forms {·; ·} divided by the scalar N , h·; ·i is positive definite. To prove that the form h·; ·i is G-invariant, we should show that hv; wi = hg0v; g0wi for all v; w 2 V when g0 is an element of G. Indeed, 1 X 1 X hg v; g wi = fg(g v); g(g w)g = fgg v; gg wg 0 0 N 0 0 N 0 0 g2G g2G Since right multiplication by g0 is a bijection on G, gg0 runs over every element in G. Therefore, we can write the equation above as follows: 1 X 1 X hg v; g wi = fgg v; gg wg = fg0v; g0wg: 0 0 N 0 0 N g2G g02G 6 SANG HOON KIM The only difference between the equation we get and the original equation of the form is just the change in the order of the sum in the original equation. 1 X hg v; g wi = fg0v; g0wg 0 0 N g02G 1 X = fgv; gwg N g2G = hv; wi: The form h·; ·i is a G-invariant, positive definite Hermitian form. Now we prove the next theorem, which leads to remarkable corollaries. Theorem 3.8. Every matrix representation M : G ! GLn of a finite group G is conjugate to a unitary representation. In other words, given M, there is a matrix −1 P 2 GLn such that PMgP 2 Un for every g 2 G. Proof. Let V = Cn and a basis be the standard basis. Any homomorphism M : G ! GLn is the matrix representation associated to a representation ρ. By Theorem 3.5, there exists a G-invariant form h·; ·i on V ; we choose an orthonormal basis for V with respect to the form.
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