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Introduction to Representation Theory and Characters

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INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS HANMING ZHANG Abstract. In this paper, we will first build up a background for representa- tion theory. We will then discuss some interesting topics in representation the- ory, including the basic properties of a representation, G-invariant subspaces, characters, and regular representations. Contents 1. Groups 1 2. The Hermitian Product 2 3. Representations 3 4. Irreducible Representations 4 5. Characters 6 6. Regular Representation 9 Acknowledgments 10 References 10 1. Groups We begin with a short discussion of groups, which will be important for the rest of the paper. Definition 1.1. A group is a set G with an associative binary operation for which there exists an identity element with respect to which every element of G has its own inverse. The order of a group is the number of elements contained in the set G. Example 1.2. An example of a group would be the collection of all invertible n×n matrices with matrix multiplication as the operation. Each matrix has an inverse, and the identity matrix is the identity of the group. This is the general linear group or GLn(F ), where F is the field from which the entries of the matrices are taken. Example 1.3. Another example would be the set f xm j m 2 Zg under the relation xn = 1 with multiplication as the operation and 1 as the identity element. This group is called the cyclic group of order n. The element x is known as the generator of this group. Theorem 1.4. Cancellation Law : Let f,g,h be elements of a group G. If fg = fh, then g = h and if gf = hf, then g = h. 1 2 HANMING ZHANG Proof. Assume fg = fh. Let f −1 be the inverse of f. Multiply both sides of fg = fh byf −1 on the left. This yields g = f −1fg = f −1fh = h: The other case is similar. Definition 1.5. A subset S of a group G is called a subgroup if it has the following properties: (i) Closure: If a 2 S and b 2 S, then ab 2 S. (ii) Identity: eG 2 S, where eG is the identity of G. (iii) Inverses: If a 2 S, then a−1 2 S. Note that since all elements of S are also elements of G, all elements of S satisfy the properties of G. Definition 1.6. Two elements g and g0 of a group G are called conjugate if g0 = aga−1 for some a in G. A conjugacy class is a subset of the group such that all elements of the conjugacy class are conjugates of each other. Definition 1.7. Let G and H be groups. A group homomorphism is a mapping φ : G ! H such that φ(ab) = φ(a)φ(b) for all a; b 2 G. Proposition 1.8. Let eG be the entity element of a group G. Then a group homomorphismφ : −1 −1 G ! H satisfies φ(eG) = eH and φ(a ) = φ(a) . Proof. Since eG = eGeG and by the definition of a group homomorphism, φ(eG)eH = φ(eG) = φ(eGeG) = φ(eG)φ(eG): Then by the cancellation law, we have eH = φ(eG). Next, for all a in G, −1 −1 φ(a)φ(a ) = φ(aa ) = φ(eG) = eH : −1 −1 −1 Also, φ(a)φ(a) = eH . Therefore, φ(a)φ(a ) = φ(a)φ(a) , which implies −1 −1 φ(a ) = φ(a) . 2. The Hermitian Product Definition 2.1. For any two vectors X = (x1; x2; :::; xn) and Y = (y1; y2; :::; yn) in the vector space Cn, the standard hermitian product is a map h; i: V × V ! C defined by the formula hX; Y i =x ¯1y1 +x ¯2y2 + ::: +x ¯nyn; wherea ¯ is the complex conjugate of a. The general hermitian form is defined by the following properties: (i) Linearity in the second variable: hX; cY i = chX; Y i and hX; Y1 + Y2i = hX; Y1i + hX; Y2i (ii) Conjugate symmetry: hX; Y i =c ¯hY; Xie The standard hermitian product is a hermitian form. INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS 3 Definition 2.2. Two vectors v and w are orthogonal if hv; wi = 0; alternatively, this relation can be written as v ? w. If two subspaces W and U are such that 8 w 2 W and u 2 U; hw; ui = 0; then W and U are orthogonal; this relation can also be written as W ? U. Also, the term W ? refers to the set of all vectors orthogonal to W . In addition, an orthonormal basis is a basis B = (v1; v2; :::; Vn) such that hvi; vii = 1 and hvi; vji = 0 if i 6= j Theorem 2.3. If W is a subspace of a vector space V and U = W ?, then the vector space V is a direct sum of W and U or V = W ⊕ U. We will only prove this theorem for the case when dim W = 1, and omit the rest of the proof. Proof. We first need to prove that W and U only share the zero vector. This is already clear, since they can only share the zero vector because U = W ?. Next we need to prove that for any x in V , x can be written as a sum of elements of W and U. In other words, we want x = aw + u for some scalar a, w in W and u in U. It is enough to prove x−aw is in U for some scalar a. This is true if and only if hx − aw; wi = 0. Solving for a using the properties of the hermitian product, we hx;wi have a = hw;wi . This solution will ensure that x − aw is in U. Definition 2.4. A linear operator T is said to be unitary if 8 v; w; hv; wi = hT (v);T (w)i Definition 2.5. A complex matrix P is unitary if P¯tP = I, where P¯ is the matrix with entries that are the complex conjugates of the entires of P and P¯t is the transpose of that matrix. The set of all such matrices together with the operation matrix multiplication forms the unitary group Un. 3. Representations Definition 3.1. A matrix representation R of a group G is a group homomorphism R : G ! GLn(F ), where F is a field. For any g in G, Rg will denote the image of g. If we do not consider a specific basis, then we can talk of the concept of a representation of a group on a finite-dimensional vector space V . The group of invertible linear operators on V is denoted by GL(V ), with composition of functions as the binary opeartion. In addition, if we do choose a specific basis B, then there will be an isomorphism between this group and the group of invertible matrices GL(V ) ! GLn(F ), which maps the linear operator to the matrix of the linear operator with respect to the basis B. We will define representation of a group G on V as a group homomorphism ρ : G ! GL(V ). The dimension of the representation ρ is defined to be the dimension of V . Similarly to matrix representations, we will use ρg to denote the image of the representation. Consider a representation ρ. If we also choose a basis B, then for any g in the group, the matrix Rg of ρg gives a matrix representation of the group. In addition, for each vector vi in the basis B, ρg(vi) is equal to the linear combination of the 4 HANMING ZHANG th vectors of B where the coefficients are the i column of Rg, which is a n×n matrix. This is true for every vector in B, so we can write this relationship as ρg(B) = BRg. Example 3.2. Let G8 be the cyclic group of order 8 with x as its generator. Let 2 Rot(θ) be rotation by θ. Then we can define a representation ρ : G8 ! GL(R ) 2nπ xn 7! Rot( ): 8 Note that if n = k8, where k is an integer, then the image will be the identity; this means ρ is a well-defined function. Definition 3.3. Two representations ρ : G ! GL(V ) and ρ0 : G ! GL(V 0) of a group G are called isomorphic if there exists an isomorphism T : V ! V 0 between the two vector spaces V and V 0 such that 0 ρg(T (v)) = T (ρg(v)); 8 v 2 V and g 2 G: We can also talk of isomorphism classes, which are just collections of representations that are isomorphic to each other. Corollary 3.4. If two representations ρ : G ! GL(V ) and ρ0 : G ! GL(V 0) are 0 isomorphic such that T : V ! V is an isomorphism, and if B = fv1; v2; :::; vng and 0 0 B = fT (v1);T (v2);T (v3); :::T (vn)g are corresponding bases for V and V , then the corresponding matrix representations R with respect to B and R0 with respect to B0 will have the same image for each member of the group. 0 Proof. We want to show that for all g in G, Rg = Rg. Let B = fv1; v2; :::; vng be a 0 0 0 basis for V and let B be the basis for V such that B = fT (v1);T (v2);T (v3); :::T (vn)g. We know by the definition of isomorphic representations that there exists an iso- morphism T : V ! V 0 such that 0 (3.5) T (ρg(vi)) = ρg(T (vi)): 0 0 Let Rg and Rg be matrix representations of ρ and ρ with respect to bases B and 0 0 0 0 B .
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