Representations of Finite Groups I (Math 240A) (Robert Boltje, UCSC, Fall 2014) Contents 1 Representations and Characters 1 2 Orthogonality Relations 13 3 Algebraic Integers 26 4 Burnside's paqb-Theorem 33 5 The Group Algebra and its Modules 39 6 The Tensor Product 50 7 Induction 58 8 Frobenius Groups 70 9 Elementary Clifford Theory 75 10 Artin's and Brauer's Induction Thoerems 82 1 Representations and Characters Throughout this section, F denotes a field and G a finite group. 1.1 Definition A (matrix) representation of G over F of degree n 2 N is a group homomorphism ∆: G ! GLn(F ). The representation ∆ is called faithful if ∆ is injective. Two representations ∆: G ! GLn(F ) and Γ: G ! GLm(F ) are called equivalent if n = m and if there exists an invertible matrix −1 S 2 GLn(F ) such that Γ(g) = S∆(g)S for all g 2 G. In this case we often write Γ = S∆S−1 for short. 1.2 Remark In the literature, sometimes a representation of G over F is defined as a pair (V; ρ) where V is a finite-dimensional F -vector space and ρ: G ! AutF (V )(= GL(V )) is a group homomorphism into the group of F -linear automorphisms of V . These two concepts can be translated into each other. Choosing an F -basis of V we obtain a group isomorphism ∼ AutF (V ) ! GLn(F ) (where n = dimF V ) and composing ρ with this iso- ∼ morphism we obtain a representation ∆: G ! AutF (V ) ! GLn(F ). If we choose another basis then we obtain equivalent representations. Conversely, if ∆: G ! GLn(F ) is a representation, we choose any n-dimensional F - vector space V and a basis of V (e.g. V = F n with the canonical basis) ∼ and obtain a homomorphism ρ: G ! GLn(F ) ! AutF (V ). Representations (V; ρ) and (W; σ) are called equivalent if there exists an F -linear isomorphism ': V ! W such that σ(g) = ' ◦ ρ(g) ◦ '−1 for all g 2 G. The above con- structions induce mutually inverse bijections between the set of equivalence classes of representations (V; ρ) and the set of equivalence classes of matrix representations ∆ of G over F . If one takes V := F n, the space of column vectors together with the standard basis (e1; : : : ; en) then the matrix representation ∆ corresponds to (F n; ρ) with ρ(g)v = ∆(g)v. 1.3 Examples (a) The trivial representation is the homomorphism ∆: G ! GL1(F ), g 7! 1. n 2 −1 (b) For n > 2 let D2n = hσ; τ j σ = 1; τ = 1; τστ = σ i be the dihedral group of order 2n. We define a faithful representation ∆ of D2n over R by cos φ − sin φ! −1 0! ∆(σ) := and ∆(τ) := ; sin φ cos φ 0 1 1 where φ = 2π=n. Note that ∆(σ) is the counterclockwise rotation about 2π=n and τ is the reflection about the vertical axis in R2. (c) The map ∆: Sym(n) ! GLn(F ), which maps σ 2 Sym(n) to the permutation matrix of σ is a faithful representation of Sym(n) of degree n over any field. It is called the natural representation of Sym(n). (The permutation matrix of σ has in its i-th colum the canonical basis vector eσ(i), i.e., ∆(σ) maps ei to eσ(i).) (d) If ∆: G ! GLn(F ) is a representation and H 6 G then the restriction of ∆ to H, G i ∆ Res (∆): H qqqqqqqqqq G qqqqqqqqqq GL (F ) qqqqq qqqqq H qqqqqqqq qqqqqqqq n qqqqqqqqqq qqqqqqqqqq is a representation of H over F of the same degree. Here, i: H ! G denotes the inclusion map. More generally, if f : H ! G is an arbitrary group homomorphism between two arbitrary finite groups then also f ∆ Res (∆): H qqqqqqqqqq G qqqqqqqqqq GL (F ) qqqqq qqqqq f qqqqqqqq qqqqqqqq n qqqqqqqqqq qqqqqqqqqq is a representation of H over F of degree n. (e) If G = fg1; : : : ; gng and if we define the homomorphism f : G ! Sym(n) by (f(g))(i) = j, when ggi = gj, then f ∆ G qqqqqqqqqq Sym(n) qqqqqqqqqq GL (F ) ; qqqqq qqqqq qqqqqqqq qqqqqqqq n qqqqqqqqqq qqqqqqqqqq with ∆ as in (c), is a representation, called the regular representation of G. If we rearrange the ordering of the elements g1; : : : ; gn we obtain an equivalent representation. (f) If ∆1 : G ! GLn1 (F ) and ∆2 : GLn2 (F ) are representations of G over F then also their direct sum ! ∆1(g) 0 ∆1 ⊕ ∆2 : G ! GLn1+n2 (F ) ; g 7! 0 ∆2(g) is a representation of G over F . Since ∆ (g) 0 ! 0 I ! ∆ (g) 0 ! 0 I ! 2 = n2 1 n1 ;; 0 ∆1(g) In1 0 0 ∆2(g) In2 0 we see that ∆1 ⊕ ∆2 and ∆2 ⊕ ∆1 are equivalent. (g) Let E=F be a Galois extension of degree n with Galois group G. Then G 6 AutF (E) and we obtain a representaion (E; i) in the sense of Remark 1.2, with i: G ! AutF (E) the inclusion. 2 1.4 Definition Let ∆: G ! GLn(F ) be a representation. (a) We say that ∆ is decomposable if it is equivalent to ∆1 ⊕ ∆2 for some representations ∆1 and ∆2 of G over F . Otherwise it is called indecomposable. (b) We say that ∆ is reducible if it is equivalent to a block upper triangular representation, i.e., to a representation of the form A(g) B(g)! g 7! 0 C(g) where A(g) 2 Matn1 (F ), B(g) 2 Matn1×n2 (F ), C(g) 2 Matn2 (F ), and where n1; n2 2 N are independent of g 2 G. Otherwise, it is called irreducible. 1.5 Remark Let ∆: G ! GLn(F ) be a representation. (a) If ∆ is decomposable then ∆ is also reducible. Thus, if ∆ is irreducible it is also indecomposable. (b) A subspace U of F n is called invariant under ∆ (or ∆-stable) if ∆(g)u 2 U for all g 2 G and u 2 U. More generally, if (V; ρ) is a rep- resentation corresponding to ∆, a subspace U of V is called invariant under ∆ (or ∆-stable) if (ρ(g))(u) 2 U for all g 2 G and u 2 U. Note that ∆ is decomposable if and only if V has a decomposition V = U ⊕ W into two non-zero ∆-invariant subspaces U and V . And ∆ is reducible if V has a ∆-invariant subspace U with f0g 6= U 6= V . In the following remark we recollect a few results from linear algebra. 1.6 Remark Let V be a finite-dimensional C-vector space. (a) A map (−; −): V × V ! C is called a hermitian scalar product on V if the following holds for all x; x0; y; y0 2 V and λ 2 C: (x + x0; y) = (x; y) + (x0; y) ; (x; y + y0) = (x; y) + (x; y0) ; (λx, y) = λ(x; y) ; (x; λy) = λ¯(x; y) ; (y; x) = (x; y) ; (x; x) 2 R>0 ; (x; x) = 0 () x = 0 : 3 (b) Let (−; −) be a hermitian scalar product on V . And let f : V ! V be an automorphism of V . Recall that f is called unitary with respect to (−; −) if (f(x); f(y)) = (x; y) for all x; y 2 V . The following results hold: (i) The vector space V has an orthonormal basis with respect to (−; −), i.e., a basis (v1; : : : ; vn) such that (vi; vj) = δi;j for i; j 2 f1; : : : ; ng. (ii) An automorphism f of V is unitary if and only if the representing matrix A of f with respect to an orthonormal basis is unitary (i.e., AA∗ = 1, where A∗ = At). (c) The complex vector space Cn has the standard hermitian scalar prod- uct (x; y) = x1y¯1 + x2y¯2 + ··· + xny¯n : The standard basis (e1; : : : ; en) is an orthonormal basis with respect to this product. An automorphism f : Cn ! Cn is unitary with respect to (−; −) if and only if the unique matrix A 2 GLn(C) with the property f(x) = Ax, for all x 2 Cn, is a unitary matrix, since A is the representing matrix of f with respect to (e1; : : : ; en). (d) The set of unitary n × n-matrices form a subgroup U(n) of GLn(C). Every unitary matrix is diagonalizable. Its eigenvalues are complex numbers of absolute value 1. 1.7 Proposition Every complex representation ∆: G ! GLn(C) is equiv- alent to a unitary representation, i.e., to a representation Γ: G ! GLn(C) with Γ(g) 2 U(n) for all g 2 G. Proof For x; y 2 Cn define hx; yi := X(∆(g)x; ∆(g)y) ; g2G where (−; −) is the standard hermitian scalar product on Cn. It is easy to check that < −; − > is again a hermitian scalar product on Cn and we have h∆(g)x; ∆(g)yi = X (∆(h)∆(g)x; ∆(h)∆(g)y) = X(∆(k)x; ∆(k)y) = hx; yi ; h2G k2G for all x; y 2 Cn and g 2 G. Thus, ∆(g) is unitary with respect to h−; −i, for every g 2 G. Let S 2 GLn(C) be a matrix such that its columns (v1; : : : ; vn) form an orthonormal basis of Cn with respect to h−; −i. Then the represent- −1 ing matrix S ∆(g)S of ∆(g) with respect to (v1; : : : ; vn) is unitary. Thus, the representation Γ = S−1∆S is unitary. 4 1.8 Corollary Let ∆: G ! GLn(C) be a representation and let g 2 G be an element of order k. Furthermore, set ζ := e2πi=k 2 C, a primitive k-th root of unity.