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Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials

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Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials 1/29 Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials Don Taylor The University of Sydney Nelson, 7–13 January 2018 2/29 Origins Suppose that G is a finite group and for every element g G we have 2 an indeterminate xg . What are the factors of the group determinant ¡ ¢ det x 1 ? gh¡ g,h This was a question posed by Dedekind. Frobenius discovered character theory (in 1896) when he set out to answer it. Characters of abelian groups had been used in number theory but Frobenius developed the theory for nonabelian groups. 3/29 Representations Matrices Characters ² ² A linear representation of a group G is a homomorphism ½ : G GL(V ), where GL(V ) is the group of all invertible linear ! transformations of the vector space V , which we assume to have finite dimension over the field C of complex numbers. The dimension n of V is called the degree of ½. If (ei )1 i n is a basis of V , there are functions ai j : G C such that · · ! X ½(x)e j ai j (x)ei . Æ i The matrices A(x) ¡a (x)¢ define a homomorphism A : G GL(n,C) Æ i j ! from G to the group of all invertible n n matrices over C. £ The character of ½ is the function G C that maps x G to Tr(½(x)), ! 2 the trace of ½(x), namely the sum of the diagonal elements of A(x). 4/29 Early applications Frobenius first defined characters as solutions to certain equations. A year later he established the connection with matrix representations. He then used character theory to establish structural properties of finite groups. Another early success of character theory was Burnside’s 1904 proof that groups of order pa qb (p and q primes) are soluble. My aim is to introduce you to enough character theory today so that we can prove this result tomorrow. 5/29 Outline I Maschke’s theorem I Irreduciblity I Schur’s lemma I Orthogonality relations I The number of irreducible characters I The character table of a finite group I Examples 6/29 Some background reading Charles W. Curtis. Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer, volume 15 of History of Mathematics. American Mathematical Society, 1999. Walter Feit. Characters of finite groups. W. A. Benjamin, 1967. I. Martin Isaacs. Character theory of finite groups. Academic Press, 1976. Jean-Pierre Serre. Linear representations of finite groups. Springer-Verlag, 1977. 7/29 Similarity Let ½ and ½0 be two linear representations of G with spaces V and V 0 and let A and A0 be the corresponding matrix representations. We say that ½ and ½0 are isomorphic if there is a linear isomorphism ¼ : V V 0 such that the diagram ! ½(x) V V ¼ ¼ V 0 V 0 ½0(x) commutes for all x G; that is, ¼½(x) ½0(x)¼. 2 Æ 1 This is equivalent to the condition A0(x) PA(x)P ¡ , where P is the Æ matrix of ¼. That is, linear representations are isomorphic if and only if the corresponding matrix representations are similar. Isomorphic representations have the same character. 8/29 Example: representations of degree 1 A representation of degree 1 of a group G is a homomorphism ½ : G C£ and in this case ½ is its own character. ! For x G, the image ½(x) is a root of unity and hence ½(x) 1. 2 j j Æ For every group G there is the principal representation 1G defined by 1 (x) 1 for all x G. G Æ 2 Example (Cyclic groups) If G is a cyclic group generated by an element x of order d and if ³ exp(2¼i/d) is a primitive d th root of unity, then for 0 k d Æ · Ç there is a representation (character) Âk of degree 1 such that  (x) ³k . k Æ These representations are pairwise non-isomorphic. 9/29 Example: the regular representation Let C[G] be the vector space of dimension G with basis (ex )x G . j j 2 For x G, let R (x) be the automorphism of C[G] such that 2 G R (x)e e . G y Æ x y This is the regular representation of G. Its degree is G . j j Let reg be the character of R . For x G we have G G 2 8 G x 1 <j j Æ regG (x) Æ :0 x 1 6Æ 10/29 Example: permutation representations Suppose that G acts on a finite set ¡ of size n. That is, for each x G 2 there is a permutation ® x® of ¡ such that 7! 1® ® and x(y®) (x y)® for all x, y G, ® ¡. Æ Æ 2 2 Let V be the vector space with basis (e®)® ¡ 2 For x G, let ½(x) be the automorphism of V such that 2 ½(x)e e . ® Æ x® The resulting linear representation of G is the permutation representation associated with ¡. The value of the character  of ½ at x is the number of fixed points of x; that is, Â(x) Fix (x) . Æ j ¡ j 11/29 Subrepresentations Let ½ : G GL(V ) be a linear representation of G and let W be a ! subspace of V . Suppose that W is G-invariant; that is, for w W , ½(x)w W 2 2 for all x G. 2 The restriction ½ (x) of ½(x) to W defines a linear representation jW ½ : G GL(W ) of G in W called a subrepresentation of V . jW ! 12/29 Maschke’s Theorem Theorem (Maschke) Let ½ : G GL(V ) be a linear representation of G in V and let W be a ! subspace of V invariant under G. Then there is a complement W ± of W in V which is also G-invariant. 13/29 Proof Let W 0 be an arbitrary complement of W in V and let p be the corresponding projection of V onto W . Form the ‘average’ of the conjugates of p by G: 1 X 1 p± ½(x)p½(x)¡ . Æ G x G j j 2 Since p maps V into W and ½(x) fixes W we see that p± maps V into W . 1 On the other hand, if w W then ½(x)¡ w W , whence 2 2 1 1 1 p½(x)¡ w ½(x)¡ w, i.e. ½(x)p½(x)¡ w w and so p±w w. Æ Æ Æ Thus p± is a projection onto W and we define W ± kerp±. Then W ± is a Æ complement to W . Furthermore 1 1 X 1 1 1 X 1 ½(y)p±½(y)¡ ½(y)½(x)p½(x)¡ ½(y)¡ ½(yx)p½(yx)¡ p±. Æ G Æ G Æ j j x j j x If w W ±, then p±w 0 and so p±½(y)w ½(y)p±w 0, whence 2 Æ Æ Æ ½(y)w W ±. This shows that W ± is G-invariant, as required. 2 14/29 Irreducible representations and characters A linear representation ½ : G GL(V ) of G is is irreducible (or simple) ! if V 0 and the only G-invariant subspaces of V are 0 and V . 6Æ In view of Maschke’s theorem this is equivalent to saying that V is not the direct sum of two proper subrepresentations. Theorem Every representation is a direct sum of irreducible representations. Proof. Induction and Maschke’s theorem. A character is irreducible if it is not the sum of two other characters. 15/29 Elementary properties of characters If  is the character of a representation ½ of G of degree n, then I Â(1) n Æ 1 I Â(x¡ ) Â(x) Æ 1 I Â(x yx¡ ) Â(y); i.e.  is constant on conjugacy classes Æ For x G, Â(x) is the sum of the eigenvalues ¸1, ¸2,..., ¸n of ½(x). 2 1 The ¸ are roots of unity and hence ¸¡ ¸ . i i Æ i If A and B are matrices, Tr(AB) Tr(BA). Æ Let ½ : G GL(V ) and ½ : G GL(V ) be linear representations of G 1 ! 1 2 ! 2 and let Â1 and Â2 be their characters. Then I the character of the direct sum V1 V2 is Â1 Â2; © Å I the character of the tensor product V1 V2 is Â1Â2. ­ 16/29 Schur’s lemma Theorem (Schur’s lemma) Let ½ : G GL(V ) and ½ : G GL(V ) be irreducible representations 1 ! 1 2 ! 2 of G and let f : V V be a linear transformation such that 1 ! 2 ½ (x)f f ½ (x) for all x G. 2 Æ 1 2 1 If ½ and ½ are not isomorphic, f 0. 1 2 Æ 2 If V V and ½ ½ , then f is a scalar multiple of the identity. 1 Æ 2 1 Æ 2 17/29 Proof ½ : G GL(V ), ½ : G GL(V ), f : V V , ½ (x)f f ½ (x) 1 ! 1 2 ! 2 1 ! 2 2 Æ 1 1 If w ker f , then f ½ (x)w ½ (x)f w 0 and so ½ (x)w ker f for 2 1 Æ 2 Æ 1 2 all x G, which proves that ker f is G-invariant. 2 Similarly if v im f , then v f w for some w V , whence 2 Æ 2 1 ½ (x)v ½ (x)f w f ½ (x)w im f and so im f is also G-invariant. 2 Æ 2 Æ 1 2 But V and V are irreducible, hence f 0 implies ker f 0 and 1 2 6Æ Æ im f V , which means that f is an isomorphism.
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