Character theory of finite groups Author: David Chen Institute: School of Mathematics and Statistics, University of Melbourne Background An example Characters 2 6 −1 The representation theory of groups is the study of representing groups Consider G = D6 = r, s j s = r = e, rs = sr , the dihedral group The character function of a representation is simply the trace of the as vector space automorphisms or matrices. That is, a representation of of order 12. The symmetries of a hexagon can be characterised by the automorphism in V . We say the character is irreducible if the a group on a vector space V is a group homomorphism r : G ! GL(V ). action of G on its vertices. As illustrated in Figure 1, r is a rotation and corresponding G-module is irreducible. Here are some useful We say V is a G-module as it has a (left) G-linear action gv ! r(g)v.A s is a reflection on the hexagon. properties of characters: G-module is irreducible if no proper subspace of it is a G-module ([1]). • c(e) is equal to the dimension of the corresponding G-module. r An example of such a module is the trivial G-module with B C • Characters are class functions, so c(g) = c(h) if g and h are conjug- r r representation: ate. r : G ! GL1(C), r(g) = 1 s A s s D s • For any g 2 G, let n = c(e), m = ord(g). Then c (g) = w1 + ··· + We consider the case where G is a finite group and the underlying field wn, where wi is an m-th root of unity. Hence their values are always r r is complex. Then we can show that there exist a finite number of F E algebraic integers. r irreducible G-modules V1, V2, . ., Vk such that every irreducible • c(g) = c(g−1) as the underlying field is C. G G V r s -module is isomorphic to one of them. In addition, every -module Figure 1: The symmetries of a hexagon (left) and how and acts on it (right). • The number of distinct irreducible characters is equal to the number is isomorphic to a direct sum by Maschke’s theorem: of conjugacy classes of G. The character table of G is shown in Table 1. Here, c1 is the trivial ∼ • Let U, V be G-modules. If c is the character of U and y the character V = d1V1 ⊕ d2V2 ⊕ · · · ⊕ dk Vk character and CG(g) is the centraliser of g. Notice that 2 of V , then c + y is the character of U ⊕ V and cy is the character of c2 r = c(e) = 1, so its kernel hri is a normal subgroup of G and where d1, d2, . ., dn are non-negative integers representing the hence G non-simple. U ⊗ V . multiplicities of V1, V2, . ., Vk . As a consequence of the complete reducibility of G-modules, there 2 3 g e r r r s sr exists irreducible characters c1, c2, . ., ck such that every character has Burnside’s Theorem jCG(g)j 12 4 4 12 6 6 the form: c = d1c1 + d2c2 + ··· + dk ck Character theory provides us another perspective to think about group c1 1 1 1 1 1 1 theoretic problems. For example, Burnside’s theorem is hard to prove c2 1 1 1 1 −1 −1 where d1, d2, . ., dk are non-negative integers. purely with group theory, but it can be proved (relatively) easily with c3 1 1 −1 −1 1 −1 We can also define an inner product on the characters. Let characters. It states that every group of order paqb is not simple, where c4 1 1 −1 −1 −1 1 [g1], [g2], ... , [gk ] be all the distinct conjugacy classes of G. Then we p q a b a + b ≥ 2 c5 2 −1 1 −2 0 0 have: , are primes and , are non-negative integers with ([2]). k − − 1 c(gi )y(gi ) Furthermore, such groups are solvable, which means that there is a c6 2 1 1 2 0 0 hc, yi = c(g)y(g) = jGj ∑ ∑ jC (g )j chain of subgroups of G g2G i=1 G i Table 1: Character table of G = D6 The second formula uses the fact that characters are class functions, so feg = G G ... G = G 0 E 1 E E n Now this group has order 12 = 22 × 31, so it should be solvable by they have the same value over any given conjugacy class of G. The Burnside’s theorem (left). Indeed, irreducible characters are orthonormal with respect to this inner where for all i, Gi is a normal subgroup of Gi+1 and Gi+1/Gi is cyclic product. Therefore, by orthonormal projection, the multiplicities of the with prime orders. 3 feg E hr i E hri E G irreducible characters can be computed easily with di = hc, ci i. ∼ 3 ∼ 3 ∼ G/hri = C2 hri/hr i = C3 hr i/feg = C2 The irreducible characters are usually presented in a table, which References displays the following properties: So all Gi+1/Gi are cyclic with prime orders. [1] William Fulton and Joe Harris. Representation theory : a first course. • The rows correspond to the distinct irreducible characters, while the Graduate texts in mathematics Readings in mathematics: 129. New columns represent the distinct conjugacy classes of G. York: Springer-Verlag, 1991. ISBN: 0387974954. Acknowledgements • The number of rows is equal to the number of columns, as the num- ber of irreducible characters is equal to the number of conjugacy [2] G. D. James and M. W. Liebeck. Representations and charac- I would like to express my deepest appreciation to my supervisor Dr classes. ters of groups. London: Cambridge University Press, 2001. ISBN: David Ridout, for his patient guidance and insightful suggestions 0521812054. throughout the vacation scholarship program. Thanks also to the • The rows (and columns) of the table are orthogonal, as a result vacation scholarship office for giving me the opportunity to participate shown above: hci , cj i = di,j in the program and offering assistance during the program. David Chen School of Mathematics and Statistics, University of Melbourne Character theory of finite groups.