Representation Theory and the McKay Correspondence Bachelorthesis Ralf Bendermacher 4038053 [email protected] Supervisor: Dr. J. Stienstra Utrecht University July 15, 2014 1 Contents 1 introduction 3 2 Representation Theory 4 2.1 Representations . .4 2.2 The group algebra . .5 2.3 CG-modules . .6 2.3.1 CG-submodules . .7 2.3.2 Irreducible CG-modules . .7 2.3.3 The regular CG-module . .7 2.3.4 CG-homomorphisms . .8 2.4 Characters . 10 2.5 Conjugacy classes . 15 3 Unitary matrices 16 4 Complex numbers 17 4.1 isomorphism SO(2) and U(1) . 17 5 Quaternions 18 5.1 H as a unital ring . 18 5.2 Representation of H ............................. 20 5.3 Rotations . 21 5.4 Composition of rotations . 24 6 The Platonic Solids 26 6.1 Symmetry groups of the solids . 26 6.2 Dual solids . 27 6.3 Tetrahedron . 27 6.4 The Cube and the Octahedron . 28 6.5 The Dodecahedron and the Icosahedron . 30 6.6 Groups . 30 7 Binary groups 32 8 The character tables of A4, S4 and A5 35 9 The symmetry groups 36 9.1 Octahedral group . 36 9.2 Tetrahedral group . 39 9.3 Icosahedral group . 41 9.4 Note on the Coxeter-Dynkin diagrams . 44 2 1 introduction We start by giving an introduction to representation theory and then apply it to the symmetry groups belonging to the 5 platonic solids; the tetrahedron, the cube, the oc- tahedron, the dodecahedron and the icosahedron. These 5 solids constitute 3 important symmetry groups, namely the tetrahedral group, octahedral group and the icosahedral group. Each symmetry group is isomorphic to some permutation group. For these 3 permutation groups we can find the character table, which we shall learn is a matrix that contains all the important information about a group. A character of a group G is an important concept in representation theory and is a function from G to C. The most interesting characters of a group are the irreducible characters. The number of ir- reducible characters of each group equals the number of conjugacy classes of that group. The symmetry groups of platonic solids consist of the rotations which leave the solid invariant and hence are subgroups of SO(3). We shall see that there exist a surjective grouphomomorphism between SU(2) and SO(3); φ : SU(2) ! SO(3) The kernel of this homomorphism is {−1; 1g. When we take the pre-images of subgroups of SO(3) under this homomorphism φ, we get subgroups of SU(2) which we shall call binary groups (since they contain twice as many elements as the original subgroup of SO(3)). Hence to each symmetry group of a platonic solid we can associate a binary group. For a binary group G∗ of a group G, we have that; G∗={−1; 1g ∼= G Using the known character tables of the symmetry groups of the platonic solids we can construct the character tables for the corresponding binary groups. Each of the 3 groups has a special "natural" character. This character is fully determined by the rotation angles of the corresponding group elements. Using this character we can construct a new inproduct on the space of all functions from G to C. We use this new inproduct to make a Gram-matrix, from which we can construct a so called Dynkin-Coxeter diagram of a group. These diagrams turn out to be important graphs in the theory of Lie groups. John McKay (1939) was a mathematician that discovered a link between the Dynkin graphs belonging to the representations of the binary groups associated with the pla- tonic solids (representation theory) and geometrical structures of these groups (singu- larity theory). This link is called the McKay correspondence. 3 2 Representation Theory 2.1 Representations Recall that GL(n; C) is the group of all invertible n×n matrices with entries in C. Now we define a representation of a group G over C. Definition 2.1. A representation of a group G over C is a homomorphism ρ from G to GL(n; C), for some n (which is the degree of the representation). We recall that a homomorphism between two groups (G; •) and (H; ◦) is a function: φ: G −! H which satisfies φ (a • b) = φ (a) ◦ φ (b) for all a; b 2 G. For a representation ρ of G then it must hold that: 1. ρ (g • h) = ρ (g) ρ (h) for all g; h 2 G 2. ρ (e) = In (identity matrix) 3. ρ (g−1) = ρ (g)−1 for all g 2 G When we are given a representation of a group we can simply transform this repre- sentation into another one. Let S be an arbitrary invertible n × n matrix and let ρ be a representation of a group G of degree n. Now we define φ : G −! GL(n; C), g 7! S−1(ρ (g))S. This clearly is another representation of G of degree n. Definition 2.2. Let ρ and φ be two representations of G of degree n, then we say that ρ equivalent to φ if there exist an invertible n × n S such that: ρ (g) = S−1(φ (g))S; 8 g 2 G Equivalence of representations is easilly seen to be an equivalence relation on the set of representations of G. Let ρ, φ and θ be representations of a group G: • Every representation is equivalent to itself (take S = In) • If ρ is equivalent to φ, then φ is equivalent to ρ (take S−1) • If ρ is equivalent to φ and φ is equivalent to θ, then ρ is equivalent to θ, (take S1S2) Like every other function a representation of a group G has a kernel defined as Ker ρ = fg 2 G j ρ (g) = Ing. We call a representation faithful when Ker ρ = feg. 4 2.2 The group algebra Let G be a finite group with elements g1; : : : ; gn. Using the elements of G as a basis we can construct a vectorspace over C. The elements of this vectorspace CG are of the form λ1g1 + ::: + λngn with λ1; : : : ; λn 2 Cg. Now let n X u = λigi i=1 and n X v = µigi i=1 be elements of CG and λ 2 C. We then define addition and scalar multiplication on the elements of CG as follows: n X u + v = (λi + µi)gi i=1 n X λu = (λλi)gi i=1 We now easily check that CG is indeed a vectorspace over C with dimension equal to n (the order of our group G). The basis g1; g2; : : : ; gn that we used is called the natural basis of CG. Definition 2.3 (The group algebra). Let X a = λgg g2G X b = µhh h2G be elements of the vectorspace CG, then the vectorspace CG together with the multipli- cation defined by; ! ! X X X ab = λgg µhh = λgµh(gh) g2G h2G g;h2G is named the group algebra of G over C. As one can check the group algebra CG satisfies the following properties, for all r; s; t 2 CG and λ 2 C: 1. rs 2 CG; 5 2. r(st) = (rs)t; 3. re = er = r; 4.( λr)s = λ(rs) = r(λs); 5.( r + s)t = rt + st; 6. r(s + t) = rs + rt; 7. r0 = 0r = 0 2.3 CG-modules One of the most used concepts in representation theory is that of the CG-module. Definition 2.4. Let V be a vector space over C and let G be a group. We call V a CG-module if we can multiply vectors of V with group elements of G (gv, v 2 V and g 2 G). Such that the following conditions are satisfied for all u; v 2 V; λ 2 C and g; h 2 G; 1. gv 2 V ; 2. (hg)v = h(gv); 3. ev = v; 4. g(λv) = λ(gv); 5. g(u + v) = gu + gv. Now let ρ : G −! GL(n; C) be a representation of a group G. Then for every g 2 G we find an invertible n × n matrix ρ(g). Let V = Cn be the vectorspace that contains T all the vectors (c1; : : : ; cn) , with ci 2 C for 1 ≤ i ≤ n. Since ρ(g) 2 GL(n; C) we can multiply with elements of V using matrix multiplication (ρ(g)v, v inV and g 2 G). Now let u; v 2 V; λ 2 C and g; h 2 G, we easily check that this multiplication satisfies the following properties; • ρ(g)v lies in V . • Since ρ is a homomorphism, it holds that ρ(gh)v = ρ(g)ρ(h)v. • Because ρ(e) = In, we have ρ(e)v = v. • ρ(g)(λv) = λ(ρ(g)v) (due to the known properties of matrixmultiplication). • ρ(g)(v + u) = ρ(g)(v) + ρ(g)(u) (again from matrix multiplication). These are precisely the conditions for Cn to be a CG-module. So representations of a group G correspond to CG-modules. Definition 2.5. Let V be a CG-module and let B be a bases of V . For each g 2 G we denote [g]B for the matrix of the endomorphism g : V ! V; v 7! gv of V , relative to the bases B. 6 We can now construct a representation of a group G from a given CG-module. Theorem 2.6. Let V be a CG-module and let B be a bases of V .