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Representations of Algebras and Finite Groups: an Introduction Representations of Algebras and Finite Groups: An Introduction Ambar N. Sengupta May, 2010 2 Ambar N. Sengupta Contents Preface . .7 1 Basic Concepts and Constructions 9 1.1 Representations of Groups . 10 1.2 Representations and their Morphisms . 11 1.3 Sums, Products, and Tensor Products . 12 1.4 Change of Field . 12 1.5 Invariant Subspaces and Quotients . 13 1.6 Dual Representations . 14 1.7 Irreducible Representations . 15 1.8 Character of a Representation . 16 1.9 Unitarity . 19 Exercises . 22 2 Basic Examples 25 2.1 Cyclic Groups . 26 2.2 Dihedral Groups . 28 2.3 The symmetric group S4 ..................... 32 Exercises . 36 3 The Group Algebra 39 3.1 Definition of F[G]......................... 39 3.2 Representations of G and F[G].................. 40 3.3 The center of F[G]........................ 41 3.4 A glimpse of some results . 42 3.5 F[G] is semisimple . 44 Exercises . 46 3 4 Ambar N. Sengupta 4 Semisimple Modules and Rings: Structure and Representa- tions 47 4.1 Schur's Lemma . 47 4.2 Semisimple Modules . 49 4.3 Structure of Semisimple Rings . 55 4.3.1 Simple Rings . 58 4.4 Semisimple Algebras as Matrix Algebras . 62 4.5 Idempotents . 63 4.6 Modules over Semisimple Rings . 71 Exercises . 76 5 The Regular Representation 79 5.1 Structure of the Regular Representation . 79 5.2 Representations of abelian groups . 82 Exercises . 82 6 Characters of Finite Groups 83 6.1 Definition and Basic Properties . 83 6.2 Character of the Regular Representation . 84 6.3 Fourier Expansion . 87 6.4 Orthogonality Relations . 90 6.5 The Invariant Inner Product . 92 6.6 The Invariant Inner Product on Function Spaces . 93 6.7 Orthogonality Revisited . 98 Exercises . 99 7 Some Arithmetic 103 7.1 Characters as Algebraic Integers . 103 7.2 Dimension of Irreducible Representations . 103 7.3 Rationality . 103 8 Representations of Sn 105 8.1 Conjugacy Classes and Young Tableaux . 106 8.2 Construction of Irreducible Representations of Sn ....... 108 8.3 Some properties of Young tableaux . 112 8.4 Orthogonality of Young symmetrizers . 116 Representations of Algebras and Finite Groups 5 9 Commutants 119 9.1 The Commutant . 119 9.2 The Double Commutant . 121 Exercises . 123 10 Decomposing a Module using the Commutant 125 10.1 Joint Decomposition . 125 10.2 Decomposition by the Commutant . 128 10.3 Submodules relative to the Commutant . 132 Exercises . 136 11 Schur-Weyl Duality 139 ⊗n 11.1 The Commutant for Sn acting on V ............. 139 11.2 Schur-Weyl Character Duality I . 141 11.3 Schur-Weyl Character Duality II . 143 Exercises . 149 12 Representations of Unitary Groups 151 12.1 The Haar Integral and Orthogonality of Characters . 152 12.1.1 The Weyl Integration Formula . 153 12.1.2 Schur Orthogonality . 154 12.2 Characters of Irreducible Representations . 155 12.2.1 Weights . 155 12.2.2 The Weyl Character Formula . 156 12.2.3 Weyl dimensional formula . 160 12.2.4 Representations with given weights . 161 12.3 Characters of Sn from characters of U(N)........... 163 13 Frobenius Induction 167 13.1 Construction of the Induced Representation . 167 13.2 Universality of the Induced Representation . 168 13.3 Character of the Induced Representation . 170 14 Representations of Clifford Algebras 173 14.1 Clifford Algebra . 173 14.1.1 Formal Construction . 174 14.1.2 The Center of Cd ..................... 175 14.2 Semisimple Structure of the Clifford Algebra . 176 6 Ambar N. Sengupta 14.2.1 Structure of Cd for d ≤ 2................. 177 14.2.2 Structure of Cd for even d ................ 180 14.2.3 Structure of Cd for odd d ................. 184 14.3 Representations . 188 14.3.1 Representations with Endomorphisms . 189 14.3.2 Representations on Exterior Algebras . 190 14.4 Superalgebra structure . 192 14.4.1 Superalgebras . 192 14.4.2 Clifford algebras as superlagebras . 194 14.4.3 Tensor product decomposition . 194 14.4.4 Semisimple structure . 195 Exercises . 196 Appendix . 198 14.5 Some basic algebraic structures . 199 14.6 Fields . 202 14.7 Vector Spaces over Division Rings . 204 14.8 Modules over Rings . 206 14.9 Tensor Products . 208 End Notes . 209 Bibliography . 211 Representations of Algebras and Finite Groups 7 Preface These notes describe the basic ideas of the theory of representations of finite groups. Most of the essential structural results of the theory follow imme- diately from the structure theory of semisimple algebras, and so this topic occupies a long chapter. Material not covered here include the theory of induced representations. The arithmetic properties of group characters are also not dealt with in detail. It is not the purpose of these notes to give comprehensive accounts of all aspects of the topics covered. The objective is to see the theory of rep- resentations of finite groups as a coherent narrative, building some general structural theory and applying the ideas thus developed to the case of the symmetric group Sn. It is also not the objective to present a very efficient and fast route through the theory. For many of the ideas we pause the exam- ine the same set of results from several different points of view. For example, in Chapter 10, we develop the the theory of decomposition of a module with respect to the commutant ring in three distinct ways. I wish to thank Thierry L´evyfor many useful comments. The present version also includes some of several changes suggested by referees. These notes have been begun at a time when I have been receiving support from the US National Science Foundation grant DMS-0601141. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. 8 Ambar N. Sengupta Chapter 1 Basic Concepts and Constructions A group is an abstract mathematical object, a set with elements and an operation satisfying certain axioms. A representation of a group realizes the elements of the group concretely as geometric symmetries. The same group will generally have many different such representations. Thus, even a group which arises naturally and is defined as a set of symmetries may have representations as geometric symmetries at different levels. In quantum physics the group of rotations in three-dimensional space gives rise to symmetries of a complex Hilbert space whose rays represent states of a physical system; the same abstract group appears once, classically, in the avatar of rotations in space and then expresses itself at the level of a more `implicate order' in the quantum theory as unitary transformations on Hilbert spaces. In this chapter we (i) introduce the basic concepts, defining group repre- sentations, irreducibility and characters, (ii) carry out certain useful standard constructions with representations, and (iii) present a result or two of interest which follow very quickly from the basic notions. All through this chapter G denotes a group, and F a field. We will work with vector spaces, denoted V , W , E, F , over the field F. 9 10 Ambar N. Sengupta 1.1 Representations of Groups A representation ρ of a group G on a vector space V associates to each element x 2 G an invertible linear map ρ(x): V ! V : v 7! ρ(x)v such that ρ(xy) = ρ(x)ρ(y) for all x; y 2 G. (1.1) ρ(e) = I; where I : V ! V is the identity map. Here, our vector space V is over a field F. We denote by EndF(V ) the ring of endomorphisms of a vector space V . A representation ρ of G on V is thus a map ρ : G ! EndF(V ) satisfying (1.1) and such that ρ(x) is invertible for every x 2 G. A complex representation is a representation on a vector space over the field C of complex numbers. The homomorphism condition (1.1) implies ρ(e) = I; ρ(x−1) = ρ(x)−1 for all x 2 G. We will often say `the representation E' instead of `the representation ρ on the vector space E'. If V is finite-dimensional with basis b1; :::; bn, then the matrix 2 3 ρ(g)11 ρ(g)12 : : : ρ(g)1n 6ρ(g) ρ(g) : : : ρ(g) 7 6 21 22 2n7 6 . 7 (1.2) 4 . 5 ρ(g)n1 ρ(g)n2 : : : ρ(g)nn of ρ(g) 2 EndF(V ) is often useful to explicitly express the representation or to work with it. Indeed, with some basis in mind, we will often not make a distinction between ρ(x) and its matrix form. Representations of Algebras and Finite Groups 11 Sometimes the term `matrix element' is used to mean a function on G which arises from a reprsentation ρ as G ! F : x 7! hfjρ(g)jvi; where jvi is a vector in the representation space of ρ, and hfj is in the dual space. Consider the group Sn of permutations of [n] = f1; :::; ng, acting on the vector space Fn by permutation of coordinates: n n def Sn × F ! F : σ; (v1; :::; vn) 7! R(σ)(v1; :::; vn) = (vσ−1(1); :::; vσ−1(n)): Another way to understand this is by specifying R(σ)ej = eσ(j) for all j 2 [n]. n Here ej is the j-th vector in the standard basis of F ; it has 1 in the j-th 4 entry and 0 in all other entries. Thus, for example, for S4 acting on F , the 4 matrix for R (134) relative to the standard basis of F , is 20 0 0 13 60 1 0 07 R (134) = 6 7 41 0 0 05 0 0 1 0 1.2 Representations and their Morphisms If ρ and ρ0 are representations of G on vector spaces E and E0 over F, and A : E ! E0 is a linear map such that ρ0(x) ◦ A = A ◦ ρ(x) for all x 2 G (1.3) then we can consider A to be a morphism from the representation ρ to the representation ρ0.
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