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Modern Group Theory MAT 4199/5145 Fall 2017 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Contents Preface 4 1 Representation theory of finite groups5 1.1 Basic concepts...................................5 1.1.1 Representations..............................5 1.1.2 Examples.................................7 1.1.3 Intertwining operators..........................9 1.1.4 Direct sums and Maschke's Theorem.................. 11 1.1.5 The adjoint representation....................... 12 1.1.6 Matrix coefficients............................ 13 1.1.7 Tensor products............................. 15 1.1.8 Cyclic and invariant vectors....................... 18 1.2 Schur's lemma and the commutant....................... 19 1.2.1 Schur's lemma............................... 19 1.2.2 Multiplicities and isotypic components................. 21 1.2.3 Finite-dimensional algebras....................... 23 1.2.4 The commutant.............................. 25 1.2.5 Intertwiners as invariant elements.................... 28 1.3 Characters and the projection formula..................... 29 1.3.1 The trace................................. 29 1.3.2 Central functions and characters..................... 30 1.3.3 Central projection formulas....................... 32 1.4 Permutation representations........................... 38 1.4.1 Wielandt's lemma............................. 39 1.4.2 Symmetric actions and Gelfand's lemma................ 41 1.4.3 Frobenius reciprocity for permutation representations........ 42 1.4.4 The structure of the commutant of a permutation representation... 47 1.5 The group algebra and the Fourier transform.................. 50 1.5.1 The group algebra............................ 50 1.5.2 The Fourier transform.......................... 54 1.5.3 Algebras of bi-K-invariant functions.................. 57 1.6 Induced representations............................. 61 1.6.1 Definitions and examples......................... 61 1.6.2 First properties of induced representations............... 63 1.6.3 Frobenius reciprocity........................... 65 2 Contents 3 1.6.4 Mackey's lemma and the intertwining number theorem........ 66 2 The theory of Gelfand{Tsetlin bases 69 2.1 Algebras of conjugacy invariant functions.................... 69 2.1.1 Conjugacy invariant functions...................... 69 2.1.2 Multiplicity-free subgroups........................ 73 2.2 Gelfand{Tsetlin bases.............................. 74 2.2.1 Branching graphs and Gelfand{Tsetlin bases.............. 74 2.2.2 Gelfand{Tsetlin algebras......................... 76 3 The Okounkov{Vershik approach 80 3.1 The Young poset................................. 80 3.1.1 Partitions and conjugacy classes in Sn ................. 80 3.1.2 Young diagrams.............................. 81 3.1.3 Young tableaux.............................. 82 3.1.4 Coxeter generators............................ 83 3.1.5 The content of a tableau......................... 86 3.1.6 The Young poset............................. 87 3.2 The Young{Jucys{Murphy elements and a Gelfand{Tsetlin basis for Sn ... 89 3.2.1 The Young{Jucys{Murphy elements................... 90 3.2.2 Marked permutations........................... 91 3.2.3 Olshanskii's Theorem........................... 93 3.3 The spectrum of the YJM elements and the branching graph of Sn ..... 96 3.3.1 The weight of a Young basis vector................... 97 3.3.2 The spectrum of the YJM elements.................. 98 3.3.3 Spec(n) = Cont(n)............................ 100 3.4 The irreducible representations of Sn ...................... 105 3.4.1 Young's seminormal form........................ 105 3.4.2 Young's orthogonal form......................... 107 3.4.3 The Young seminormal units....................... 110 3.4.4 The Theorem of Jucys and Murphy................... 113 4 Further directions 114 4.1 Schur{Weyl duality............................... 114 4.2 Categorification.................................. 115 4.2.1 Symmetric functions........................... 115 4.2.2 The Grothendieck group......................... 117 4.2.3 Categorification of the algebra of symmetric functions........ 118 4.2.4 The Heisenberg algebra.......................... 119 4.2.5 Categorification of bosonic Fock space................. 119 4.2.6 Categorification of the basic representation.............. 121 4.2.7 Going even further............................ 122 Index 123 Preface These are notes for the course Modern Group Theory (MAT 4199/5145) at the University of Ottawa. Since the pioneering works of Frobenius, Schur, and Young more than a hundred years ago, the representation theory of the symmetric group has developed into a huge area of study, with applications to algebra, combinatorics, category theory, and mathematical physics. In this course, we will cover the representation theory of the symmetric group following modern techniques developed by Vershik, Olshankii, and Okounkov. Using techniques from algebra, combinatorics, and category theory, we will cover the following topics. • Representation theory of finite groups. We will begin the course with an introduction to the representation theory of finite groups. This will include a discussion of irreducible representations, tensor products, Schur's lemma, characters, permutation representa- tions, group algebras, and Frobenius reciprocity. • The theory of Gelfand-Tsetlin bases. We will discuss branching rules for representations of symmetric groups and see how such branching rules allow one to obtain particularly nice bases for irreducible representations. • The Okounkov-Vershik approach. We will discuss the combinatorics of Young tableaux, Jucys-Murphy elements, and the Okounkov-Vershik approach to the representation theory of symmetric groups. Acknowledgements: These notes closely follow the book [CSST10], which is the recom- mended textbook for the course. Alistair Savage Course website: http://alistairsavage.ca/mat5145 4 Chapter 1 Representation theory of finite groups In this chapter we discuss some basic facts about the representation theory of finite groups. While we will focus on the symmetric groups later in the course, we work mostly with arbitrary finite groups in this chapter. We closely follow the presentation in [CSST10, Ch. 1]. Throughout this chapter, G will denote a finite group and V; W will denote finite- dimensional complex vector spaces. Unless otherwise specified, we will always work over the field of complex numbers. So the term vector space means complex vector space. 1.1 Basic concepts In this section we give the main definitions related to representations of finite groups and discuss some examples. 1.1.1 Representations Recall that the general linear group GL(V ) := fT : V ! V : T is an invertible linear mapg is a group under composition. Its identity element is the identity map IV . A (linear) representation of G on V is a group homomorphism σ : G ! GL(V ): The name arises from the fact that elements g of G are \represented" by linear transformati- ons σ(g) of V . When we wish to make the vector space V explicit, we will sometimes denote the representation by (σ; V ), or simply by V (with the homomorphism σ understood). The dimension of the representation σ is the dimension of V . A subspace W ≤ V is said to be σ-invariant (or G-invariant, when the representation σ is understood) if σ(g)W ⊆ W; for all g 2 G (i.e. σ(g)w 2 W for all g 2 G; w 2 W ): 5 6 Chapter 1. Representation theory of finite groups If this is the case, then (σjW ;W ) is also a representation of G. We say that σjW is a subrepresentation of G. (Note that we will use the notation ≤ for subspaces and subgroups. We reserve the symbol ⊆ for set inclusion.) Note that the trivial spaces V and f0g are always invariant. A nonzero representation (σ; V ) is irreducible if V has no nontrivial invariant subspaces; otherwise we say it is reducible. If (σ; V ) is a representation of G and K ≤ G is a subgroup, the restriction of σ from G G G to K, denoted ResK σ (or ResK V ) is the representation of K of V defined by the restriction σjK : K ! GL(V ). A unitary space is a vector space V endowed with a Hermitian scalar product. Recall that a Hermitian scalar product (or Hermitian inner product) is a map h·; ·iV : V × V ! C such that, for all u; v; w 2 V and α 2 C, we have (a) hu + v; wi = hu; wi + hv; wi, (b) hu; v + wi = hu; vi + hu; wi, (c) hαu; vi = αhu; vi, (d) hv; αvi =α ¯hu; vi, (e) hu; vi = hv; ui, (f) hu; ui ≥ 0, with equality only if u = 0, wherez ¯ denotes the complex conjugate of z 2 C. From now on, the term scalar product will mean Hermitian scalar product. Suppose V is a unitary space and T : V ! V is a linear operator. The adjoint operator T ∗ is defined by ∗ hT u; viV = hu; T viV ; for all u; v 2 V: (1.1) (See Exercise 1.1.1.) If V is a unitary space, then a linear operator T : V ! V is unitary if it preserves the scalar product, i.e. if hT u; T viV = hu; viV ; for all u; v 2 V: (More generally, T : V ! W is unitary if hT u; T viW = hu; viV for all u; v 2 V .) All unitary operators are invertible (Exercise 1.1.2). Furthermore, T 2 GL(V ) is a unitary operator if and only if T −1 = T ∗ (Exercise 1.1.3). Suppose V is a unitary space. A representation (σ; V ) is unitary if σ(g) is a unitary operator for all g 2 G or, in other words, if σ(g−1) = σ(g)∗ for all g 2 G. We way a representation (σ; V ) is unitarizable if there exists a scalar product on V with respect to which σ is unitary. Lemma 1.1.1. Every finite-dimensional representation of a finite group is unitarizable. Proof. Let (·; ·) be an arbitrary scalar product on V . (See Exercise 1.1.4.) Then define a new scalar product on V by X hu; vi = (σ(g)u; σ(g)v); for all u; v 2 V: g2G 1.1. Basic concepts 7 Then, for all h 2 G and u; v 2 V , we have X hσ(h)u; σ(h)vi = (σ(gh)u; σ(gh)v) g2G X = (σ(s)u; σ(s)v) (setting s = gh) s2G = hu; vi: Hence the representation is unitary with respect to h·; ·i.
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