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The Symmetric Group, Its Representations, and Combinatorics

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The Symmetric Group, Its Representations, and Combinatorics The Symmetric Group, its Representations, and Combinatorics Judith M. Alcock-Zeilinger Lecture Notes 2018/19 Tubingen¨ Course Overview: In this course, we'll be examining the symmetric group and its representations from a combinatorial view point. We will begin by defining the symmetric group Sn in a combinatorial way (as a permutation group) and in an algebraic way (as a Coxeter group). We then move on to study some general results of the representation theory of finite groups using the theory of characters. Thereafter, we once again lay our focus on the symmetric group and study its representation. The method used here follows that of Vershik and Okounkov, and the central result is that the Bratteli diagram of the symmetric group (giving a relation between its irreducible representations) is isomorphic to the Young lattice. In doing so, we will be able to intruduce Young tableaux in a natural way, and we will see that the number of Young tableaux of a given shape λ is the dimension of the irreducible representation corresponding to λ. Then, we discuss several results pertaining to the representation theory of the symmetric group from a combinatorial viewpoint. We will use a typical combinatorial tool, namely a proof by bijection, which was also already implemented in the Vershik-Okounkov method, without explicitly saying so. In particular, we discuss the Robinson-Schensted algorithm which allows us to proof that the sum of the (dimensions of the irreducible representations of the symmetric group)2 is the order of the group. We use this result to discuss how one can arrive at a general formula for the number of Young tableaux of size n. Lastly, we focus on the famous hook length formula giving the number of Young tableaux of a certain shape λ. We will follow the bijective proof by Novelli, Pak and Stoyanovskii to prove this result. 1 Contents Contents 2 Quick reference: Notes4 List of Exercises and Examples5 1 The symmetric group Sn 7 1.1 Combinatorial and algebraic definition of Sn ....................... 7 1.1.1 Combinatorial definition of Sn .......................... 7 1.1.2 Algebraic definition of Sn (Sn as a Coxeter group)............... 8 1.1.3 S3 as a Coxeter group............................... 14 1.1.4 Geometric definition of Sn: Example for n = 3 ................. 15 1.2 Transpositions of Sn .................................... 16 1.2.1 Graphs: definition and basic results ....................... 17 2 Representations of finite groups 20 2.1 Subrepresentations..................................... 21 2.2 (Left) regular representation................................ 27 2.3 Irreducible representations................................. 28 2.4 Equivalent representations................................. 29 2.4.1 Schur's Lemma................................... 30 3 Characters & conjugacy classes 32 3.1 Character of a representation: General properties.................... 32 3.2 The regular representation and irreducible representations............... 36 3.3 Conjugacy class of a group element............................ 38 3.4 Characters as class functions ............................... 41 4 Representations of the symmetric group Sn 47 4.1 Inductive chain & restricted representations....................... 47 4.2 Bratteli diagram ...................................... 50 4.3 Young-Jucys-Murphy elements .............................. 52 4.4 Spectrum of a representation ............................... 53 4.4.1 Bratteli diagram of Sn up to level 3 ....................... 56 2 4.4.2 The action of the Coxeter generators on the Young basis............ 58 4.4.3 Equivalence relation between spectrum vectors ................. 61 4.4.4 Spectrum vectors are content vectors....................... 64 4.5 Young tableaux, their contents and equivalence relations................ 66 4.5.1 Young tableaux................................... 67 4.5.2 Content vector of a Young tableau........................ 68 4.5.3 Equivalence relation between Young tableaux.................. 72 4.6 Main result: A bijection between the Bratteli diagram of the symmetric groups and the Young lattice...................................... 74 4.7 Irreducible representations of Sn and Young tableaux.................. 76 5 Robinson{Schensted correspondence and emergent results 78 5.1 The Robinson{Schensted correspondence......................... 79 5.1.1 P -symbol of a permutation ............................ 80 5.1.2 Q-symbol of a permutation ............................ 82 5.2 The inverse mapping to the RS correspondence..................... 82 5.3 The number of Young tableaux of size n: A roadmap to a general formula . 84 6 Hook length formula 89 6.1 Hook length......................................... 89 6.2 A probabilistic proof.................................... 90 6.2.1 A false prababilistic proof............................. 91 6.2.2 Outline of a correct prababilistic proof...................... 92 6.3 A bijective proof ...................................... 96 6.3.1 Bijection strategy.................................. 97 6.3.2 The Novelli{Pak{Stoyanovskii correspondence.................. 97 6.3.3 Defining the inverse mapping to the NPS-correspondence . 101 6.3.4 Proving that the SPN-algorithm is well-defined . 109 6.3.5 Proving that SPN = NPS−1 ............................117 References 122 3 Quick reference: Notes Note 1.1 : Subgroups, cosets, index of a subgroup and Lagrange's theorem ......... 9 Note 1.2 : Coxeter groups as reflection groups ......................... 15 Note 2.1 : Why representations?................................. 21 Note 2.2 : Why irreducible representations?........................... 28 Note 3.1 : Unitary representations................................ 33 Note 3.2 : Number of irreducible representations........................ 38 Note 3.3 : Number of inequivalent irreducible representations................. 46 Note 4.1 : Gelfand-Tsetlin basis of V' .............................. 52 Note 4.2 : Bijection between spectra α and chains T ...................... 54 Note 4.3 : Bratteli diagram of the symmetric groups and the Young lattice | Part I . 57 Note 4.4 : Bratteli diagram of the symmetric groups and the Young lattice | Part II . 76 Note 5.1 : Bijective proofs in combinatorics........................... 78 Note 6.1 : NPS-algorithm..................................... 98 Note 6.2 : SPN-algorithm | Part I ...............................103 Note 6.3 : SPN-algorithm | Part II...............................107 Note 6.4 : Potential problems with the SPN-algorithm.....................109 4 List of Exercises and Examples Exercise 1.1 : Showing that σ H for every i = j ...................... 13 i 62 j 6 Example 1.1 : (Vertex-) labelled graph ............................. 18 3 Example 2.1 : Defining/permutation representation of S3 on R | Definition . 21 3 Example 2.2 : Defining/permutation representation of S3 on R | Subrepresentations . 22 3 Example 2.3 : Defining/permutation representation of S3 on R | Reducing representations 25 Example 2.4 : Left regular representation of S3: group element (123) ............ 27 Exercise 2.1 : Left regular representation of S3 ......................... 28 Exercise 3.1 : Multiplication table of S3 ............................ 37 Exercise 3.2 : Conjugacy classes and cycle structure of permutations ............ 39 Example 3.1 : Young diagrams of size 4............................. 40 Example 3.2 : Iterative construction of Young diagrams.................... 40 Example 4.1 : Restricting the 2-dimensional irreducible representation of S3 to S2 . 48 Exercise 4.1 : Verifying relations between YJM elements and Coxeter generators . 52 Example 4.2 : GZ basis for the 2-dimensional irreducible representation of S3 . 53 Example 4.3 : Spectrum of GZ basis for the 2-dimensional irreducible representation of S3 54 Example 4.4 : Spectrum of GZ basis for the 1-dimensional irreducible representation1 of S3 54 Exercise 4.2 : GZ basis and spectra of S2 ............................ 55 Exercise 4.3 : Relation between spectrum vectors is an equivalence relation . 62 ∼ Example 4.5 : Young tableaux of size 4............................. 67 Example 4.6 : Young tableaux and paths in the Young lattice ................ 68 Example 4.7 : Content of a Young tableau ........................... 69 Exercise 4.4 : Relation between Young tableaux is an equivalence relation . 73 ≈ Example 5.1 : Permutations in S3 in 2-line notation...................... 80 Example 5.2 : Constructing the P -symbol of ρ = (134)(2569)(78) .............. 80 Example 5.3 : Constructing the Q-symbol of ρ = (134)(2569)(78) .............. 82 Example 5.4 : Reconstructing ρ from (P; Q) .......................... 83 Example 5.5 : Telephone number problem for 4 phones.................... 86 Exercise 5.1 : A formula for the number of Young tableaux in . 87 Yn Example 6.1 : Hook lengths of cells in a Young diagram.................... 89 Example 6.2 : Number of Young tableaux of a certain shape................. 90 Example 6.3 : Outer corners of a Young diagram........................ 93 5 Example 6.4 : Cell-ordering of a Young tableau | I...................... 98 Example 6.5 : Cell-ordering of a Young tableau | II ..................... 98 Exercise 6.1 : Showing that the NPS-algorithm yields a hook tableau............ 99 Example 6.6 : NPS-algorithm for a given tableau .......................100 Example 6.7 : SPN-algorithm | I................................102 Example 6.8 : SPN-algorithm | II ...............................104 Example 6.9 : SPN-algorithm | determining the correct candidate
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