DOCSLIB.ORG

Lecture Notes: Group Theory and Its Applications in Physics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes: Group Theory and Its Applications in Physics . Lecture notes: Group theory and its applications in physics Boris Gutkin Faculty of Physics, University Duisburg-Essen School of Physics, Georgia Tech |||||||||||||||||||||- version 2.3 printed April 29, 2016 comments to: [email protected] Contents Part 1. Finite and discrete groups 1 Lecture 1. Symmetries in Physics3 1. Classical physics3 2. Continues symmetries & Noether theorem4 3. Quantum mechanics6 4. Continues symmetries6 5. Discrete symmetries6 Lecture 2. Basics of Group Theory9 1. Groups definitions9 2. Subgroups 10 3. Conjugate classes. Normal subgroups 11 4. Point groups 12 5. Non-special transformations 13 Lecture 3. Representation Theory I 15 1. Basic notions 15 2. Schur's lemmas 16 3. Orthogonality theorem 17 Lecture 4. Representation Theory II 19 1. Characters 19 2. Classification of irreducible representations 19 3. How to find characters of irreducible representations? 20 4. Dual orthogonality relationship 20 5. Three types of representations 21 6. Representations of cross products 22 7. Representations of subgroups 22 8. Appendix. Proof of the theorem (4.1) 23 Lecture 5. Applications I. Vibration modes 25 1. Vibration spectrum of molecules 25 2. 3-dimensional symmetries 30 Lecture 6. Applications II. Quantum Mechanics 31 1. Spectral decomposition 31 2. Perturbation theory 32 3. Selection rules 33 4. Appendix A: Density of states or why singlets are rare in the energy spectrum of systems with high symmetry 34 5. Appendix B: Spectral statistics or why compex representation matter 35 References 35 References 35 ii LECTURE 0. CONTENTS Lecture 7. Applications III. Energy Band Structure 37 1. Lattice symmetries 37 2. Band structure 38 3. Band structure of graphene 40 References 41 References 41 Part 2. Continuous groups 43 Lecture 8. Lie groups & Lie algebras 45 1. Basic definitions and properties 45 2. Representations 46 Lecture 9. SU(2), SO(3) and their representations 49 1. Connection between SU(2) and SO(3) 49 2. Representations of SO(3) and SU(2) 50 3. Applications 51 4. Spinors 53 5. Product representations of SO(3) 55 6. Clebsch-Gordan series 55 7. Clebsch-Gordan coefficients. Adding angular momenta 55 8. Wigner-Eckart theorem 56 9. Applications 58 Lecture 10. Representations of simple algebras, general construction. Application to SU(3) 61 1. Adjoint representation and Killing form 61 2. Cartan sub-algebra and roots 61 3. Main properties of root systems 62 4. Building up representations of g 62 5. Representations of su(3) 63 Lecture 11. Strong interactions: flavor SU(3) 67 1. Internal symmetries: isospin 67 2. Internal symmetries: strangeness 69 3. Quarks 70 References 73 References 73 Lecture 12. Many particle systems. Young tableaux 75 1. Irreducible representations of Sn 75 2. Irreducible representations of SU(N) 78 References 78 References 78 Lecture 13. Classification of semi-simple Lie algebras 79 1. Semisimple algebras 79 2. Roots 79 3. A B C D E F G classification 81 4. Representation 83 Lecture 14. Infinite dimensional symmetries. String theory. 85 2015-01-24 iii Lecture notes: Group theory Part 1 Finite and discrete groups LECTURE 1 Symmetries in Physics 1. Classical physics What does it mean that a dynamical system has symmetries? Consider standard dynamical equa- tions: d @L @L Z t2 (1.1) = ;S = L(q; q_)dt dt @q_ @q t1 for a time evolution of a number of classical particles. To find solution q(t) we have to solve a complicated (system) of differential equations, which in exact form only in few exceptional cases (of integrable systems) is possible. On the other hand, it often happens that given an arbitrary solution q(t) one can construct a family of solutions by application of certain transformation to it: q(t; ) := g · q(t); such that q(t) = q(t; 0): In such a case the transformations g, form a symmetry group G of the system. By the definition −1 g1 ·g2 2 G if g1 ; g2 2 G and there exists an inverse transformation g for each g 2 G. In math- ematical language the last two properties in combination with the associativity of transformations imply that G is a group. x g ’ γ γ t1t 2 t’1t’ 2 t Figure 1. Symmetry transformation. Example 1.1. Continuous symmetries: Translational symmetries q(t; ) = q(t)+. Rotation symmetries q(t; ) = R() · q(t), R()RT () = I. The existence of such symmetries is directly follows from the invariance of the action S under the transformation g: (1.2) S(q(t)) = S(g · q(t)): This is clear, since g transforms extremum of S into another extremum, see figure1. In its own turn the invariance of the action would follow from the invariance of the Lagrangian: (1.3) L(q(t); q_(t))dt = L (g · q(t); g · q_(t)) d(g · t): 3 4 Lecture 1. SYMMETRIES IN PHYSICS Here we can also add an additional dF which changes transformed action by a constant (and therefore does not affect the above arguments). This leads to the following (sufficient) symmetry condition: d(g · t) dF (1.4) L(q(t); q_(t)) = L(g · q(t); g · q_(t)) + : dt dt To summarize, the transformations g leaving Ldt invariant up to a closed form dF form a group of symmetries of the system. They transform one solution of the dynamical equation to another: q(t) ! q(t; ); q(t; 0) = q(t): Note that typically we do not know explicitly any solution of this family. Nevertheless, knowl- edge of symmetries of the system might be very useful: With any continues symmetry of a Lagrangian system we can associate a conserved quantity. 2. Continues symmetries & Noether theorem Let g be a continues symmetry, i.e., 2 R. Consider an infinitesimal transformation q ! q + η(q) q(t; ) = q(t) + η(q(t)) which leaves L invariant up to a full time derivative, see (1.4): @L @L (2.1) L(q(t) + η(t); q_(t) + η_(t)) = L(q(t); q_(t)) + η(t) + η_(t) : @q @q_ It follows immediately from eq. (1.1) that: @L dJ (2.2) J = η(t) + F (t); = 0: @q_ dt From this also follows that dJ (2.3) fH; J g = η = 0; η dt @f @g @f @g where ff; gg = @q @p − @p @q is the Poisson bracket. Currents Jη form Lie algebra with the Poisson brackets playing the role of \commutator operation". Symmetry transformations including time. Considering more general transformation: 0 0 t ! t + φ(q; t) = t ; qi ! qi + ηi(q; t) = qi leaving S invariant (one can also consider φ and η depending on time derivatives of q = (q1; : : : qn)) we can get a generalized version of Noether theorem: X @L @L dJ (2.4) J = η (q; t) + L − q_ φ(q; t) + F (t); = 0: @q_ i @q_ i dt i i i Proof : If the system is invariant under the above transformation dq (2.5) L q ; i dt + dF = i dt dq0 = L q0; i dt0 = L (q + η (q; t); q_ (1 − φ(q; t)) + η_ (q; t)) (1 + φ(q; t))dt: i dt0 i i i i symmPhys - 2016-01-04 2. CONTINUES SYMMETRIES & NOETHER THEOREM 5 After expansion of the right hand side we have: X _ _ (2.6) [@qi Lηi + @q_i L(_ηi − q_iφ)] + Lφ = F i Now using the equation of motion (1.1) the left part of this equation can be written down as full time derivative: 2 3 d X 6 @L @L 7 6 η + L − q_ φ7 : dt 6@q_ i @q_ i 7 i 4 i i 5 | {z } Hamiltonian P 2 P Applications L = i mq_ i =2 + i6=j V (jqi − qjj) A) Shift in space: (x) (x) X (x) qi ! qi + , Jx = pi (Momentum). i B) Rotations: qi ! Rx;y;zqi: 0 cos() sin() 01 00 −1 01 2 (2.7) Rz = @− sin() cos() 0A = I + @1 0 0A + O( ) 0 0 1 0 0 0 X (x) (y) (y) (x) (2.8) Jz = −pi qi + pi qi (Angular Momentum). i C) Shift in time: X @L t ! t + , J = L − q_ = H (Energy). t @q_ i i i D) Some less standard example for the potential V (q) ∼ 1=jqj2. The transformation: 2 t ! λ t; qi ! λqi is an obvious symmetry of the action (consider Ldt). In the infinitesimal form λ = 1 + this symmetry is given by: 2 t ! λ t(1 + 2); qi ! qi(1 + ); implying that X 2 J = (1=2)dt qi − 2Ht; J = const: This leads to the conclusion that X 2 2 qi = 2Et + t · const: Depending on the sign of energy the particle either flies away with a constant radial velocity or collapses to the center in a finite time! symmPhys - 2016-01-04 6 Lecture 1. SYMMETRIES IN PHYSICS 3. Quantum mechanics Now consider action of symmetry group in the framework of quantum mechanics. By the same principle, if Ψ(x; t) is solution of the Schrodinger equation: (3.1) i~@tΨ(x; t) = HbΨ(x; t) the function Ψ0(x; t) = g · Ψ(x; t) ≡ Ψ(g · x; t) is also solution of the same equation. This is automatically satisfied if (3.2) g · Hb = Hb · g: 4. Continues symmetries As in the classical case we can consider an infinitesimal group action: (4.1) g · Ψ(x) ≈ Ψ(x) + JbΨ(x): By (3.2) we then immediately obtain: dJb (4.2) [H;b Jb] = = 0: dt Example 1.2. Translation and rotation symmetries: (a) Translation symmetry Ψ(x + ) = Ψ(x) + ∂xΨ; −i~Jb = −i~@x =p; ^ dtp^ = 0: (b) Rotation symmetry Ψ(x + y; y − x) = Ψ(x) + (y@x − x@y)Ψ ; −i~Jb = L;b dtLb = 0 5.
Load more

Recommended publications
  • SCHUR-WEYL DUALITY Contents Introduction 1 1. Representation
    SCHUR-WEYL DUALITY JAMES STEVENS Contents Introduction 1 1. Representation Theory of Finite Groups 2 1.1. Preliminaries 2 1.2. Group Algebra 4 1.3. Character Theory 5 2. Irreducible Representations of the Symmetric Group 8 2.1. Specht Modules 8 2.2. Dimension Formulas 11 2.3. The RSK-Correspondence 12 3. Schur-Weyl Duality 13 3.1. Representations of Lie Groups and Lie Algebras 13 3.2. Schur-Weyl Duality for GL(V ) 15 3.3. Schur Functors and Algebraic Representations 16 3.4. Other Cases of Schur-Weyl Duality 17 Appendix A. Semisimple Algebras and Double Centralizer Theorem 19 Acknowledgments 20 References 21 Introduction. In this paper, we build up to one of the remarkable results in representation theory called Schur-Weyl Duality. It connects the irreducible rep- resentations of the symmetric group to irreducible algebraic representations of the general linear group of a complex vector space. We do so in three sections: (1) In Section 1, we develop some of the general theory of representations of finite groups. In particular, we have a subsection on character theory. We will see that the simple notion of a character has tremendous consequences that would be very difficult to show otherwise. Also, we introduce the group algebra which will be vital in Section 2. (2) In Section 2, we narrow our focus down to irreducible representations of the symmetric group. We will show that the irreducible representations of Sn up to isomorphism are in bijection with partitions of n via a construc- tion through certain elements of the group algebra.
    [Show full text]
  • Higher-Spin Gauge Fields and Duality
    Higher-Spin Gauge Fields and Duality D. Franciaa and C.M. Hullb a Dipartimento di Fisica, Universit`adi Roma Tre, INFN, Sezione di Roma III, via della Vasca Navale 84, I-00146 Roma, Italy bTheoretical Physics Group, Blackett Laboratory, Imperial College of Science and Technology, London SW7 2BZ, U.K. Abstract. We review the construction of free gauge theories for gauge fields in arbitrary representations of the Lorentz group in D dimensions. We describe the multi-form calculus which gives the natural geometric framework for these theories. We also discuss duality transformations that give different field theory representations of the same physical degrees of freedom, and discuss the example of gravity in D dimensions and its dual realisations in detail. Based on the lecture presented by C.M. Hull at the First Solvay Workshop on Higher-Spin Gauge Theories, held in Brussels on May 12-14, 2004 36 Francia, Hull 1 Introduction Tensor fields in exotic higher-spin representations of the Lorentz group arise as massive modes in string theory, and limits in which such fields might become massless are of particular interest. In such cases, these would have to become higher-spin gauge fields with appropriate gauge invariance. Such exotic gauge fields can also arise as dual representations of more familiar gauge theories [1], [2]. The purpose here is to review the formulation of such exotic gauge theories that was developed in collaboration with Paul de Medeiros in [3], [4]. Free massless particles in D-dimensional Minkowski space are classified by representations of the little group SO(D − 2).
    [Show full text]
  • Math.NT] 13 Apr 2017 Rn DMS-1502834
    DIFFERENTIAL OPERATORS AND FAMILIES OF AUTOMORPHIC FORMS ON UNITARY GROUPS OF ARBITRARY SIGNATURE E. EISCHEN, J. FINTZEN, E. MANTOVAN, AND I. VARMA Abstract. In the 1970’s, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz’s approach to p-adic differential operators, his strategy has influenced four decades of devel- opments. Prior papers employing Katz’s and Serre’s ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a ≠ b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series.
    [Show full text]
  • Symmetries of Tensors
    Symmetries of Tensors A Dissertation Submitted to The Faculty of the Graduate School of the University of Minnesota by Andrew Schaffer Berget In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Advised by Professor Victor Reiner August 2009 c Andrew Schaffer Berget 2009 Acknowledgments I would first like to thank my advisor. Vic, thank you for listening to all my ideas, good and bad and turning me into a mathematician. Your patience and generosity throughout the many years has made it possible for me to succeed. Thank you for editing this thesis. I would also like to particularly thank Jay Goldman, who noticed that I had some (that is, ε > 0 amount of) mathematical talent before anyone else. He ensured that I found my way to Vic’s summer REU program as well as to graduate school, when I probably would not have otherwise. Other mathematicians who I would like to thank are Dennis Stanton, Ezra Miller, Igor Pak, Peter Webb and Paul Garrett. I would also like to thank Jos´eDias da Silva, whose life work inspired most of what is written in these pages. His generosity during my visit to Lisbon in October 2008 was greatly appreciated. My graduate school friends also helped me make it here: Brendon Rhoades, Tyler Whitehouse, Jo˜ao Pedro Boavida, Scott Wilson, Molly Maxwell, Dumitru Stamate, Alex Rossi Miller, David Treumann and Alex Hanhart. Thank you all. I would not have made it through graduate school without the occasional lapse into debauchery that could only be provided by Chris and Peter.
    [Show full text]
  • Schur Functors (A Project for Class)
    Schur Functors (a project for class) R. Vandermolen 1 Introduction In this presentation we will be discussing the Schur functor. For a complex vector space V , the Schur functor gives many irreducible representations of GL(V ), and other important subgroups of GL(V ), it will not be the purpose of this presentation to give this deep of an explanation. We will be concerning ourselves with the basic concepts needed to understand the Schur functor, and finish off by showing that the Schur functor does indeed help us decompose the nth tensor products V ⊗n. We will, for the most part, follow the presentation given in [1], and in turn we will use German gothic font for the symmetric group, and use diagrams in many occasions. With this said, we will not reprove major theorems from [1], instead we will focus our efforts on “filling in" the details of the presentation. We create new lemmas and deliver proofs of these new lemmas in efforts to deliver a fuller exposition on these Schur functors. Before we can begin a discussion of the Schur functor (or Weyl's module), we will need a familiarity with multilinear algebra. 1.1 Preliminaries Now we will deliver the definitions that will be of most interest to us in this presentation. We begin with the definition of the n-th tensor product. Definition 1. For a finite dimensional complex vector space V , and an natural number n, we define the complex n-th tensor product as follows V ⊗n := F (V n)= ∼ where F (V n) is the free abelian group on V n, considered as a set, and ∼ is the equivalence relation defined for v1; :::; vn; v^1; :::; v^n; w1; :::; wn 2 V , as (v1; :::; vn) ∼ (w1; :::; wn) whenever there exists a k 2 C such that k(v1; ::; vn) = (w1; :::; wn), and (v1; ::; vi; :::vn) + (v1; :::; v^i; :::; vn) ∼ (v1; ::; vi +v ^i; ::; vn) One can easily verify that this is indeed an equivalence relation.
    [Show full text]
  • Arxiv:0911.5452V1 [Math.CO] 29 Nov 2009
    TABLEAUX IN THE WHITNEY MODULE OF A MATROID ANDREW BERGET Abstract. The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of the matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom (also known as Schu- bert, or shifted) matroids. The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a for- mula for the multiplicity hook shapes in this representation in terms of the no broken circuit sets. 1. Introduction and Motivation If V is a complex vector space of dimension k we let V denote the exterior algebra of V , T n(V ) the n-fold tensor product V ⊗n and T (V ) the tensor algebra n V n≥0 T (V ). For the moment we will only be concerned with the GL(V )-module structure of T ( V ). We begin by seeing how tableaux describe a basis for the N C-vector space T ( V ). Recall that theV irreducible polynomial representations of GL(V ) are indexed by partitions λ with lengthV at most dim V . We denote the irreducible representation with highest weight λ by Sλ(V ). It follows from the Weyl character formula that the dimension of Sλ(V ) is the number of column strict tableaux of shape λ with entries in [dim V ] := {1, 2,..., dim V }.
    [Show full text]
  • Schur Functors and Schur Complexes
    ADVANCES m MATIFEMATICS 44,207-278 (1982) Schur Functors and Schur Complexes KAAN AKIN Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 DAVID A. BUCHSBAUM* Brandeis University, Waltham, Massachusetts 02154 AND JERZY WEYMAN Polish Academy of Sciences, Institute of Mathematics, N. Copernicus University Town’, Poland The study of Schur functors has a relatively long history. Its main impetus derived from representation theory, originally in characteristic zero. Over the years, however, with the development of modular representations, and algebraic geometry over fields of positive characteristic, the need for a theory of universal polynomial functors increased and, since the mid-1960s, approaches to a characteristic-free treatment of Schur functors have been developing (see, for instance, the recent book of Green [ 1 l] in which the treatments by Carter and Lusztig [6], Higman [12], and Towber [20], among others, are discussed). Our own interest in such a treatment was awakened by the work of Lascoux [ 141 on resolutions of determinantal ideals. Although his thesis treated only the characteristic zero case, it suggested that a general and elementary theory of Schur functors could be developed using only the rudiments of multilinear algebra (involving the Hopf algebra structures of the symmetric, exterior, and divided power algebras). Moreover, this elementary development admitted of a natural generalization to the idea of Schur complexes, whose usefulness was demonstrated, for instance, in our construction of a universal minimal * The author acknowledges the partial support of NSF Grants MC8 80-03118 and MCS 79-14790, and the support of the Sonderforschungsbereich “Theoretische Mathematik,” University of Bonn, during the preparation of this article.
    [Show full text]
  • Representation Theory of the Symmetric Group: Basic Elements
    REPRESENTATION THEORY OF THE SYMMETRIC GROUP: BASIC ELEMENTS BY GRAHAM GILL, FOR MAT1196F Since Cayley’s theorem implies that every finite group G is isomorphic to a subgroup of SjGj, understanding the representation theory of the finite symmetric groups is likely to yield productive tools and results for the representation theory of finite groups in general. In these notes we will examine the basic results for the representation theory of Sn. We will describe all irreducible representations of Sn and their characters, as well as giving a complete method for decomposing a representation of Sn into its irreducible factors, in terms of certain elements of the group algebra CSn known as Young symmetrizers. No such general method exists for general finite groups, so in a sense the fundamental problems of representation theory have been answered in the case of finite symmetric groups. The Group Algebra CG We first review some facts regarding the group algebra CG where G is a finite group. If A(G) is the algebra of all complex valued functions on G with multipli- cation given by the convolution X ¡ ¢ f ¤ f 0(g) = f (g0) f 0 g0¡1g ; g02G then CG is isomorphic as a complex algebra with A(G), via the map CG !A(G), 0 P g 7! e , where e : G ! C is the function g 7! ± 0 . If a = a g 2 CG, then g g g;g gP2G g a is sent to the function g 7! ag. Henceforth we will write a = a(g)g. Then ³ ¡ ¢´ g2G P 0 0 P P 0 0¡1 ab = 0 a(g)b(g )gg = 0 a(g )b g g g, so that as an element g;g 2G g2G g 2G¡ ¢ P 0 0¡1 in A(G), (ab)(g) := a ¤ b(g) = g02G a(g )b g g .
    [Show full text]
  • SCHUR-WEYL DUALITY Contents 1. Some Background 1 2. Plan of The
    SCHUR-WEYL DUALITY QUANG DAO Abstract. We will switch gears this week and talk about the relationship between irreducible representations of the symmetric group Sk and irreducible finite-dimensional representations of the general linear groups GLn. This is known as Schur-Weyl duality. Along the way, we will introduce some key ingredients in the proof such as the Lie algebra gln and the Double Commu- tant Theorem. Schur-Weyl duality also gives rise to the Schur functor, which generalizes the constructions of the symmetric and exterior powers. We will comment on this generalization and work out some non-trivial cases by hand. Contents 1. Some Background 1 2. Plan of the Proof 2 3. Double Commutant Theorem 3 4. Proof of Schur-Weyl Duality 4 5. Schur Functor and Examples 5 References 6 1. Some Background In order to describe the theorems and their proofs, we need some terminology from module theory and Lie theory. Definition 1.1. A simple module is a non-zero module with no non-zero proper submodule. Definition 1.2. A semi-simple module is a module that can be written as a direct sum of simple module. Proposition 1.3. Submodules and quotient modules of a semi-simple module is semi-simple. Remark. In the language of representation theory, any representation of a finite group is a semi-simple module, and simple modules correspond to irreducible rep- resentations. The only definition we need from Lie theory is that of a Lie algebra. Date: July 29, 2018. 1 2 QUANG DAO Definition 1.4. Let g be a vector space over a field k and let [,] : g × g ! g be a skew-symmetric bilinear map.
    [Show full text]
  • The Symmetric Group and Young Diagrams Proseminar: Algebra, Topology and Group Theory in Physics, Organised by Prof
    The Symmetric Group and Young Diagrams Proseminar: Algebra, Topology and Group Theory in Physics, organised by Prof. Matthias Gaberdiel Martina Niggli, Department of Physics ETH Zurich, [email protected] Assistant: Dr. Mahdi Godazgar March 26, 2018 Contents 1 Introduction 1 2 The symmetric group 2 2.1 Conjugacy classes . .2 3 Partitions and Young diagrams 3 3.1 Motivation . .3 3.2 Young tableaux . .3 3.3 Interlude: regular representation and group algebra . .4 3.4 Young symmetrizer . .4 3.5 Example: S3 .........................................5 4 Frobenius formula 7 4.1 Dimension of an irreducible representation of the symmetric group . .8 5 Application: identical particles 9 5.1 Example: three identical particles that obey parastatistics . 10 6 Conclusions 10 7 References 11 1 Introduction The symmetric group plays an important role in some aspects of Lie theory. Therefore we want to understand its irreducible representations. In this paper we will give a complete description of the irreducible representations of the symmetric group, that is, to provide a construction of the irreducible representations via Young symmetrizers and a formula for the characters of the irreducible representations, the Frobenius formula. Also, a physical application of the theory is of interest. We will study a quantum mechanical system of identical particles and examine the physical meaning of the irreducible representations of the symmetric group. 1 2 The symmetric group The symmetric group Sn is the set of invertible transformations of the set of natural numbers f1; : : : ; ng with the group operation being the composition of transformations. The composition of transforma- tions is associative, there is an identity element being the trivial transformation and for every transfor- mation there is an inverse transformation such that the composition of the two is equal to the trivial transformation.
    [Show full text]
  • REPRESENTATIONS of the SYMMETRIC GROUP Contents 1. Introduction 1 2. Basic Definitions and Complete Reducibility 2 3. Characters
    REPRESENTATIONS OF THE SYMMETRIC GROUP DAPHNE KAO Abstract. I will introduce the topic of representation theory of finite groups by investigating representations of S3 and S4 using character theory. Then I will generalize these examples by describing all irreducible representations of any symmetric group on n letters. Finally, I will briefly discuss how to dis- cover irreducible representations of any group using Schur Functors, which are constructed using the irreducible representations of Sn. This paper assumes familiarity with group theory, FG-modules, linear algebra, and category the- ory. Contents 1. Introduction 1 2. Basic Definitions and Complete Reducibility 2 3. Characters and Hands-on Examples Involving Them 4 4. Representations of Sn 9 5. Schur Functors 13 Acknowledgments 14 References 14 1. Introduction A representation of a group G is a homomorphism ρ from G into the general linear group of some vector space V . When such a map exists, if we write g · v rather than ρ(g)(v) to denote the image of a vector v in V under the automorphism ρ(g), then we can think of V as a CG-module, since for any vectors v and w, any group elements g and h, and any scalar c, the following properties hold: (1) ρ(g)(v) = g · v is in V . (2) ρ(1) = I 2 GL(V ) by the homomorphism property, so ρ(1)(v) = I(v) = v, which means 1 · v = v. (3) (ρ(g) ◦ ρ(h))(v) = ρ(g)(ρ(h)(v)), so (gh) · v = g · (h · v). (4) ρ(g)(cv) = cρ(g)(v) by linearity, so g · cv = c(g · v).
    [Show full text]
  • Representation Theory of Combinatorial Categories
    Representation Theory of Combinatorial Categories by John D. Wiltshire-Gordon A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2016 Doctoral Committee: Associate Professor David E. Speyer, Chair Professor William Fulton Professor Jeffrey C. Lagarias Associate Professor Yaoyun Shi Assistant Professor Andrew Snowden c John D. Wiltshire-Gordon 2016 All Rights Reserved ACKNOWLEDGEMENTS Thanks to my helpful, knowledgeable, and wise advisor, David Speyer. Many thanks as well to Thomas Church and Benson Farb for their inspirational paper [CF13] and their encouragement when I was an undergraduate at the University of Chicago and after. Thank you to my fellow graduate students at the University of Michigan for providing a stimulating mathematical environment. Thanks to the NSF for support through a graduate research fellowship, and the Rackham Graduate School for support through a predoctoral fellowship. It was a pleasure to discuss the ideas of this thesis in helpful conversations with Daniel Barter, E. Hunter Brooks, Thomas Church, Jordan Ellenberg, Pavel Etingof, Benson Farb, Nic Ford, Trevor Hyde, Gene Kopp, Visu Makam, Jeremy Miller, Andrew Putman, Victor Reiner, Julian Rosen, Steven Sam, Andrew Snowden, Benjamin Steinberg, Phil Tosteson, and Jennifer Wilson. Finally, my heartfelt thanks go to Hannah Fehlner-Peach, my parents, and my siblings for their continued encouragement and support. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS :::::::::::::::::::::::::: ii LIST OF FIGURES ::::::::::::::::::::::::::::::: vi LIST OF TABLES :::::::::::::::::::::::::::::::: vii ABSTRACT ::::::::::::::::::::::::::::::::::: viii CHAPTER I. Introduction ..............................1 1.0.1 This thesis in broad strokes . .3 1.0.2 Outline . .4 1.1 What is a representation? .
    [Show full text]