DIFFERENT VERSIONS of the IMPRIMITIVITY THEOREM 1 Introduction
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The Theory of Induced Representations in Field Theory
QMW{PH{95{? hep-th/yymmnnn The Theory of Induced Representations in Field Theory Jose´ M. Figueroa-O'Farrill ABSTRACT These are some notes on Mackey's theory of induced representations. They are based on lectures at the ITP in Stony Brook in the Spring of 1987. They were intended as an exercise in the theory of homogeneous spaces (as principal bundles). They were never meant for distribution. x1 Introduction The theory of group representations is by no means a closed chapter in mathematics. Although for some large classes of groups all representations are more or less completely classified, this is not the case for all groups. The theory of induced representations is a method of obtaining representations of a topological group starting from a representation of a subgroup. The classic example and one of fundamental importance in physics is the Wigner construc- tion of representations of the Poincar´egroup. Later Mackey systematized this construction and made it applicable to a large class of groups. The method of induced representations appears geometrically very natural when expressed in the context of (homogeneous) vector bundles over a coset manifold. In fact, the representation of a group G induced from a represen- tation of a subgroup H will be decomposed as a \direct integral" indexed by the elements of the space of cosets G=H. The induced representation of G will be carried by the completion of a suitable subspace of the space of sections through a given vector bundle over G=H. In x2 we review the basic notions about coset manifolds emphasizing their connections with principal fibre bundles. -
Base Size of Finite Primitive Solvable Permutation Groups
Base size of finite primitive solvable permutation groups Ayan Maiti Under the supervision of Prof. Pal Hegedus A thesis presented for the partial fulfilment towards the degree of Masters of Science in Mathematics Mathematics and its Application Central European University Hungary Declaration of Authorship I, Ayan Maiti, declare that this thesis titled, "Base size of finite primitive solvable per- mutation groups" and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a masters degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Ayan Maiti Date: 20th May, 2016 1 Abstract The content of this thesis report is based on the bounds of the base size of affine type primitive permutation groups, the bound was conjectured by Pyber and later was proved by Akos Seress. The the primary focus of this thesis is to understand the basic idea and the proof given by Akos Seress. -
REPRESENTATION THEORY. WEEK 4 1. Induced Modules Let B ⊂ a Be
REPRESENTATION THEORY. WEEK 4 VERA SERGANOVA 1. Induced modules Let B ⊂ A be rings and M be a B-module. Then one can construct induced A module IndB M = A ⊗B M as the quotient of a free abelian group with generators from A × M by relations (a1 + a2) × m − a1 × m − a2 × m, a × (m1 + m2) − a × m1 − a × m2, ab × m − a × bm, and A acts on A ⊗B M by left multiplication. Note that j : M → A ⊗B M defined by j (m) = 1 ⊗ m is a homomorphism of B-modules. Lemma 1.1. Let N be an A-module, then for ϕ ∈ HomB (M, N) there exists a unique ψ ∈ HomA (A ⊗B M, N) such that ψ ◦ j = ϕ. Proof. Clearly, ψ must satisfy the relation ψ (a ⊗ m)= aψ (1 ⊗ m)= aϕ (m) . It is trivial to check that ψ is well defined. Exercise. Prove that for any B-module M there exists a unique A-module satisfying the conditions of Lemma 1.1. Corollary 1.2. (Frobenius reciprocity.) For any B-module M and A-module N there is an isomorphism of abelian groups ∼ HomB (M, N) = HomA (A ⊗B M, N) . F Example. Let k ⊂ F be a field extension. Then induction Indk is an exact functor from the category of vector spaces over k to the category of vector spaces over F , in the sense that the short exact sequence 0 → V1 → V2 → V3 → 0 becomes an exact sequence 0 → F ⊗k V1⊗→ F ⊗k V2 → F ⊗k V3 → 0. Date: September 27, 2005. -
Representations of Finite Groups’ Iordan Ganev 13 August 2012 Eugene, Oregon
Notes for ‘Representations of Finite Groups’ Iordan Ganev 13 August 2012 Eugene, Oregon Contents 1 Introduction 2 2 Functions on finite sets 2 3 The group algebra C[G] 4 4 Induced representations 5 5 The Hecke algebra H(G; K) 7 6 Characters and the Frobenius character formula 9 7 Exercises 12 1 1 Introduction The following notes were written in preparation for the first talk of a week-long workshop on categorical representation theory. We focus on basic constructions in the representation theory of finite groups. The participants are likely familiar with much of the material in this talk; we hope that this review provides perspectives that will precipitate a better understanding of later talks of the workshop. 2 Functions on finite sets Let X be a finite set of size n. Let C[X] denote the vector space of complex-valued functions on X. In what follows, C[X] will be endowed with various algebra structures, depending on the nature of X. The simplest algebra structure is pointwise multiplication, and in this case we can identify C[X] with the algebra C × C × · · · × C (n times). To emphasize pointwise multiplication, we write (C[X], ptwise). A C[X]-module is the same as X-graded vector space, or a vector bundle on X. To see this, let V be a C[X]-module and let δx 2 C[X] denote the delta function at x. Observe that δ if x = y δ · δ = x x y 0 if x 6= y It follows that each δx acts as a projection onto a subspace Vx of V and Vx \ Vy = 0 if x 6= y. -
Representation Theory with a Perspective from Category Theory
Representation Theory with a Perspective from Category Theory Joshua Wong Mentor: Saad Slaoui 1 Contents 1 Introduction3 2 Representations of Finite Groups4 2.1 Basic Definitions.................................4 2.2 Character Theory.................................7 3 Frobenius Reciprocity8 4 A View from Category Theory 10 4.1 A Note on Tensor Products........................... 10 4.2 Adjunction.................................... 10 4.3 Restriction and extension of scalars....................... 12 5 Acknowledgements 14 2 1 Introduction Oftentimes, it is better to understand an algebraic structure by representing its elements as maps on another space. For example, Cayley's Theorem tells us that every finite group is isomorphic to a subgroup of some symmetric group. In particular, representing groups as linear maps on some vector space allows us to translate group theory problems to linear algebra problems. In this paper, we will go over some introductory representation theory, which will allow us to reach an interesting result known as Frobenius Reciprocity. Afterwards, we will examine Frobenius Reciprocity from the perspective of category theory. 3 2 Representations of Finite Groups 2.1 Basic Definitions Definition 2.1.1 (Representation). A representation of a group G on a finite-dimensional vector space is a homomorphism φ : G ! GL(V ) where GL(V ) is the group of invertible linear endomorphisms on V . The degree of the representation is defined to be the dimension of the underlying vector space. Note that some people refer to V as the representation of G if it is clear what the underlying homomorphism is. Furthermore, if it is clear what the representation is from context, we will use g instead of φ(g). -
Induced Representations and Frobenius Reciprocity 1
Induced Representations and Frobenius Reciprocity Math G4344, Spring 2012 1 Generalities about Induced Representations For any group G and subgroup H, we get a restriction functor G ResH : Rep(G) ! Rep(H) that gives a representation of H from a representation of G, by simply restricting the group action to H. This is an exact functor. We would like to be able to go in the other direction, building up representations of G from representations of its subgroups. What we want is an induction functor G IndH : Rep(H) ! Rep(G) that should be an adjoint to the restriction functor. This adjointness relation is called \Frobenius Reciprocity". Unfortunately, there are two possible adjoints, Ind could be a left-adjoint G G HomG(IndH (V );W ) = HomH (V; ResH (W )) or a right-adjoint. Let's call the right-adjoint \coInd" G G HomG(V; coIndH (W )) = HomH (ResH (V );W ) Given an algebra A, with sub-algebra B, tensor product ⊗ and Hom of modules provide such functors. Since A HomA(A ⊗B V; W ) = HomB(V; ResB(W )) we can get a left-adjoint version of induction using the tensor product and since A HomA(V; HomB(A; W )) = HomB(ResB(V );W ) we can get a right-adjoint version of induction using Hom. So Definition 1 (Induction for finite groups). For H a subgroup of G, both finite groups, and V a representation of H, one has representations of G defined by G IndH (V ) = CG ⊗CH V and G coIndH (V ) = HomCH (CG; V ) and 1 G Theorem 1 (Frobenius reciprocity for finite groups). -
Induced Representations
MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 37 (b) f is balanced in the sense that f(xs, v)=f(x, sv) for all x M, s S, v V . In other words,∈ f is bilinear∈ ∈ and balanced. (2) For any other bilinear, balanced mapping g : M V W there is a unique homomorphism g : M V W so× that→g = g f ⊗S → ◦ Let BiLin(M V, W ) denote the set of all balanced bilinear maps ×S M V W . Then the universal property! says that ! × → BiLin(M V, W ) = Hom(M V, W ) ×S ∼ ⊗S On the other hand the definitions of balanced and bilinear imply that BiLin(M V, W ) = Hom (V,Hom(M, W )) ×S ∼ S The balance bilinear map φ : M V W corresponds to its adjoint ×S → φ : V Hom(M, W ) given by φ(v)(x)=φ(x, v). → (1) φ(x, v) is linear in x iff φ(v) Hom(M, W ). This is clear. ∈ ! (2) φ(x, v) is linear in v iff φ! is linear, i.e., gives a homomorphism of abelian groups V Hom(! M, W ). This is also clear. → (3) Finally, φ is balance iff φ!(xs, v)=φ(x, sv) iff φ(sv)(x)=φ(v)(xs) = [sφ(v)](x) iff φs = sφ. ! ! ! In the case when M is an R-S-bimodule we just need to observe the obvious fact! that!φ is an R-homomorphism in the first coordinate iff φ(V ) Hom (M, W ). ⊆ R The adjunction formula follows from these observations. -
Induced Representations, Intertwining Operators, and Transfer for Real Groups
Induced Representations, Intertwining Operators and Transfer James Arthur This paper is dedicated to George Mackey. Abstract. Induced representations and intertwining operators give rise to distributions that represent critical terms in the trace formula. We shall de- scribe a conjectural relationship between these distributions and the Langlands- Shelstad-Kottwitz transfer of functions Our goal is to describe a conjectural relationship between intertwining oper- ators and the transfer of functions. One side of the proposed identity is a linear combination of traces tr RP (πw) ◦ IP (π; f) ; where IP (π; f) is the value of an induced representation at a test function f, and RP (πw) is a standard self-intertwining operator for the representation. Distribu- tions of this sort are critical terms in the trace formula. The other side is defined by the transfer of f to an endoscopic group. The endoscopic transfer of functions represents a sophisticated theory, some of it still conjectural, for comparing trace formulas on different groups [L2]. This is a central theme in the general study of automorphic forms. There are a number of relatively recent ideas of Langlands, Shelstad and Kot- twitz, which will have to be taken for granted in the statement of such an identity. On the other hand, induced representations and intertwining operators go back many years. They were lifelong preoccupations of George Mackey [M1]{[M5], whose investigations anticipated their future role as basic objects in modern rep- resentation theory. It is therefore fitting to devote the first part of the article to a historical introduction. I shall try to describe in elementary terms how Mackey's initial ideas are reflected in some of the ways the later theory developed. -
APPENDIX A. INDUCED REPRESENTATIONS It Is Our Purpose Here to Provide a Brief Summary of the Main Ideas and Results on Induced R
APPENDIX A. INDUCED REPRESENTATIONS It is our purpose here to provide a brief summary of the main ideas and results on induced representations which are fundamental to the subject matter of this entire treatise. The idea of forming induced representations -- that is, of inducing a representation of a group G from a representation of a subgroup H goes back to Frobenius in the last century. His work of course was in the case G finite. The possibility of forming induced representations for infinite groups was later explored (usually in primitive or ad hoc fashion) by many authors, most notably by the Russians Gelfand, Namiark, Graev, etc. The subject was finally givena firm footing by Mackey in the 1940's and early 1950's. It was a fundamental achievement, without which the theory of group representations might still be crawling around on its hands and knees. We begin with the notion of a quasi-invariant measure. Let G be a locally compact group and suppose X is a right Borel G-space. That is, there is a Borel map XxG + X, (x,g) + x•g, such that: x•e = x, Vx E X, and or said otherwise g + (g: x + x•g) is an anti-homomorphism of G into the group of Borel automorphisms of X. Next suppose that X carries a cr finite positive Borel measure ~· We say that ~ is quasi-invariant under the action of G if for every g E G, the measures ~ and ~·g are equivalent. By ~·g we mean the measure that assigns to a Borel set E ~X the value (~·g)(E) = ~(E•g), E•g = {x•g: x E E}. -
George Mackey and His Work on Representation Theory and Foundations of Physics
Contemporary Mathematics George Mackey and His Work on Representation Theory and Foundations of Physics V. S. Varadarajan To the memory of George Mackey Abstract. This article is a retrospective view of the work of George Mackey and its impact on the mathematics of his time and ours. The principal themes of his work–systems of imprimitivity, induced representations, projective rep- resentations, Borel cohomology, metaplectic representations, and so on, are examined and shown to have been central in the development of the theory of unitary representations of arbitrary separable locally compact groups. Also examined are his contributions to the foundations of quantum mechanics, in particular the circle of ideas that led to the famous Mackey-Gleason theorem and a significant sharpening of von Neumann’s theorem on the impossibil- ity of obtaining the results of quantum mechanics by a mechanism of hidden variables. Contents 1. Introduction 2 2. Stone-Von Neumann theorem, systems of imprimitivity, and the imprimitivity theorem 3 3. Semi direct products and the little group method. Representations of finite length and orbit schemes 8 4. Super Lie groups and their systems of imprimitivity 11 5. Intertwining numbers, double coset decompositions, and irreducibility of induced representations 13 6. Projective representations, Borel cohomology of groups, and the metaplectic representation 15 7. Foundations of Physics: hidden variables and the Mackey-Gleason theorem 21 2000 Mathematics Subject Classification. 22D10, 81P10. Key words and phrases. Stone-von Neumann theorem, systems of imprimitivity, unitary rep- resentations, induced representations, semi direct products, super Lie groups, intertwining num- bers, double coset decompositions, projective representations, Borel cohomology, Baruer groups, metaplectic representation, hidden variables, Mackey-Gleason theorem, quantum logics, canonical commutation rules, quantum information theory, quantum stochastic calculus. -
Covariant Quantum Fields Via Lorentz Group Representation of Weyl
Covariant Quantum Fields via Lorentz Group Representation of Weyl Operators Radhakrishnan Balu Army Research Laboratory Adelphi, MD, 21005-5069, USA [email protected] Department of Mathematics & Norbert Wiener Center for Harmonic Analysis and Applications, University of Maryland, College Park, MD 20742. [email protected] Received: date / Accepted: September 11, 2019 Abstract The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covari- ant version of the calculus we construct representations of Poinca`re group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations arXiv:1908.09180v2 [math-ph] 9 Sep 2019 of Poinca`re group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hilbert space formed from the sections of the bun- dle. Our Weyl operators are constructed on symmetric Fock space of this space and the corresponding annihilation, creation, and conservation operators are syn- thesized in the usual fashion in relativistic theories for space-like, time-like, and light-like fields. We achieve this by constructing transitive systems of imprimitivity (second-quantized SI), which are dynamical systems with trajectories dense in the configuration space, by induced representations. We provide the details of the field operators for the case of massive Bosons as the rest are similar in construction and indicate the ways to construct adapted processes paving way for building covariant quantum stochastic calculus. -
Imprimitive Permutation Groups
Imprimitive Permutation Groups Edward Dobson Department of Mathematics and Statistics Mississippi State University and IAM, University of Primorska [email protected] 1 Introduction The O'Nan-Scott Theorem together with the Classification of the Finite Simple Groups is a powerful tool that give the structure of all primitive permutation groups, as well as their actions. This has allowed for the solution to many classical problems, and has opened the door to a deeper understanding of imprimitive permutation groups, as primitive permutation groups are the building blocks of imprimitive permutation groups. We first give a more or less standard introduction to imprimitive groups, and then move to less well-known techniques, with an emphasis on studying automorphism groups of graphs. A few words about these lecture notes. The lecture notes are an \expanded" version of the lecture - some of the lecture will be basically exactly these lecture notes, but in many cases the proofs of some background results (typically those that in my view are those whose proofs are primarily checking certain computations) are given in these lecture notes but will not be given in the lectures due to time constraints. Also, the material is organized into sections by topic, not by lecture. 2 Basic Results on Imprimitive Groups Definition 2.1 Let G be a transitive group acting on X. A subset B ⊆ X is a block of G if whenever g 2 G, then g(B) \ B = ; or B. If B = fxg for some x 2 X or B = X, then B is a trivial block. Any other block is nontrivial.