The Origin of Representation Theory
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REPRESENTATIONS of FINITE GROUPS 1. Definition And
REPRESENTATIONS OF FINITE GROUPS 1. Definition and Introduction 1.1. Definitions. Let V be a vector space over the field C of complex numbers and let GL(V ) be the group of isomorphisms of V onto itself. An element a of GL(V ) is a linear mapping of V into V which has a linear inverse a−1. When V has dimension n and has n a finite basis (ei)i=1, each map a : V ! V is defined by a square matrix (aij) of order n. The coefficients aij are complex numbers. They are obtained by expressing the image a(ej) in terms of the basis (ei): X a(ej) = aijei: i Remark 1.1. Saying that a is an isomorphism is equivalent to saying that the determinant det(a) = det(aij) of a is not zero. The group GL(V ) is thus identified with the group of invertible square matrices of order n. Suppose G is a finite group with identity element 1. A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism ρ : G ! GL(V ) of G to the group of automorphisms of V . We say such a map gives V the structure of a G-module. The dimension V is called the degree of ρ. Remark 1.2. When there is little ambiguity of the map ρ, we sometimes call V itself a representation of G. In this vein we often write gv_ or gv for ρ(g)v. Remark 1.3. Since ρ is a homomorphism, we have equality ρ(st) = ρ(s)ρ(t) for s; t 2 G: In particular, we have ρ(1) = 1; ; ρ(s−1) = ρ(s)−1: A map ' between two representations V and W of G is a vector space map ' : V ! W such that ' V −−−−! W ? ? g? ?g (1.1) y y V −−−−! W ' commutes for all g 2 G. -
Notes on the Finite Fourier Transform
Notes on the Finite Fourier Transform Peter Woit Department of Mathematics, Columbia University [email protected] April 28, 2020 1 Introduction In this final section of the course we'll discuss a topic which is in some sense much simpler than the cases of Fourier series for functions on S1 and the Fourier transform for functions on R, since it will involve functions on a finite set, and thus just algebra and no need for the complexities of analysis. This topic has important applications in the approximate computation of Fourier series (which we won't cover), and in number theory (which we'll say a little bit about). 2 The group Z(N) What we'll be doing is simplifying the topic of Fourier series by replacing the multiplicative group S1 = fz 2 C : jzj = 1g by the finite subgroup Z(N) = fz 2 C : zN = 1g If we write z = reiθ, then zN = 1 =) rN eiθN = 1 =) r = 1; θN = k2π (k 2 Z) k =) θ = 2π k = 0; 1; 2; ··· ;N − 1 N mod 2π The group Z(N) is thus explicitly given by the set i 2π i2 2π i(N−1) 2π Z(N) = f1; e N ; e N ; ··· ; e N g Geometrically, these are the points on the unit circle one gets by starting at 1 2π and dividing it into N sectors with equal angles N . (Draw a picture, N = 6) The set Z(N) is a group, with 1 identity element 1 (k = 0) inverse 2πi k −1 −2πi k 2πi (N−k) (e N ) = e N = e N multiplication law ik 2π il 2π i(k+l) 2π e N e N = e N One can equally well write this group as the additive group (Z=NZ; +) of inte- gers mod N, with isomorphism Z(N) $ Z=NZ ik 2π e N $ [k]N 1 $ [0]N −ik 2π e N $ −[k]N = [−k]N = [N − k]N 3 Fourier analysis on Z(N) An abstract point of view on the theory of Fourier series is that it is based on exploiting the existence of a particular othonormal basis of functions on the group S1 that are eigenfunctions of the linear transformations given by rotations. -
An Introduction to the Trace Formula
Clay Mathematics Proceedings Volume 4, 2005 An Introduction to the Trace Formula James Arthur Contents Foreword 3 Part I. The Unrefined Trace Formula 7 1. The Selberg trace formula for compact quotient 7 2. Algebraic groups and adeles 11 3. Simple examples 15 4. Noncompact quotient and parabolic subgroups 20 5. Roots and weights 24 6. Statement and discussion of a theorem 29 7. Eisenstein series 31 8. On the proof of the theorem 37 9. Qualitative behaviour of J T (f) 46 10. The coarse geometric expansion 53 11. Weighted orbital integrals 56 12. Cuspidal automorphic data 64 13. A truncation operator 68 14. The coarse spectral expansion 74 15. Weighted characters 81 Part II. Refinements and Applications 89 16. The first problem of refinement 89 17. (G, M)-families 93 18. Localbehaviourofweightedorbitalintegrals 102 19. The fine geometric expansion 109 20. Application of a Paley-Wiener theorem 116 21. The fine spectral expansion 126 22. The problem of invariance 139 23. The invariant trace formula 145 24. AclosedformulaforthetracesofHeckeoperators 157 25. Inner forms of GL(n) 166 Supported in part by NSERC Discovery Grant A3483. c 2005 Clay Mathematics Institute 1 2 JAMES ARTHUR 26. Functoriality and base change for GL(n) 180 27. The problem of stability 192 28. Localspectraltransferandnormalization 204 29. The stable trace formula 216 30. Representationsofclassicalgroups 234 Afterword: beyond endoscopy 251 References 258 Foreword These notes are an attempt to provide an entry into a subject that has not been very accessible. The problems of exposition are twofold. It is important to present motivation and background for the kind of problems that the trace formula is designed to solve. -
Weak Representation Theory in the Calculus of Relations Jeremy F
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2006 Weak representation theory in the calculus of relations Jeremy F. Alm Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Alm, Jeremy F., "Weak representation theory in the calculus of relations " (2006). Retrospective Theses and Dissertations. 1795. https://lib.dr.iastate.edu/rtd/1795 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Weak representation theory in the calculus of relations by Jeremy F. Aim A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Roger Maddux, Major Professor Maria Axenovich Paul Sacks Jonathan Smith William Robinson Iowa State University Ames, Iowa 2006 Copyright © Jeremy F. Aim, 2006. All rights reserved. UMI Number: 3217250 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. -
Geometric Representation Theory, Fall 2005
GEOMETRIC REPRESENTATION THEORY, FALL 2005 Some references 1) J. -P. Serre, Complex semi-simple Lie algebras. 2) T. Springer, Linear algebraic groups. 3) A course on D-modules by J. Bernstein, availiable at www.math.uchicago.edu/∼arinkin/langlands. 4) J. Dixmier, Enveloping algebras. 1. Basics of the category O 1.1. Refresher on semi-simple Lie algebras. In this course we will work with an alge- braically closed ground field of characteristic 0, which may as well be assumed equal to C Let g be a semi-simple Lie algebra. We will fix a Borel subalgebra b ⊂ g (also sometimes denoted b+) and an opposite Borel subalgebra b−. The intersection b+ ∩ b− is a Cartan subalgebra, denoted h. We will denote by n, and n− the unipotent radicals of b and b−, respectively. We have n = [b, b], and h ' b/n. (I.e., h is better to think of as a quotient of b, rather than a subalgebra.) The eigenvalues of h acting on n are by definition the positive roots of g; this set will be denoted by ∆+. We will denote by Q+ the sub-semigroup of h∗ equal to the positive span of ∆+ (i.e., Q+ is the set of eigenvalues of h under the adjoint action on U(n)). For λ, µ ∈ h∗ we shall say that λ ≥ µ if λ − µ ∈ Q+. We denote by P + the sub-semigroup of dominant integral weights, i.e., those λ that satisfy hλ, αˇi ∈ Z+ for all α ∈ ∆+. + For α ∈ ∆ , we will denote by nα the corresponding eigen-space. -
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). -
Representing Groups on Graphs
Representing groups on graphs Sagarmoy Dutta and Piyush P Kurur Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India 208016 {sagarmoy,ppk}@cse.iitk.ac.in Abstract In this paper we formulate and study the problem of representing groups on graphs. We show that with respect to polynomial time turing reducibility, both abelian and solvable group representability are all equivalent to graph isomorphism, even when the group is presented as a permutation group via generators. On the other hand, the representability problem for general groups on trees is equivalent to checking, given a group G and n, whether a nontrivial homomorphism from G to Sn exists. There does not seem to be a polynomial time algorithm for this problem, in spite of the fact that tree isomorphism has polynomial time algorithm. 1 Introduction Representation theory of groups is a vast and successful branch of mathematics with applica- tions ranging from fundamental physics to computer graphics and coding theory [5]. Recently representation theory has seen quite a few applications in computer science as well. In this article, we study some of the questions related to representation of finite groups on graphs. A representation of a group G usually means a linear representation, i.e. a homomorphism from the group G to the group GL (V ) of invertible linear transformations on a vector space V . Notice that GL (V ) is the set of symmetries or automorphisms of the vector space V . In general, by a representation of G on an object X, we mean a homomorphism from G to the automorphism group of X. -
Groups for Which It Is Easy to Detect Graphical Regular Representations
CC Creative Commons ADAM Logo Also available at http://adam-journal.eu The Art of Discrete and Applied Mathematics nn (year) 1–x Groups for which it is easy to detect graphical regular representations Dave Witte Morris , Joy Morris Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta. T1K 3M4, Canada Gabriel Verret Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Received day month year, accepted xx xx xx, published online xx xx xx Abstract We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G; S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism ' of G such that '(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product Zp o Zp is not DRR-detecting, for every odd prime p. We also show that if G1 and G2 are nontrivial groups that admit a digraphical regular representation and either gcd jG1j; jG2j = 1, or G2 is not DRR-detecting, then the direct product G1 × G2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations. Keywords: Cayley graph, GRR, DRR, automorphism group, normalizer Math. Subj. Class.: 05C25,20B05 1 Introduction All groups and graphs in this paper are finite. Recall [1] that a digraph Γ is said to be a digraphical regular representation (DRR) of a group G if the automorphism group of Γ is isomorphic to G and acts regularly on the vertex set of Γ. -
The Conjugate Dimension of Algebraic Numbers
THE CONJUGATE DIMENSION OF ALGEBRAIC NUMBERS NEIL BERRY, ARTURAS¯ DUBICKAS, NOAM D. ELKIES, BJORN POONEN, AND CHRIS SMYTH Abstract. We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Q-dimension of the space spanned by its conjugates. For all but seven nonnegative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GLn(Q); this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when Q is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension Q(!`) of Q. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number. 1. Introduction Let Q be an algebraic closure of the field Q of rational numbers, and let α 2 Q. Let α1; : : : ; αd 2 Q be the conjugates of α over Q, with α1 = α. Then d equals the degree d(α) := [Q(α): Q], the dimension of the Q-vector space spanned by the powers of α. In contrast, we define the conjugate dimension n = n(α) of α as the dimension of the Q-vector space spanned by fα1; : : : ; αdg. -
Introuduction to Representation Theory of Lie Algebras
Introduction to Representation Theory of Lie Algebras Tom Gannon May 2020 These are the notes for the summer 2020 mini course on the representation theory of Lie algebras. We'll first define Lie groups, and then discuss why the study of representations of simply connected Lie groups reduces to studying representations of their Lie algebras (obtained as the tangent spaces of the groups at the identity). We'll then discuss a very important class of Lie algebras, called semisimple Lie algebras, and we'll examine the repre- sentation theory of two of the most basic Lie algebras: sl2 and sl3. Using these examples, we will develop the vocabulary needed to classify representations of all semisimple Lie algebras! Please email me any typos you find in these notes! Thanks to Arun Debray, Joakim Færgeman, and Aaron Mazel-Gee for doing this. Also{thanks to Saad Slaoui and Max Riestenberg for agreeing to be teaching assistants for this course and for many, many helpful edits. Contents 1 From Lie Groups to Lie Algebras 2 1.1 Lie Groups and Their Representations . .2 1.2 Exercises . .4 1.3 Bonus Exercises . .4 2 Examples and Semisimple Lie Algebras 5 2.1 The Bracket Structure on Lie Algebras . .5 2.2 Ideals and Simplicity of Lie Algebras . .6 2.3 Exercises . .7 2.4 Bonus Exercises . .7 3 Representation Theory of sl2 8 3.1 Diagonalizability . .8 3.2 Classification of the Irreducible Representations of sl2 .............8 3.3 Bonus Exercises . 10 4 Representation Theory of sl3 11 4.1 The Generalization of Eigenvalues . -
Some Applications of the Fourier Transform in Algebraic Coding Theory
Contemporary Mathematics Some Applications of the Fourier Transform in Algebraic Coding Theory Jay A. Wood In memory of Shoshichi Kobayashi, 1932{2012 Abstract. This expository article describes two uses of the Fourier transform that are of interest in algebraic coding theory: the MacWilliams identities and the decomposition of a semi-simple group algebra of a finite group into a direct sum of matrix rings. Introduction Because much of the material covered in my lectures at the ASReCoM CIMPA research school has already appeared in print ([18], an earlier research school spon- sored by CIMPA), the organizers requested that I discuss more broadly the role of the Fourier transform in coding theory. I have chosen to discuss two aspects of the Fourier transform of particular interest to me: the well-known use of the complex Fourier transform in proving the MacWilliams identities, and the less-well-known use of the Fourier transform as a way of understanding the decomposition of a semi-simple group algebra into a sum of matrix rings. In the latter, the Fourier transform essentially acts as a change of basis matrix. I make no claims of originality; the content is discussed in various forms in such sources as [1,4, 15, 16 ], to whose authors I express my thanks. The material is organized into three parts: the theory over the complex numbers needed to prove the MacWilliams identities, the theory over general fields relevant to decomposing group algebras, and examples over finite fields. Each part is further divided into numbered sections. Personal Remark. This article is dedicated to the memory of Professor Shoshichi Kobayashi, my doctoral advisor, who died 29 August 2012. -
Effective Noether Irreducibility Forms and Applications*
Appears in Journal of Computer and System Sciences, 50/2 pp. 274{295 (1995). Effective Noether Irreducibility Forms and Applications* Erich Kaltofen Department of Computer Science, Rensselaer Polytechnic Institute Troy, New York 12180-3590; Inter-Net: [email protected] Abstract. Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring, and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial, and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class . The factorization results also extend to the case where the coefficient field is a function field.