Introduction to Representation Theory
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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. -
Frobenius Recip-Rocity and Extensions of Nilpotent Lie Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 298, Number I, November 1986 FROBENIUS RECIP-ROCITY AND EXTENSIONS OF NILPOTENT LIE GROUPS JEFFREY FOX ABSTRACT. In §1 we use COO-vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson multiplicity formula for compact nilmanifolds. In §2 we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In §3 we give an application of the previous results to obtain a reduction formula for solvable Lie groups. Introduction. In the representation theory of rings the fact that the tensor product functor and Hom functor are adjoints is of fundamental importance and at the time very simple and elegant. When this adjointness property is translated into a statement about representations of finite groups via the group ring, the classical Frobenius reciprocity theorem emerges. From this point of view, Frobenius reciproc- ity, aside from being a useful tool, provides a very natural explanation for many phenomena in representation theory. When G is a Lie group and r is a discrete cocompact subgroup, a main problem in harmonic analysis is to describe the decomposition of ind~(1)-the quasiregular representation. When G is semisimple the problem is enormously difficult, and little is known in general. However, if G is nilpotent, a rather detailed description of ind¥(l) is available, thanks to the work of Moore, Howe, Richardson, and Corwin and Greenleaf [M, H-I, R, C-G]. If G is solvable Howe has developed an inductive method of describing ind~(l) [H-2]. -
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. -
Pioneers of Representation Theory, by Charles W
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 37, Number 3, Pages 359{362 S 0273-0979(00)00867-3 Article electronically published on February 16, 2000 Pioneers of representation theory, by Charles W. Curtis, Amer. Math. Soc., Prov- idence, RI, 1999, xvi + 287 pp., $49.00, ISBN 0-8218-9002-6 The theory of linear representations of finite groups emerged in a series of papers by Frobenius appearing in 1896{97. This was at first couched in the language of characters but soon evolved into the formulation now considered standard, in which characters give the traces of representing linear transformations. There were of course antecedents in the number-theoretic work of Lagrange, Gauss, and others| especially Dedekind, whose correspondence with Frobenius suggested a way to move from characters of abelian groups to characters of arbitrary finite groups. In the past century this theory has developed in many interesting directions. Besides being a natural tool in the study of the structure of finite groups, it has turned up in many branches of mathematics and has found extensive applications in chemistry and physics. Marking the end of the first century of the subject, the book under review offers a somewhat unusual blend of history, biography, and mathematical exposition. Before discussing the book itself, it may be worthwhile to pose a general question: Does one need to know anything about the history of mathematics (or the lives of individual mathematicians) in order to appreciate the subject matter? Most of us are complacent about quoting the usual sloppy misattributions of famous theorems, even if we are finicky about the details of proofs. -
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). -
Induction and Restriction As Adjoint Functors on Representations of Locally Compact Groups
Internat. J. Math. & Math. Scl. 61 VOL. 16 NO. (1993) 61-66 INDUCTION AND RESTRICTION AS ADJOINT FUNCTORS ON REPRESENTATIONS OF LOCALLY COMPACT GROUPS ROBERT A. BEKES and PETER J. HILTON Department of Mathematics Department of Mathematical Sciences Santa Clara University State University of New York at Binghamton Santa Clara, CA 95053 Bingharnton, NY 13902-6000 (Received April 16, 1992) ABSTRACT. In this paper the Frobenius Reciprocity Theorem for locally compact groups is looked at from a category theoretic point of view. KEY WORDS AND PHRASES. Locally compact group, Frobenius Reciprocity, category theory. 1991 AMS SUBJECT CLASSIFICATION CODES. 22D30, 18A23 I. INTRODUCTION. In [I] C. C. Moore proves a global version of the Frobenius Reciprocity Theorem for locally compact groups in the case that the coset space has an invariant measure. This result, for arbitrary closed subgroups, was obtained by A. Kleppner [2], using different methods. We show how a slight modification of Moore's original proof yields the general result. It is this global version of the reciprocity theorem that is the basis for our categorical approach. (See [3] for a description of the necessary category-theoretical concepts.) We begin by setting up the machinery necessary to discuss the reciprocity theorem. Next we show how, using the global version of the theorem, the functors of induction and restriction are adjoint. The proofs of these results are at the end of the paper. 2. THE MAIN RFULTS. Throughout G is a separable locally compact group and K is a closed subgroup. Let G/K denote the space of right cosets of K in G and for s E G, we write for the coset Ks. -
Permutation Groups Containing a Regular Abelian Subgroup: The
PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP: THE TANGLED HISTORY OF TWO MISTAKES OF BURNSIDE MARK WILDON Abstract. A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2- transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this note we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data. 1. Introduction In 1911, writing in §252 of the second edition of his influential textbook [6], Burnside claimed a proof of the following theorem. Theorem 1.1. Let G be a transitive permutation group of composite prime- power degree containing a regular cyclic subgroup. Either G is imprimitive or G is 2-transitive. An error in the penultimate sentence of Burnside’s proof was noted in [7, page 24], where Neumann remarks ‘Nevertheless, the theorem is certainly true and can be proved by similar character-theoretic methods to those that Burnside employed’. In §3 we present the correct part of Burnside’s arXiv:1705.07502v2 [math.GR] 11 Jun 2017 proof in today’s language. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Math 126 Lecture 4. Basic Facts in Representation Theory
Math 126 Lecture 4. Basic facts in representation theory. Notice. Definition of a representation of a group. The theory of group representations is the creation of Frobenius: Georg Frobenius lived from 1849 to 1917 Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Frobenius.html Matrix form of a representation. Equivalence of two representations. Invariant subspaces. Irreducible representations. One dimensional representations. Representations of cyclic groups. Direct sums. Tensor product. Unitary representations. Averaging over the group. Maschke’s theorem. Heinrich Maschke 1853 - 1908 Schur’s lemma. Issai Schur Biography of Schur. Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Russian Empire (now Belarus) Died: 10 Jan 1941 in Tel Aviv, Palestine (now Israel) Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius. -
A Brief History of an Important Classical Theorem
Advances in Group Theory and Applications c 2016 AGTA - www.advgrouptheory.com/journal 2 (2016), pp. 121–124 ISSN: 2499-1287 DOI: 10.4399/97888548970148 A Brief History of an Important Classical Theorem L.A. Kurdachenko — I.Ya.Subbotin (Received Nov. 6, 2016 – Communicated by Francesco de Giovanni) Mathematics Subject Classification (2010): 20F14, 20F19 Keywords: Schur theorem; Baer theorem; Neumann theorem This note is a by-product of the authors’ investigation of relations between the lower and upper central series in a group. There are some famous theorems laid at the foundation of any research in this area. These theorems are so well-known that, by the established tra- dition, they usually were named in honor of the mathematicians who first formulated and proved them. However, this did not always hap- pened. Our intention here is to talk about one among the fundamen- tal results of infinite group theory which is linked to Issai Schur. Issai Schur (January 10, 1875 in Mogilev, Belarus — January 10, 1941 in Tel Aviv, Israel) was a very famous mathematician who proved many important and interesting results, first in algebra and number theory. There is a biography sketch of I. Schur wonderfully written by J.J. O’Connor and E.F. Robertson [6]. The authors documented that I. Schur actively and successfully worked in many branches of mathematics, such as for example, finite groups, matrices, alge- braic equations (where he gave examples of equations with alterna- ting Galois groups), number theory, divergent series, integral equa- tions, function theory. We cannot add anything important to this, saturated with factual and emotional details, article of O’Connor 122 L.A. -
Interview with Martin Gardner Page 602
ISSN 0002-9920 of the American Mathematical Society june/july 2005 Volume 52, Number 6 Interview with Martin Gardner page 602 On the Notices Publication of Krieger's Translation of Weil' s 1940 Letter page 612 Taichung, Taiwan Meeting page 699 ., ' Martin Gardner (see page 611) AMERICAN MATHEMATICAL SOCIETY A Mathematical Gift, I, II, Ill The interplay between topology, functions, geometry, and algebra Shigeyuki Morita, Tokyo Institute of Technology, Japan, Koji Shiga, Yokohama, Japan, Toshikazu Sunada, Tohoku University, Sendai, Japan and Kenji Ueno, Kyoto University, Japan This three-volume set succinctly addresses the interplay between topology, functions, geometry, and algebra. Bringing the beauty and fun of mathematics to the classroom, the authors offer serious mathematics in an engaging style. Included are exercises and many figures illustrating the main concepts. It is suitable for advanced high-school students, graduate students, and researchers. The three-volume set includes A Mathematical Gift, I, II, and III. For a complete description, go to www.ams.org/bookstore-getitem/item=mawrld-gset Mathematical World, Volume 19; 2005; 136 pages; Softcover; ISBN 0-8218-3282-4; List US$29;AII AMS members US$23; Order code MAWRLD/19 Mathematical World, Volume 20; 2005; 128 pages; Softcover; ISBN 0-8218-3283-2; List US$29;AII AMS members US$23; Order code MAWRLD/20 Mathematical World, Volume 23; 2005; approximately 128 pages; Softcover; ISBN 0-8218-3284-0; List US$29;AII AMS members US$23; Order code MAWRLD/23 Set: Mathematical World, Volumes 19, 20, and 23; 2005; Softcover; ISBN 0-8218-3859-8; List US$75;AII AMS members US$60; Order code MAWRLD-GSET Also available as individual volumes .. -
Letters from William Burnside to Robert Fricke: Automorphic Functions, and the Emergence of the Burnside Problem
LETTERS FROM WILLIAM BURNSIDE TO ROBERT FRICKE: AUTOMORPHIC FUNCTIONS, AND THE EMERGENCE OF THE BURNSIDE PROBLEM CLEMENS ADELMANN AND EBERHARD H.-A. GERBRACHT Abstract. Two letters from William Burnside have recently been found in the Nachlass of Robert Fricke that contain instances of Burnside's Problem prior to its first publication. We present these letters as a whole to the public for the first time. We draw a picture of these two mathematicians and describe their activities leading to their correspondence. We thus gain an insight into their respective motivations, reactions, and attitudes, which may sharpen the current understanding of professional and social interactions of the mathemat- ical community at the turn of the 20th century. 1. The simple group of order 504 { a first meeting of minds Until 1902, when the publication list of the then fifty-year-old William Burnside already encompassed 90 papers, only one of these had appeared in a non-British journal: a three-page article [5] entitled \Note on the simple group of order 504" in volume 52 of Mathematische Annalen in 1898. In this paper Burnside deduced a presentation of that group in terms of generators and relations,1 which is based on the fact that it is isomorphic to the group of linear fractional transformations with coefficients in the finite field with 8 elements. The proof presented is very concise and terse, consisting only of a succession of algebraic identities, while calculations in the concrete group of transformations are omitted. In the very same volume of Mathematische Annalen, only one issue later, there can be found a paper [18] by Robert Fricke entitled \Ueber eine einfache Gruppe von 504 Operationen" (On a simple group of 504 operations), which is on exactly the same subject.