DOCSLIB.ORG

Representation Theory: a Combinatorial Viewpoint

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Representation Theory: a Combinatorial Viewpoint Cambridge7A FM 2014/8/25 15:39 Page i #1 CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 147 Editorial Board B. BOLLOBAS,´ W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO REPRESENTATION THEORY: A COMBINATORIAL VIEWPOINT This book discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. The first chapter provides a detailed account of necessary representation-theoretic background. An important highlight of this book is an innovative treatment of the Robinson–Schensted–Knuth correspondence and its dual by extending Viennot’s geometric ideas. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. The treatment of Schur–Weyl duality reveals the directness and simplicity of Schur’s original treatment of the subject. This book is suitable for graduate students, advanced undergraduates and non-specialists with a background in mathematics or physics. Amritanshu Prasad is a mathematician at The Institute of Mathematical Sciences, Chennai. He obtained his PhD from the University of Chicago, where he worked on automorphic forms and representations of p-adic groups. His current research interests include representation theory, combinatorics, harmonic analysis and number theory. Prasad has extensive experience in teaching mathematics to undergraduate and graduate students in the US, Canada and India. He has been an associate of the Indian Academy of Sciences and was awarded the Young Scientist Medal by the Indian National Science Academy. Downloaded from https://www.cambridge.org/core. Cambridge University Main, on 09 Jul 2018 at 04:21:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139976824 Cambridge7A FM 2014/8/25 15:39 Page ii #2 cambridge studies in advanced mathematics Editorial Board: B. Bollobas,´ W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: www.cambridge.org/mathematics. Already published 107 K. Kodaira Complex analysis 108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic analysis on finite groups 109 H. Geiges An introduction to contact topology 110 J. Faraut Analysis on Lie groups: An introduction 111 E. Park Complex topological K-theory 112 D. W. Stroock Partial differential equations for probabilists 113 A. Kirillov, Jr An introduction to Lie groups and Lie algebras 114 F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II 115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics 116 D. Applebaum L´evyprocesses and stochastic calculus (2nd Edition) 117 T. Szamuely Galois groups and fundamental groups 118 G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices 119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields 120 P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths 121 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups 122 S. Kalikow & R. McCutcheon An outline of ergodic theory 123 G. F. Lawler & V. Limic Random walk: A modern introduction 124 K. Lux & H. Pahlings Representations of groups 125 K. S. Kedlaya p-adic differential equations 126 R. Beals & R. Wong Special functions 127 E. de Faria & W. de Melo Mathematical aspects of quantum field theory 128 A. Terras Zeta functions of graphs 129 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I 130 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II 131 D. A. Craven The theory of fusion systems 132 J. Va¨an¨ anen¨ Models and games 133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type 134 P. Li Geometric analysis 135 F. Maggi Sets of finite perimeter and geometric variational problems 136 M. Brodmann & R. Y. Sharp Local cohomology (2nd Edition) 137 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I 138 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, II 139 B. Helffer Spectral theory and its applications 140 R. Pemantle & M. C. Wilson Analytic combinatorics in several variables 141 B. Branner & N. Fagella Quasiconformal surgery in holomorphic dynamics 142 R. M. Dudley Uniform central limit theorems (2nd Edition) 143 T. Leinster Basic category theory 144 I. Arzhantsev, U. Derenthal, J. Hausen & A. Laface Cox rings 145 M. Viana Lectures on Lyapunov exponents 146 C. Bishop & Y. Peres Fractal sets in probability and analysis Downloaded from https://www.cambridge.org/core. Cambridge University Main, on 09 Jul 2018 at 04:21:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139976824 Cambridge7A FM 2014/8/25 15:39 Page iii #3 Representation Theory A Combinatorial Viewpoint Amritanshu Prasad The Institute of Mathematical Sciences, Chennai Downloaded from https://www.cambridge.org/core. Cambridge University Main, on 09 Jul 2018 at 04:21:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139976824 Cambridge7A FM 2014/8/25 15:39 Page iv #4 Cambridge House, 4381/4 Ansari Road, Daryaganj, Delhi 110002, India Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107082052 c Amritanshu Prasad 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in India A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Prasad, Amritanshu, author. Representation theory : a combinatorial viewpoint / Amritanshu Prasad. pages cm Includes bibliographical references and index. Summary: “Discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups”—Provided by publisher. ISBN 978-1-107-08205-2 (hardback) 1. Combinatorial group theory. 2. Representations of groups. 3. Symmetry groups. 4. Symmetric functions. I. Title. QA182.5.P73 2015 515’.7223—dc23 2014024621 ISBN 978-1-107-08205-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Downloaded from https://www.cambridge.org/core. Cambridge University Main, on 09 Jul 2018 at 04:21:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139976824 Cambridge7A FM 2014/8/25 15:39 Page v #5 Contents List of Tables page vii Preface page ix 1 Basic Concepts of Representation Theory 1 1.1 Representations and Modules 1 1.2 Invariant Subspaces and Simplicity 5 1.3 Complete Reducibility 7 1.4 Maschke’s Theorem 11 1.5 Decomposing the Regular Module 13 1.6 Tensor Products 19 1.7 Characters 22 1.8 Representations over Complex Numbers 29 2 Permutation Representations 32 2.1 Group Actions and Permutation Representations 32 2.2 Permutations 34 2.3 Partition Representations 39 2.4 Intertwining Permutation Representations 41 2.5 Subset Representations 44 2.6 Intertwining Partition Representations 46 3 The RSK Correspondence 51 3.1 Semistandard Young Tableaux 51 3.2 The RSK Correspondence 56 3.3 Classification of Simple Representations of S n 68 4 Character Twists 70 4.1 Inversions and the Sign Character 70 4.2 Twisting by a Multiplicative Character 73 4.3 Conjugate of a Partition 75 4.4 Twisting by the Sign Character 79 v Downloaded from https://www.cambridge.org/core. Cambridge University Main, on 09 Jul 2018 at 04:21:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139976824 Cambridge7A FM 2014/8/25 15:39 Page vi #6 vi Contents 4.5 The Dual RSK Correspondence 80 4.6 Representations of Alternating Groups 83 5 Symmetric Functions 96 5.1 The Ring of Symmetric Functions 96 5.2 Other Bases for Homogeneous Symmetric Functions 98 5.3 Specialization to m Variables 107 5.4 Schur Functions and the Frobenius Character Formula 110 5.5 Frobenius’ Characteristic Function 117 5.6 Branching Rules 119 5.7 Littlewood–Richardson Coefficients 120 5.8 The Hook–Length Formula 124 5.9 The Involution sλ 7! sλ0 127 5.10 The Jacobi–Trudi Identities 129 5.11 The Recursive Murnaghan–Nakayama Formula 132 5.12 Character Values of Alternating Groups 136 6 Representations of General Linear Groups 141 6.1 Polynomial Representations 141 6.2 Schur Algebras 142 6.3 Schur Algebras and Symmetric Groups 148 6.4 Modules of a Commutant 150 6.5 Characters of the Simple Representations 153 6.6 Polynomial Representations of the Torus 155 6.7 Weight Space Decompositions 158 Hints and Solutions to Selected Exercises 160 Suggestions for Further Reading 182 References 185 Index 189 Downloaded from https://www.cambridge.org/core. Cambridge University Main, on 09 Jul 2018 at 04:21:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
Load more

Recommended publications
  • Contents 1 Root Systems
    Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations.
    [Show full text]
  • Sums of Multiplicative Characters with Additive Convolutions
    Sums of multiplicative characters with additive convolutions ∗ I. D. Shkredov, A. S. Volostnov Annotation. In the paper we obtain new estimates for binary and ternary sums of multiplica- tive characters with additive convolutions of characteristic functions of sets, having small additive doubling. In particular, we improve a result of M.–C. Chang. The proof uses Croot–Sisask almost periodicity lemma. 1 Introduction Let p be a prime number, Fp be the prime field and χ be a nontrivial multi- plicative character modulo p. In the paper we consider a problem of obtaining good upper bounds for the exponential sum χ(a + b) , (1) a∈XA, b∈B arXiv:1606.00358v1 [math.NT] 1 Jun 2016 where A, B are arbitrary subsets of the field Fp. Exponential sums of such a type were studied by various authors, see e.g. [2], [4], [8]–[10]. There is a well– known hypothesis on sums (1) which is called the graph Paley conjecture, see the history of the question in [2] or [13], for example. Conjecture (Paley graph). Let δ > 0 be a real number, A, B F be ⊂ p arbitrary sets with A > pδ and B > pδ. Then there exists a number | | | | ∗This work is supported by grant Russian Scientific Foundation RSF 14–11–00433. 1 I. D. Shkredov, A. S. Volostnov 2 τ = τ(δ) such that for any sufficiently large prime number p and all nontrivial characters χ the following holds χ(a + b) <p−τ A B . (2) | || | a∈A, b∈B X Let us say a few words about the name of the hypothesis.
    [Show full text]
  • NOTES for NUMBER THEORY COURSE 1. Unique Factorization
    NOTES FOR NUMBER THEORY COURSE 1. Unique factorization 1.1. All the rings we consider are assumed to have multiplicative unit 1 and almost always they will be commutative. N, Z, Q, R, C will denote the natural numbers, integers, rational numbers, real numbers and complex numbers respectively. A number α 2 C is called an algebraic number, if there exists a polynomial p(x) 2 Q[x] with p(α) = 0. We shall let Q¯ be the set of all algebraic numbers. Fact: \C and Q¯ are algebraically closed". IF R is a ring R[x1; ··· ; xn] will denote the ring of polynomials in n variables with coeffi- cients in R. The letter k will usually denote a field. If R ⊆ S are rings, and α1; ··· ; αk are elements of S, we shall let R[α1; ··· ; αn] be the subring of S generated by R and α1; ··· ; αn. Here are somep examples of rings R of the type we willp be interested in: R = Z, R = k[x], R = Z[i]( i = −1), R = Z[!](! = e2πi=3), R = Z[ 3], or more generally let R = Z[α], where α is an algebraic number. 1.2. First definitions: principal ideals, prime ideals... An element u 2 R is called an unit if there exists v 2 R such that uv = 1. Such a v is necessarily unique (Why?) and is called the inverse of u. The set of units in R will be denoted by U(R). The units in Z are 1 and −1. There are six units in Z[!] (the sixth roots of unity).
    [Show full text]
  • Complex Numbers
    APPENDIX Complex Numbers The complex Bumbers are a set of objects which can be added and multiplied, the sum and product of two complex numbers being also a complex number, and satisfy the following conditions. (1) Every real number is a complex number, and if a, ß are real numbers, then their sum and product as complex numbers are the same as their sum and product as real numbers. (2) There is a complex number denoted by i such that i2 = - 1. (3) Every complex number can be written uniquely in the form a + bi where a, b are real numbers. (4) The ordinary laws of arithmetic concerning addition and multipli­ cation are satisfied. We list these laws: If a, ß, y are complex numbers, then (aß)y = a(ßy) and (a + ß) + Y = a + (ß + y). We have a(ß + y) = aß + ay, and (ß + y)a = ßa + ya. We have aß = ßa, and a + ß = ß + a. If 1 is the real number one, then 1a = a. If 0 is the real number zero, then Oa = o. We have a + (-l)a = o. We shall now draw consequences of these properties. With each complex number a + bi, we associate the vector (a, b) in the plane. Let a = a 1 + a2 i and ß = b1 + b2 i be two complex numbers. Then 278 COMPLEX NUMBERS [APP. I] Hence addition of complex numbers is carried out "componentwise" and corresponds to addition of vectors in the plane. For example, (2 + 3i) + ( - 1 + 5i) = 1 + 8i. In multiplying complex numbers, we use the rule i2 = - 1 to simplify a product and to put it in the form a + bio For instance, let a = 2 + 3i and ß = 1 - i.
    [Show full text]
  • IDENTITIES on QUADRATIC GAUSS SUMS the Multiplicative Group F X
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 321, Number I, September 1990 IDENTITIES ON QUADRATIC GAUSS SUMS PAUL GERARDIN AND WEN-CH 'ING WINNIE LI ABSTRACT. Given a local field F , each multiplicative character () of the split algebra F x F or of a separable quadratic extension of F has an associated generalized Gauss sum )I:. It is a complex valued function on the character group of F X x F, meromorphic in the first variable. We define a pairing between such Gauss sums and study its properties when F is a nonarchimedean local field. This has important applications to the representation theory of GL(2, F) and correspondences [GL3]. INTRODUCTION The multiplicative group F X of a local field F is a split extension of the value group IF x I by the compact group of the units. Hence, the group .91 (Fx) of continuous homomorphisms of F X in CX is a one-dimensional complex Lie group, with connected component of identity the image of C under the map S I---t (t I---t Itn. We have written It I for the normalized absolute value of t. The group F x acts on functions on F by translations: X t: J I---t /, /(x) = J(tx) , t E F • This gives an action of F X on the space sP(F) of Schwartz-Bruhat functions on F, hence also an action on the space sP' (F) of tempered distributions on F: (t.DI/) = (Dlf) . For each X in .91 (F x ), the space of tempered distributions of type X under the action of F X is one-dimensional (e.g.
    [Show full text]
  • Character Sums in Finite Fields
    COMPOSITIO MATHEMATICA A. ADOLPHSON STEVEN SPERBER Character sums in finite fields Compositio Mathematica, tome 52, no 3 (1984), p. 325-354 <http://www.numdam.org/item?id=CM_1984__52_3_325_0> © Foundation Compositio Mathematica, 1984, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 52 (1984) 325-354 © 1984 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands CHARACTER SUMS IN FINITE FIELDS A. Adolphson * and Steven Sperber ** 1. Introduction Let p be a prime, q = pa, and denote by Fqm the field of qm elements. Let ~1,...,~b: Fqm ~ ex be multiplicative characters. Composing with the norm map Nm: F§m - Fq gives multiplicative characters on Fqm: We extend these characters to Fq m by defining ~(m)i(0) = 0. Let X be an algebraic variety over Fq and gl, ... , gb regular functions on X. We define character sums Sm ( X; gl , ... , gb; ~1,...,~b)( = Sm ) by where the sum is over all x E X(Fqm), the Fqm-valued points of X. Such sums have been studied classically by Davenport [6] in the one variable case, and the Brewer and Jacobsthal sums in particular are of this type. More recently, mixed sums involving additive and multiplica- tive characters have been treated p-adically by Gross-Koblitz, Boyarsky, Robba, and Adolphson-Sperber.
    [Show full text]
  • Fourier Analysis on Finite Groups
    FOURIER ANALYSIS ON FINITE GROUPS A Thesis by JOSE MANUEL RIVERA MONTES DE OCA Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Chair of Committee, Matthew Young Committee Members, Matthew Papanikolas Riad Masri Alan Dabney Head of Department, Emil Straube August 2017 Major Subject: Mathematics Copyright 2017 Jose Manuel Rivera Montes de Oca ABSTRACT We start with some Fourier analysis on cyclic groups as the base case. This theory comes along with some basic notions about character theory. So, we develop some prop- erties of the additive characters of Z=qZ. Formulas like the matrix version of the DFT, its inverse formula and Plancherel’s theorem are proved for this case. Then, we give a constructive generalization to any finite abelian group. We also present some work on properties of characters in Z=qZ∗. All these tools are used to define and develop multi- plicative characters. In particular, we mention Dirichlet characters and Gauss and Jacobi sums. The most important result of this work is to note that Fourier transform in Z=qZ gives us a method to write a Dirichlet character in terms of additive characters. Finally, we apply some of this theory to Cayley graphs. ii DEDICATION A mi familia, por haberles tocado el trabajo más difícil. To my family, for they did the real hard work. iii CONTRIBUTORS AND FUNDING SOURCES Contributors This work was supported by a thesis committee consisting of Professor Mathew Young [advisor], Professor Mathew Papanikolas and Professor Riad Masri of the Department of Mathematics and Professor Alan Dabney of the Department of Statistics.
    [Show full text]
  • Character Sums Estimates and an Application to a Problem of Balog
    Character sums estimates and an application to a problem of Balog Tomasz Schoen∗, Ilya D. Shkredov† Abstract We prove new bounds for sums of multiplicative characters over sums of set with small doubling and applying this result we break the square–root barrier in a problem of Balog concerning products of differences in a field of prime order. 1. Introduction For a prime number p let Fp be the prime field and let χ be a nontrivial multiplicative character modulo p. We will deal with a problem of estimating the exponential sum of the form χ(a + b) , (1) a A, b B ∈X∈ where A, B are arbitrary subsets of the field Fp. Such exponential sums were studied by numerous authors, see e.g. [4], [6], [7], [9]–[11], [19], [22]. One of the most important conjecture concerning character sums is the Paley graph conjecture, see [4]. Conjecture 1 For every δ > 0 there is τ = τ(δ) > 0 such that for every prime number p>p(τ) δ δ and for any set A, B Fp with A >p and B >p we have arXiv:2004.01885v1 [math.NT] 4 Apr 2020 ⊆ | | | | τ χ(a + b) <p− A B . (2) | | | | a A, b B ∈X∈ Currently there are very few results regarding the above conjecture. The only affirmative answer was obtained in the following case 1 A >p 2 +δ, B >pδ , (3) | | | | see [9]—[11]. M.–C. Chang [4] proved a result towards the conjecture for sets A and B, where one of them has a small sumset.
    [Show full text]
  • Notes on Character Sums
    NOTES ON CHARACTER SUMS WEN WANG Abstract. In this article, the properties of character sums, including Gauss sums and Jacobi sums are investigated. 1. Introduction{Historical Notes The origin of the Gauss sum and Jacobi sum in the work of C.F. Gauss and C.G.J. Jacobi. Gauss introduced the Gauss sum in his Disquisitione Arithmeticae[Ga1] in July, 1801, and Jacobi introduced the Jacobi sum in a letter to Gauss[Ja1] dated February 8, 1827. The sum introduced by Gauss in 1801 is kX−1 e2휋imn2=k; n=0 which is now called a quadratic Gauss sum. This sum is not easy to evaluate, even in the special case that m = 1 and k is an odd positive integer.p Inp this case, Gauss was easily able to show that this sum has the value ± k or ±i k, according as k is of the form 4u + 1 or 4u + 3, respectively. Specific examples convinced Gauss that the plus sign is always correct. On August 30, 1805, Gauss wrote in his diary he was able to prove his conjecture on the sign of these sums. A few years later, Gauss[Ga2] published an evaluation of his quadratic Gauss sum for all positive integer k. In his study of primes in arithmetic progressions, G.L. Dirichlet [Di1] introduced the multiplicative character 휒 modulo k and the sum kX−1 G(휒) = e2휋imn=k: n=0 This is also called a Gauss sum, as it coincides with the quadratic Gauss sum above in the case that 휒 has order 2 and k is a prime not dividing m.
    [Show full text]
  • Explicit Multiplicative Relations Between Gauss Sums Brian J
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2002 Explicit multiplicative relations between Gauss sums Brian J. Murray Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Murray, Brian J., "Explicit multiplicative relations between Gauss sums" (2002). LSU Doctoral Dissertations. 757. https://digitalcommons.lsu.edu/gradschool_dissertations/757 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. EXPLICIT MULTIPLICATIVE RELATIONS BETWEEN GAUSS SUMS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Brian J. Murray B.A., Mathematics, Washington University in St. Louis, 1997 M.S., Louisiana State University, 1999 August 2002 Acknowledgments First and foremost, I would like to thank my advisor Dr. Paul van Wamelen for both his guidance and lack thereof. Throughout the development of this dissertation, I was involved in all decisions and was always offered support and advice. Rarely, however, was I given answers; for this, I am truly grateful. I would also like to thank Dr. van Wamelen for the informal nature of our relationship{although the last three years were challenging, they were certainly enjoyable.
    [Show full text]
  • Basic Concepts of Representation Theory Amritanshu Prasad
    Basic concepts of representation theory Amritanshu Prasad To cite this version: Amritanshu Prasad. Basic concepts of representation theory. 3rd cycle. Shillong - Inde, 2013, pp.22. cel-00963677 HAL Id: cel-00963677 https://cel.archives-ouvertes.fr/cel-00963677 Submitted on 21 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. BASIC CONCEPTS OF REPRESENTATION THEORY AMRITANSHU PRASAD 1. Representations and Modules Let K be a field, and G be a finite group. For a K-vector space V , let GL(V ) denote the group of all invertible K-linear maps V → V . Definition 1.1 (Representation). A representation of G is a pair (ρ, V ), where V is a K-vector space and ρ : G → GL(V ) is a homomorphism of groups. Definition 1.2 (Multiplicative character). A multiplicative character of G is a homomorphism χ : G → K∗. Multiplicative char- Each multiplicative character χ gives rise to a representation as follows: take V acters give rise to to the one dimensional vector space K, and take ρ to be the homomorphism which one dimensional rep- takes g ∈ G to the linear automorphism of K which multiplies each element by resentations.
    [Show full text]
  • Introduction to Characters
    Introduction to Characters Travis Scholl April 23, 2015 Abstract *These notes were taken from a course by Ralph Greenburg in Spring 2015 [Gre15] on counting points on varieties over finite fields. It also overlaps with [Ser12, Ch. VI] and [Was97]. Any mistakes should be due to me. 1 Characters This section will define the basic objects required for studying characters. Much of this theory can be generalized but for simplicity we will stick to the \hands on" approach. Definition 1.1. Let A be a finite abelian group. The dual of A to be the group A “ HompA; C˚q, that is the group of homomorphisms A Ñ C˚. This is also called the Pontryagin dual. Elements χ P A are examples of characters. In this group the trivial character χ0 is the mapp sending A to 1. Exercisep 1.2. Prove that (a) Let χ P A. Show the image χpAq is contained in the set of roots of unity: n ` p χpAq Ď tz P C | z “ 1 for some n P Z u: (b) For any fixed A, there exists an isomorphism A – A. (c) There is a canonical isomorphism A – A, i.e. thep \double dual" functor is naturally isomorphic to the identity functor on the category of finite abelian groups. x Hint. Use the structure theorem for finite abelian groups. If A is a product of cyclic groups Z{nZ, then maps out of A are uniquely determined by where a generator in each component go. Notice that there is a unique cyclic subgroup of order n in C˚.
    [Show full text]