DOCSLIB.ORG

Research Statement

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Research Statement Research Statement Jay Taylor Groups form the abstract framework for studying the natural notion of symmetry and are ubiquitous in mathematics. The building blocks for constructing all groups are the simple groups. Often one hopes to reduce a question about groups to statements involving only groups that are close to being simple. A key example of this is given by Isaacs–Malle–Navarro’s reduction of the McKay Conjecture [IMN07]. Such reductions make it essential to have a detailed understanding of simple groups. The celebrated Classification of Finite Simple Groups states that the vast majority of finite simple groups arise as central quotients of finite reductive groups. Each finite reductive group G(q) is defined with respect to a finite field Fq of q elements. The principal examples are the matrix groups GLn(q), SLn(q), and Sp2n(q), whose matrices have entries in the field Fq. It is known that the quotients PSLn(q) and PSp2n(q) of SLn(q) and Sp2n(q) by their centers are generically simple. My research concerns the character theory of finite reductive groups, which is a fundamental tool for studying finite groups. A character of a finite group G is a function G ! C defined by taking the trace of a matrix obtained from a homomorphism G ! GLn(C). The characters of a group encode an impressive amount of information and, somewhat surprisingly, are often computable even without knowing the underlying homomorphisms G ! GLn(C) from which they are defined. For these reasons, characters have become an indispensable tool for studying finite groups. The basic object of study in character theory is the set of irreducible characters Irr(G). The overarching theme of my research concerns the values of irreducible characters of finite reductive groups at unipotent elements. The study of all character values can largely be reduced to the case of values at unipotent elements, which makes this an important case to consider. Moreover, it is this case where the connections to geometry are most rich, such as through the Springer correspondence and studying singularities in the closures of unipotent conjugacy classes. An important viewpoint on character values at unipotent elements comes from Kawanaka’s gen- eralized Gelfand–Graev characters (GGGCs), which were introduced in [Kaw85; Kaw86]. These are a family of characters of G that are defined whenever the characteristic p of the field Fq is good (p > 5 is sufficient). In the last few years these characters have seen remarkable applications to the following fundamental questions: • determining the action of automorphisms of G on the set Irr(G) of irreducible characters, by work of Cabanes–Späth [CS17] and myself [Tay18a], • bounding the values of irreducible characters, by work of Bezrukavnikov–Liebeck–Shalev–Tiep [BLST17] and my recent joint work with Tiep [TT18], • and determining the `-modular decomposition matrices where ` 6= p is a prime, by work of Dudas–Malle [DM16; DM18] and my current ongoing joint work with Brunat and Dudas [BDT], This last application encodes how homomorphisms G ! GLn(C) defined over C are related to homo- morphisms G ! GLn(F`) defined over an algebraically closed field of characteristic ` > 0. All of the above applications rely on our understanding of the decomposition of a GGGC into its irreducible constituents. When p is a good prime, I established the best general statements regarding such a decomposition in [Tay16]. This generalizes previous work of Lusztig [Lus92] whose results were page 1 Jay Taylor Research Statement obtained under the assumption that the characteristic p > 0 is sufficiently large. Such assumptions on the characteristic are incredibly prohibitive when invoking the Classification of Finite Simple Groups. In general, it is difficult to say whether an irreducible character will occur as a constituent of a GGGC. Answering this question is an ongoing focus of my research. I have developed a strategy for solving this problem which involves: • relating the parameterization of the irreducible characters Irr(G) via Harish-Chandra theory to the labelling given by Lusztig [Lus84], • computing the decomposition of the Deligne–Lusztig induction map from a Levi subgroup, • evaluating the characteristic function of a character sheaf at unipotent elements, • determining the decomposition of an irreducible character as a linear combination of character- istic functions of character sheaves. In the following sections I will provide some background on finite reductive groups and elaborate on the above projects and future work in more detail. Background A finite reductive group is defined to be the fixed points GF of a connected reductive algebraic group, defined over an algebraic closure F = Fq of the finite field Fq, under a Frobenius endomorphism F : G ! G corresponding to the finite field Fq. The main example is the case where G = GLn(F) q F and F (xij ) = (xij ) for all (xij ) 2 G. We then have G is the finite general linear group GLn(q). By realizing the finite group GF as a subgroup of the algebraic group G we may use the geometry of the group G when studying GF . The Frobenius endomorphism F is the key to this relationship. A particularly convenient setting for studying the irreducible characters Irr(GF ) is to view them as F a basis for the C-vector space Class(G ) of all class functions, i.e., functions that are constant on conjugacy classes. In his seminal work [Lus85] Lusztig has introduced his geometric theory of character sheaves, which provides the theoretical framework for studying the values of irreducible characters. Each such character sheaf is a G-equivariant simple perverse sheaf on the algebraic group G. From the theory of character sheaves one obtains a new basis C(GF ) ⊆ Class(GF ) for the space of class functions. It has been conjectured by Lusztig that this basis C(GF ) is very close to the basis of F irreducible characters. For instance, if G = GLn(q) then these functions are precisely the irreducible characters. Lusztig’s conjecture has a positive solution in many cases but remains open in general. Even when the conjecture is known to hold the entries of the change of basis matrix relating Irr(GF ) and C(GF ) are not known explicitly. Evaluating Characteristic Functions of Character Sheaves Current and Previous Work For each irreducible character χ 2 Irr(GF ) and characteristic function X 2 C(GF ) there exists a scalar c(χ, X ) 2 C such that X χ = c(χ, X )X : X 2C(GF ) page 2 Jay Taylor Research Statement Computing the value of an irreducible character χ 2 Irr(GF ) at an element g 2 GF can be viewed as the following two step process: 1) for any function X 2 C(GF ) compute the value X (g), 2) for any X 2 C(GF ) with X (g) 6= 0 compute the scalar c(χ, X ). That the values of X 2 C(G) can be computed at all comes from the fact that X is obtained from a geometric object, namely a perverse sheaf. In this section I will describe some results concerning step 1 of this process when g 2 GF is a unipotent element. If O ⊆ G is an F -stable unipotent conjugacy class of G then OF = O\ GF is a union of GF -conjugacy classes. To determine X (g) explicitly involves choosing one of the GF - conjugacy classes contained in OF that satisfies desirable properties. In many, but not all, cases Shoji has made a choice of such a class whose elements are called split unipotent elements [Sho87]. Even though a theory of split unipotent elements does not exist in general, I was able to establish the following. Theorem 1 (Taylor, [Tay14]). Assume a theory of split unipotent elements exists for GF then for each function X 2 C(GF ) and unipotent element u 2 GF the value X (u) is explicitly computable via a recursive algorithm. Moreover, a theory of split unipotent elements exists if p is a good prime for G and Z(G) is connected. Future Work If p is a good prime for G then a theory of split unipotent elements exists for most groups, even without the assumption that Z(G) is connected. The most prominent case where this theory is missing is when G = Spinn(F) is a spin group. For such a group there exists a surjective homomorphism Spinn(F) ! SOn(F) onto a special orthogonal group with kernel of size 2. In other words, Spinn(F) is a double cover of SOn(F). In [Sho07] Shoji explicitly described split unipotent elements for the special orthogonal group SOn(F). The surjective homomorphism Spinn(F) ! SOn(F) restricts to a bijection between the unipotent elements of Spinn(F) and those of SOn(F). By using this natural relationship and the tools developed by Shoji I aim to tackle the following problem. Problem 2. Using the existence of split unipotent elements in special orthogonal groups develop a theory of split unipotent elements for spin groups. This problem constitutes the last major stumbling block in developing a theory of split unipotent elements when p is a good prime for G. As mentioned in the introduction, studying character values at arbitrary elements can often be reduced to the case of unipotent elements. Specifically one studies the ◦ F F values of the irreducible characters Irr(CG(s) ) at unipotent elements, where s 2 G is a semisimple ◦ element and CG(s) is its connected centralizer. When considering exceptional groups such as E8 one encounters low dimensional spin groups when ◦ studying the connected centralizers CG(s) of semisimple elements. Hence, even partial results for Problem 2 would have applications for determining character tables of exceptional groups at non- unipotent elements.
Load more

Recommended publications
  • Symmetries on the Lattice
    Symmetries On The Lattice K.Demmouche January 8, 2006 Contents Background, character theory of finite groups The cubic group on the lattice Oh Representation of Oh on Wilson loops Double group 2O and spinor Construction of operator on the lattice MOTIVATION Spectrum of non-Abelian lattice gauge theories ? Create gauge invariant spin j states on the lattice Irreducible operators Monte Carlo calculations Extract masses from time slice correlations Character theory of point groups Groups, Axioms A set G = {a, b, c, . } A1 : Multiplication ◦ : G × G → G. A2 : Associativity a, b, c ∈ G ,(a ◦ b) ◦ c = a ◦ (b ◦ c). A3 : Identity e ∈ G , a ◦ e = e ◦ a = a for all a ∈ G. −1 −1 −1 A4 : Inverse, a ∈ G there exists a ∈ G , a ◦ a = a ◦ a = e. Groups with finite number of elements → the order of the group G : nG. The point group C3v The point group C3v (Symmetry group of molecule NH3) c ¡¡AA ¡ A ¡ A Z ¡ A Z¡ A Z O ¡ Z A ¡ Z A ¡ Z A Z ¡ Z A ¡ ZA a¡ ZA b G = {Ra(π),Rb(π),Rc(π),E(2π),R~n(2π/3),R~n(−2π/3)} noted G = {A, B, C, E, D, F } respectively. Structure of Groups Subgroups: Definition A subset H of a group G that is itself a group with the same multiplication operation as G is called a subgroup of G. Example: a subgroup of C3v is the subset E, D, F Classes: Definition An element g0 of a group G is said to be ”conjugate” to another element g of G if there exists an element h of G such that g0 = hgh−1 Example: on can check that B = DCD−1 Conjugacy Class Definition A class of a group G is a set of mutually conjugate elements of G.
    [Show full text]
  • Lie Algebras of Generalized Quaternion Groups 1 Introduction 2
    Lie Algebras of Generalized Quaternion Groups Samantha Clapp Advisor: Dr. Brandon Samples Abstract Every finite group has an associated Lie algebra. Its Lie algebra can be viewed as a subspace of the group algebra with certain bracket conditions imposed on the elements. If one calculates the character table for a finite group, the structure of its associated Lie algebra can be described. In this work, we consider the family of generalized quaternion groups and describe its associated Lie algebra structure completely. 1 Introduction The Lie algebra of a group is a useful tool because it is a vector space where linear algebra is available. It is interesting to consider the Lie algebra structure associated to a specific group or family of groups. A Lie algebra is simple if its dimension is at least two and it only has f0g and itself as ideals. Some examples of simple algebras are the classical Lie algebras: sl(n), sp(n) and o(n) as well as the five exceptional finite dimensional simple Lie algebras. A direct sum of simple lie algebras is called a semi-simple Lie algebra. Therefore, it is also interesting to consider if the Lie algebra structure associated with a particular group is simple or semi-simple. In fact, the Lie algebra structure of a finite group is well known and given by a theorem of Cohen and Taylor [1]. In this theorem, they specifically describe the Lie algebra structure using character theory. That is, the associated Lie algebra structure of a finite group can be described if one calculates the character table for the finite group.
    [Show full text]
  • GROUP REPRESENTATIONS and CHARACTER THEORY Contents 1
    GROUP REPRESENTATIONS AND CHARACTER THEORY DAVID KANG Abstract. In this paper, we provide an introduction to the representation theory of finite groups. We begin by defining representations, G-linear maps, and other essential concepts before moving quickly towards initial results on irreducibility and Schur's Lemma. We then consider characters, class func- tions, and show that the character of a representation uniquely determines it up to isomorphism. Orthogonality relations are introduced shortly afterwards. Finally, we construct the character tables for a few familiar groups. Contents 1. Introduction 1 2. Preliminaries 1 3. Group Representations 2 4. Maschke's Theorem and Complete Reducibility 4 5. Schur's Lemma and Decomposition 5 6. Character Theory 7 7. Character Tables for S4 and Z3 12 Acknowledgments 13 References 14 1. Introduction The primary motivation for the study of group representations is to simplify the study of groups. Representation theory offers a powerful approach to the study of groups because it reduces many group theoretic problems to basic linear algebra calculations. To this end, we assume that the reader is already quite familiar with linear algebra and has had some exposure to group theory. With this said, we begin with a preliminary section on group theory. 2. Preliminaries Definition 2.1. A group is a set G with a binary operation satisfying (1) 8 g; h; i 2 G; (gh)i = g(hi)(associativity) (2) 9 1 2 G such that 1g = g1 = g; 8g 2 G (identity) (3) 8 g 2 G; 9 g−1 such that gg−1 = g−1g = 1 (inverses) Definition 2.2.
    [Show full text]
  • Representation Theory
    M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23.
    [Show full text]
  • Lecture Notes: Basic Group and Representation Theory
    Lecture notes: Basic group and representation theory Thomas Willwacher February 27, 2014 2 Contents 1 Introduction 5 1.1 Definitions . .6 1.2 Actions and the orbit-stabilizer Theorem . .8 1.3 Generators and relations . .9 1.4 Representations . 10 1.5 Basic properties of representations, irreducibility and complete reducibility . 11 1.6 Schur’s Lemmata . 12 2 Finite groups and finite dimensional representations 15 2.1 Character theory . 15 2.2 Algebras . 17 2.3 Existence and classification of irreducible representations . 18 2.4 How to determine the character table – Burnside’s algorithm . 20 2.5 Real and complex representations . 21 2.6 Induction, restriction and characters . 23 2.7 Exercises . 25 3 Representation theory of the symmetric groups 27 3.1 Notations . 27 3.2 Conjugacy classes . 28 3.3 Irreducible representations . 29 3.4 The Frobenius character formula . 31 3.5 The hook lengths formula . 34 3.6 Induction and restriction . 34 3.7 Schur-Weyl duality . 35 4 Lie groups, Lie algebras and their representations 37 4.1 Overview . 37 4.2 General definitions and facts about Lie algebras . 39 4.3 The theorems of Lie and Engel . 40 4.4 The Killing form and Cartan’s criteria . 41 4.5 Classification of complex simple Lie algebras . 43 4.6 Classification of real simple Lie algebras . 44 4.7 Generalities on representations of Lie algebras . 44 4.8 Representation theory of sl(2; C) ................................ 45 4.9 General structure theory of semi-simple Lie algebras . 48 4.10 Representation theory of complex semi-simple Lie algebras .
    [Show full text]
  • Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials
    1/29 Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials Don Taylor The University of Sydney Nelson, 7–13 January 2018 2/29 Origins Suppose that G is a finite group and for every element g G we have 2 an indeterminate xg . What are the factors of the group determinant ¡ ¢ det x 1 ? gh¡ g,h This was a question posed by Dedekind. Frobenius discovered character theory (in 1896) when he set out to answer it. Characters of abelian groups had been used in number theory but Frobenius developed the theory for nonabelian groups. 3/29 Representations Matrices Characters ² ² A linear representation of a group G is a homomorphism ½ : G GL(V ), where GL(V ) is the group of all invertible linear ! transformations of the vector space V , which we assume to have finite dimension over the field C of complex numbers. The dimension n of V is called the degree of ½. If (ei )1 i n is a basis of V , there are functions ai j : G C such that · · ! X ½(x)e j ai j (x)ei . Æ i The matrices A(x) ¡a (x)¢ define a homomorphism A : G GL(n,C) Æ i j ! from G to the group of all invertible n n matrices over C. £ The character of ½ is the function G C that maps x G to Tr(½(x)), ! 2 the trace of ½(x), namely the sum of the diagonal elements of A(x). 4/29 Early applications Frobenius first defined characters as solutions to certain equations.
    [Show full text]
  • Representation Theory of Symmetric Groups
    MATH 711: REPRESENTATION THEORY OF SYMMETRIC GROUPS LECTURES BY PROF. ANDREW SNOWDEN; NOTES BY ALEKSANDER HORAWA These are notes from Math 711: Representation Theory of Symmetric Groups taught by Profes- sor Andrew Snowden in Fall 2017, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from December 18, 2017. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. The first part of the course will be devoted to the representation theory of symmetric groups and the main reference for this part of the course is [Jam78]. Another standard reference is [FH91], although it focuses on the characteristic zero theory. There is not general reference for the later part of the course, but specific citations have been provided where possible. Contents 1. Preliminaries2 2. Representations of symmetric groups9 3. Young's Rule, Littlewood{Richardson Rule, Pieri Rule 28 4. Representations of GLn(C) 42 5. Representation theory in positive characteristic 54 6. Deligne interpolation categories and recent work 75 References 105 Introduction. Why study the representation theory of symmetric groups? (1) We have a natural isomorphism V ⊗ W ∼ W ⊗ V v ⊗ w w ⊗ v We then get a representation of S2 on V ⊗V for any vector space V given by τ(v⊗w) = ⊗n w ⊗ v. More generally, we get a representation of Sn on V = V ⊗ · · · ⊗ V . If V is | {z } n times Date: December 18, 2017.
    [Show full text]
  • Linear Algebraic Groups
    Appendix A Linear Algebraic Groups A.1 Linear Algebraic Groups Let us fix an algebraically closed field K. We work in the category of affine varieties over K. The purpose of this section is to give a brief exposition on the basic facts of algebraic groups. Definition A.1.1 A linear algebraic group is an affine algebraic variety G together with a unit element e 2 G and morphisms m W G G ! G and i W G ! G satisfying the group axioms (a) m.; e/ D m.e;/D for all 2 G; (b) m.; i.// D m.i./; / D e for all 2 G; (c) m.; m.; // D m.m.; /; / (associativity) for all ; ; 2 G. Often we will just write and 1 instead of m.; / and i./, respectively. Example A.1.2 The group GLn, the set of invertible nn matrices over K, is a linear algebraic group. Any Zariski closed subgroup G GLn is a linear algebraic group. In fact, every linear algebraic group is isomorphic to a Zariski closed subgroup of GLn (see Borel [1, Proposition I.1.10]). This justifies calling it a linear algebraic group. G The axioms for a linear algebraic group can also be stated in terms of the coordinate ring. The maps m W G G ! G and i W G ! G correspond to ring homomorphisms m W KŒG ! KŒG ˝ KŒG and i W KŒG ! KŒG. The unit element e 2 G corresponds to a ring homomorphism W KŒG ! K defined by f 7! f .e/. The axioms for a linear algebraic group translate to (a) .id ˝/ ı m D .
    [Show full text]
  • Representation Theory of Finite Groups Anupam Singh
    Representation Theory of Finite Groups Anupam Singh IISER, central tower, Sai Trinity building, Pashan circle, Pune 411021 INDIA E-mail address: [email protected] arXiv:1001.0462v1 [math.RT] 4 Jan 2010 Contents Chapter 1. Introduction 5 Chapter 2. Representation of a Group 7 2.1. Commutator Subgroup and One dimensional representations 10 Chapter 3. Maschke’s Theorem 11 Chapter 4. Schur’s Lemma 15 Chapter 5. Representation Theory of Finite Abelian Groups over C 17 5.1. Example of representation over Q 19 Chapter 6. The Group Algebra k[G] 21 Chapter 7. Constructing New Representations 23 7.1. SubrepresentationandSumofRepresentations 23 7.2. Adjoint Representation 23 7.3. Tensor Product of two Representations 24 7.4. Restriction of a Representation 25 Chapter 8. Matrix Elements 27 Chapter 9. Character Theory 29 Chapter 10. Orthogonality Relations 31 Chapter 11. Main Theorem of Character Theory 33 11.1. Regular Representation 33 11.2. TheNumberofIrreducibleRepresentations 33 Chapter 12. Examples 37 12.1. Groups having Large Abelian Subgroups 38 12.2. Character Table of Some groups 39 Chapter 13. Characters of Index 2 Subgroups 43 13.1. The Representation V V 43 ⊗ 13.2. Character Table of S5 44 3 4 CONTENTS 13.3. Index two subgroups 46 13.4. Character Table of A5 48 Chapter 14. Characters and Algebraic Integers 51 Chapter 15. Burnside’s pq Theorem 53 Chapter 16. Further Reading 55 16.1. RepresentationTheoryofSymmetricGroup 55 16.2. Representation Theory of GL2(Fq) and SL2(Fq) 55 16.3. Wedderburn Structure Theorem 55 16.4. Modular Representation Theory 55 Bibliography 57 CHAPTER 1 Introduction This is class notes for the course on representation theory of finite groups taught by the author at IISER Pune to undergraduate students.
    [Show full text]
  • Research Statement
    Research Statement Anders O.F. Hendrickson December 6, 2010 1 Supercharacter Theories 1.1 Background Since the time of Burnside and Frobenius, one of the most important ways to study finite groups has been via their representations. The representation theory of finite groups over C and over fields of positive characteristic, especially the work of Brauer, played a large role in the classification of finite simple groups, and today it has strong connections with the study of Lie algebras, algebraic topology, homotopy theory, and group actions on finite complexes. Let G be a finite group; then a complex representation of G is a homomorphism from G into a group of complex matrices. It turns out that almost everything to be gained from the representation theory of G over C is captured by the traces of these homomorphisms, which are called the characters of G. Sums of characters are again characters, but those characters that are not sums we call irreducible. The set Irr(G) of irreducible characters in fact is finite and forms an orthonormal basis for the space of class functions of G. Since characters are constant on conjugacy classes, the character theory of G is completely described by its character table, a square matrix whose rows are labeled by Irr(G) and columns by the conjugacy classes. Although it captures a great deal of information about the group, unfortunately to compute the character table explicitly is quite difficult in many cases. It would be very desirable to find a simpler object, easier to compute than the character table, which nevertheless captures much of the same information.
    [Show full text]
  • Computational Modular Character Theory
    Computational Modular Character Theory G. HISS C. JANSEN K. LUX R. PARKER arXiv:1901.08453v1 [math.RT] 24 Jan 2019 November 11, 1993 To Joachim Neub¨user and Herbert Pahlings Acknowledgements During the development of the MOC-system and the preparation of this book we were supported by various mathematicians and institutions. Above all, we wish to thank Joachim Neub¨user and Herbert Pahlings who brought us together at Aachen. Pahlings was the supervisor of the PhD-thesis of Lux and of the Diploma thesis of Jansen, which were con- cerned with extensions and applications of the MOC-system. Neub¨user and Pahlings provided a great working atmosphere at their institute, the Lehrstuhl D f¨ur Mathematik, and supported our project during the whole time. We therefore dedicate this book to them. It is a great pleasure for us to thank H. W. Lenstra for his imme- diate interest in a particular number theoretical problem related to the MOC-system, for solving this problem and for allowing us to include his proof of this previously unpublished result in our book. Meinolf Geck read a preliminary version of the manuscript with great care. His various remarks and observations concerning the theory of modular characters have been incorporated. Together with his detailed comments and sug- gestions on the style of the exposition, this lead to many improvements and added very much to the legibility of our book. We wish to thank him for all his work. Finally we thank Thomas Breuer for suggestions concerning abelian number fieds. We also gratefully acknowledge the financial support we received from the Deutsche Forschungsgemeinschaft and from the Bundesland Nordrhein-Westfalen in form of the Bennigsen-Foerder-Preis.
    [Show full text]
  • Algorithms for the Character Theory of the Symmetric Group
    Algorithms for the Character Theory of the Symmetric Group Omer Egecio~lu Department of Computer Science University of California Santa Barbara, CA 93106 O. Introduction The representation theory of the symmetric groups Su - aside from being extremely elegant and interesting in its own right - can be used in a number of ways to obtain informa- tion about the representation theory of other classes of groups. This theory also turns out to have applications in such diverse areas of interest as atomic physics and quantum chemistry to graph isomorphism and PI algebras. Prior to the advent of the electronic computing devices, the computational aspect of the character theory of S n had to stay essentially at the level of examples. This is because of the formidable computational difficulties that arise in the resolution of various products defined on the irreducible characters of Sn • Following the introduction of the first electronic computing machines, there emerged a number of approaches and significant computational results in this area, mainly focused on the construction of the character tables of S n • Theoretical results of Frobenius [Frobl-Frob2], Murnaghan [Murn2], Nakayama [Naka], Yamanouchi [Yama] enabled Bivins et al [BiMSW] to calculate the characters of symmetric groups of degrees up to 16 in 1950's. This was followed by the results of Comet [Cornel- Come3}, Gabriel [Gabri] among others which extended and improved these considerably. In recent years, the classification theory of finite simple groups spawned comprehensive packages such as the Character Algorithm System (CAS) [CAS1-CAS3] for handling characters of arbitrary finite groups. The development of the special case of the operations on the characters of the symmetric groups was initiated by Young's fundamental series of monographs [Young] and the isomor- phism between the multiplicative properties of group characters and the algebra of Schur functions introduced by Littlewood [Littl-Litt4].
    [Show full text]