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REPRESENTATION THEORY WEEK 9 1. Jordan-Hölder Theorem And
REPRESENTATION THEORY WEEK 9 1. Jordan-Holder¨ theorem and indecomposable modules Let M be a module satisfying ascending and descending chain conditions (ACC and DCC). In other words every increasing sequence submodules M1 ⊂ M2 ⊂ ... and any decreasing sequence M1 ⊃ M2 ⊃ ... are finite. Then it is easy to see that there exists a finite sequence M = M0 ⊃ M1 ⊃···⊃ Mk = 0 such that Mi/Mi+1 is a simple module. Such a sequence is called a Jordan-H¨older series. We say that two Jordan H¨older series M = M0 ⊃ M1 ⊃···⊃ Mk = 0, M = N0 ⊃ N1 ⊃···⊃ Nl = 0 ∼ are equivalent if k = l and for some permutation s Mi/Mi+1 = Ns(i)/Ns(i)+1. Theorem 1.1. Any two Jordan-H¨older series are equivalent. Proof. We will prove that if the statement is true for any submodule of M then it is true for M. (If M is simple, the statement is trivial.) If M1 = N1, then the ∼ statement is obvious. Otherwise, M1 + N1 = M, hence M/M1 = N1/ (M1 ∩ N1) and ∼ M/N1 = M1/ (M1 ∩ N1). Consider the series M = M0 ⊃ M1 ⊃ M1∩N1 ⊃ K1 ⊃···⊃ Ks = 0, M = N0 ⊃ N1 ⊃ N1∩M1 ⊃ K1 ⊃···⊃ Ks = 0. They are obviously equivalent, and by induction assumption the first series is equiv- alent to M = M0 ⊃ M1 ⊃ ··· ⊃ Mk = 0, and the second one is equivalent to M = N0 ⊃ N1 ⊃···⊃ Nl = {0}. Hence they are equivalent. Thus, we can define a length l (M) of a module M satisfying ACC and DCC, and if M is a proper submodule of N, then l (M) <l (N). -
Topics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order. -
Block Theory with Modules
JOURNAL OF ALGEBRA 65, 225--233 (1980) Block Theory with Modules J. L. ALPERIN* AND DAVID W. BURRY* Department of Mathematics, University of Chicago, Chicago, Illinois 60637 and Department of Mathematics, Yale University, New Haven, Connecticut 06520 Communicated by Walter Felt Received October 2, 1979 1. INTRODUCTION There are three very different approaches to block theory: the "functional" method which deals with characters, Brauer characters, Carton invariants and the like; the ring-theoretic approach with heavy emphasis on idempotents; module-theoretic methods based on relatively projective modules and the Green correspondence. Each approach contributes greatly and no one of them is sufficient for all the results. Recently, a number of fundamental concepts and results have been formulated in module-theoretic terms. These include Nagao's theorem [3] which implies the Second Main Theorem, Green's description of the defect group in terms of vertices [4, 6] and a module description of the Brauer correspondence [1]. Our purpose here is to show how much of block theory can be done by statring with module-theoretic concepts; our contribution is the methods and not any new results, though we do hope these ideas can be applied elsewhere. We begin by defining defect groups in terms of vertices and so use Green's theorem to motivate the definition. This enables us to show quickly the fact that defect groups are Sylow intersections [4, 6] and can be characterized in terms of the vertices of the indecomposable modules in the block. We then define the Brauer correspondence in terms of modules and then prove part of the First Main Theorem, Nagao's theorem in full and results on blocks and normal subgroups. -
Characterization of Δ-Small Submodule
Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.7, 2015 Characterization of δ-small submodule R. S. Wadbude, M.R.Aloney & Shubhanka Tiwari 1.Mahatma Fule Arts, Commerce and Sitaramji Chaudhari Science Mahavidyalaya, Warud. SGB Amaravati University Amravati [M.S.] 2.Technocrats Institute of Technology (excellence) Barkttulla University Bhopal. [M.P.] Abstract Pseudo Projectivity and M- Pseudo Projectivity is a generalization of Projevtevity. [2], [8] studied M-Pseudo Projective module and small M-Pseudo Projective module. In this paper we consider some generalization of small M-Pseudo Projective module, that is δ-small M-Pseudo Projective module with the help of δ-small and δ- cover. Key words: Singular module, S.F. small M-Pseudo Projective module, δ-small and projective δ- cover. Introduction: Throughout this paper R is an associative ring with unity module and all modules are unitary left R- modules. A sub module K of a module M. K ≤ M. Let M be a module, K ≤ M is said to be small in M if for every L ≤ M, the equality K + L = M implies L = M, ( denoted by 퐾 ≪ 푀 ). The concept of δ-small sub modules was introduced by Zhon [10]. A sub module K of M is said to be δ-small sub module of M (denoted by 퐾 ≪훿 푀 ) if whenever M = K+ L, with M/K is singular, then M = L. The sum of all δ-small sub module of M is denoted by δ(M). δ(M) is the reject in M of class of all singular simple modules. -
Economic Indicators and Social Networks: New Approaches to Measuring Poverty, Prices, and Impacts of Technology
Economic Indicators and Social Networks: New approaches to measuring poverty, prices, and impacts of technology by Niall Carrigan Keleher A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Information Management and Systems in the Graduate Division of the University of California, Berkeley Committee in charge: Dr. Joshua Evan Blumenstock, Chair Dr. John Chuang Dr. Jeremy Magruder Fall 2019 1 Abstract Economic Indicators and Social Networks: New approaches to measuring poverty, prices, and impacts of technology by Niall Carrigan Keleher Doctor of Philosophy in Information Management and Systems University of California, Berkeley Dr. Joshua Evan Blumenstock, Chair Collecting data to inform policy decisions is an ongoing global challenge. While some data collection has become routine, certain populations remain dicult to reach. From targeting social protection programs in densely-populated urban areas to reaching the “last mile” of infrastructure coverage, data collection and service delivery go hand-in-hand. Understanding the populations that live in urban communities as well as remote villages can help to tailor the design, targeting, and implementation of development programs. New sources of information have the potential to improve awareness of the needs and preferences of individuals, households, and communities. The goal of this dissertation is to provide multiple vantage points on the role that data, commu- nity input, and individual preferences can play in informing development policy. The empirical investigation presented in this dissertation covers two studies in Liberia and one in the Philippines. The unifying theme of the three chapters is the exploration of new sources of information about hard-to-reach populations. -
Algebraic Modules for Finite Groups
Algebraic Modules for Finite Groups David A. Craven January 2007 For the purposes of this talk, G is a finite group, K is an algebraically closed field of characteristic p, where p | |G|, and all modules are finite-dimensional. 1 Algebraic Modules The Green ring of KG-modules is defined to be the free abelian group on the basis set of all indecomposable KG-modules, with M + N defined to be equal to M ⊕ N, and the product of two modules defined as M ⊗ N. Notice that not all elements of the Green ring can be thought of as modules, since they could have negative multiplicities attached; they are virtual modules. The structure of the Green ring, while a commutative ring with a 1, is far from that of traditional commutative rings. For example, it is not an integral domain: in general, it has nilpotent elements. It is also in general infinite-dimensional. We can still, however, carry over some notions from algebraic number theory. One of those is algebraic modules. A module is said to be algebraic if it satisfies some polynomial equation in the Green ring, with co-efficients in Z. Proposition 1.1 Let M be a KG-module. Then the following are equivalent: (i) M is algebraic; (ii) M satisfies a monic polynomial equation in the Green ring with co-efficients in Z; and (iii) there are only finitely many different indecomposable summands of the (infinite-dimensional) module T (M) = M ⊕ M ⊗2 ⊕ M ⊗3 ⊕ · · · . The third equivalent condition is often the easiest to use in actually deciding if modules are algebraic or not. -
Regular and Semisimple Modules
Pacific Journal of Mathematics REGULAR AND SEMISIMPLE MODULES T. J. CHEATHAM AND J. R. SMITH Vol. 65, No. 2 October 1976 PACIFIC JOURNAL OF MATHEMATICS Vol. 65, No. 2, 1976 REGULAR AND SEMISIMPLE MODULES T. J. GHEATHAM AND J. R. SMITH A module is regular if all its submodules are (Cohn) pure. The family of all regular modules is closed under products if and only if RjJ{R) is a von Neumann regular ring. If each regular ϋJ-module is semisimple then R is a Γ-ring. An extra condition is needed for the converse* Character modules and extensions of regular and semisimple modules are investigated. 1* Introduction* Rings will be associative with identity and modules will be (left) unitary. R will denote a ring which is not assumed commutative unless specifically stated and J{R) will denote the Jacobson radical of R. Fieldhouse [5] calls a module B regular if each submodule A of B is pure in B, i.e., the inclusion 0—• A—>B remains exact upon tensor ing by any (right) iϋ-module. Regular modules have been studied under different definitions by Ware [12], Zelmanowitz [14], and Ramamurthi and Rangaswamy [9]. A module is semisimple if it is a sum of simple modules. For a subset A of a module J5, (0: A) will denote the left ideal {r eR\rx — 0 for all x e A}. 2* Products. The class of all semisimple modules is closed under products if and only if R/J(R) is a semisimple (Artinian) ring. This follows from the canonical embedding R/J(R) <=-+ ΠR/M, where the product is taken over the set of maximal left ideals M of R. -
INJECTIVE MODULES OVER a PRTNCIPAL LEFT and RIGHT IDEAL DOMAIN, \Ryith APPLICATIONS
INJECTIVE MODULES OVER A PRTNCIPAL LEFT AND RIGHT IDEAL DOMAIN, \ryITH APPLICATIONS By Alina N. Duca SUBMITTED IN PARTTAL FULFILLMENT OF THE REQUIREMENTS FOR TI{E DEGREE OF DOCTOR OFPHILOSOPHY AT UMVERSITY OF MAMTOBA WINNIPEG, MANITOBA April 3,2007 @ Copyright by Alina N. Duca, 2007 THE T.TNIVERSITY OF MANITOBA FACULTY OF GRADUATE STI]DIES ***** COPYRIGHT PERMISSION INJECTIVE MODULES OVER A PRINCIPAL LEFT AND RIGHT IDEAL DOMAIN, WITH APPLICATIONS BY Alina N. Duca A Thesis/Practicum submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfïllment of the requirement of the degree DOCTOR OF PHILOSOPHY Alina N. Duca @ 2007 Permission has been granted to the Library of the University of Manitoba to lend or sell copies of this thesidpracticum, to the National Library of Canada to microfilm this thesis and to lend or sell copies of the film, and to University MicrofïIms Inc. to pubtish an abstract of this thesiVpracticum. This reproduction or copy of this thesis has been made available by authority of the copyright owner solely for the purpose of private study and research, and may only be reproduced and copied as permitted by copyright laws or with express written authorization from the copyright owner. UNIVERSITY OF MANITOBA DEPARTMENT OF MATIIEMATICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled "Injective Modules over a Principal Left and Right ldeal Domain, with Applications" by Alina N. Duca in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dated: April3.2007 External Examiner: K.R.Goodearl Research Supervisor: T.G.Kucera Examining Committee: G.Krause Examining Committee: W.Kocay T]NTVERSITY OF MANITOBA Date: April3,2007 Author: Alina N. -
The (Cyclic) Enhanced Nilpotent Cone Via Quiver Representations
The (cyclic) enhanced nilpotent cone via quiver representations Gwyn Bellamy 1 and Magdalena Boos 2 Abstract The GL(V )-orbits in the enhanced nilpotent cone V × N (V ) are (essentially) in bi- jection with the orbits of a certain parabolic P ⊆ GL(V ) (the mirabolic subgroup) in the nilpotent cone N (V ). We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameteriza- tion generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit transla- tions between the different parametrizations are given. Contents 1 Introduction 2 1.1 A representation theoretic parameterization . .2 1.2 Parabolic conjugacy classes in the nilpotent cone . .4 1.3 Admissible D-modules . .4 1.4 Outline of the article . .5 1.5 Outlook . .5 2 Theoretical background 5 2.1 Quiver representations . .5 2.2 Representation types . .6 2.3 Group actions . .7 3 Combinatorial objects 8 3.1 (Frobenius) Partitions and Young diagrams . .8 arXiv:1609.04525v4 [math.RT] 27 Sep 2017 3.2 The affine root system of type A .........................9 3.3 Matrices . .9 3.4 Circle diagrams . 10 1School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow, G12 8QW, United Kingdom. [email protected] 2Ruhr-Universität Bochum, Faculty of Mathematics, D - 44780 Bochum, Germany. Magdalena.Boos- [email protected] 1 4 The enhanced nilpotent cone 12 4.1 Representation types . 13 4.2 The indecomposable representations . 14 4.3 Translating between the different parameterizations . -
Normal Forms for Representations of Representation-Finite Algebras
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Symbolic Computation (2001) 32, 491{497 doi:10.1006/jsco.2000.0480 Available online at http://www.idealibrary.com on Normal Forms for Representations of Representation-finite Algebras ¨ PETER DRAXLERy Fakult¨atf¨urMathematik, Universit¨atBielefeld, P. O. Box 100131, D-33501 Bielefeld, Germany Using the CREP system we show that matrix representations of representation-finite algebras can be transformed into normal forms consisting of (0; 1)-matrices. c 2001 Academic Press 1. Introduction and Main Result Why is representation theory of algebras important for applications? Besides well-known answers like the occurrence of structures like symmetries, operator algebras or quantum groups there is a less popular and more naive answer to this question. Namely, repre- sentation theory of finite-dimensional algebras deals with normal forms of matrices. But finding normal forms for matrices is an important technique in many areas where math- ematics is applied. A student probably encounters this for the first time when trying to solve a system of linear differential equations describing some physical or biological system or when trying to find the main axes of a symmetric matrix describing the inertia or the magnetic momentum of a solid body. Representation theory of finite-dimensional algebras provides a general environment to deal with problems of this shape. In our paper we will show how a symbolic computation is applied to obtain information about certain types of normal forms. The symbols we use are finite partially ordered sets and finite quivers. -
Li, Yiyang the Principal Indecomposable Modules of Sl3(K). J
Li, Yiyang The principal indecomposable modules of sl3(k). J. Algebra Number Theory Appl. 25, No. 1, 45-58 (2012). Mathematics Subject Classification 2010: *17B10, 17B45, 17B50, 17B56 Keywords: Lie algebra of a reductive algebraic group; simple module; p-character; reduced universal enveloping algebra; standard Levi form; subregular nilpotent; baby Verma module; principal indecomposable module; self-extension Reviewer: J¨orgFeldvoss (8086) Let g be a finite dimensional restricted Lie algebra over an algebraically closed field k of prime characteristic p. Then for every simple g-module S there exists a linear form χ on g, the so-called p-character of S, such that S is a simple module for a certain finite dimensional quotient Uχ(g) of the universal enveloping algebra U(g) of g. In case g is the Lie algebra of a reductive algebraic group G, the linear dual g∗ and g are G-isomorphic and so the Jordan decomposition of elements in g transfers to g∗. Hence χ can be decomposed into a sum of its semisimple and nilpotent parts. It is well known that it is enough to consider only simple Uχ(g)-modules for nilpotent χ, but, in general, the structure of these simple modules is still unkown. If g = sln(k), then one can even assume that χ is in standard Levi form. In particular χ vanishes on the standard Borel subalgebra of g. The easiest case is when χ is regular, that is, when the dimension of the G- stabilizer of χ is equal to the rank of G. In this case every simple Uχ(g)-module is isomorphic to a baby Verma module, that is, a module induced from a one-dimensional (restricted) module for the standard Borel subalgebra of g (if χ is in standard Levi form). -
Indecomposable Modules and Gelfand Rings Francois Couchot
Indecomposable modules and Gelfand rings Francois Couchot To cite this version: Francois Couchot. Indecomposable modules and Gelfand rings. Communications in Algebra, Taylor & Francis, 2007, 35, pp.231–241. 10.1080/00927870601041615. hal-00009474v2 HAL Id: hal-00009474 https://hal.archives-ouvertes.fr/hal-00009474v2 Submitted on 7 Mar 2007 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. INDECOMPOSABLE MODULES AND GELFAND RINGS FRANC¸OIS COUCHOT Abstract. It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if RP is an artinian valuation ring for each maximal prime ideal P . Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic then R is an elementary divisor ring. In this paper R is a commutative ring with unity and modules are unitary.