Dissertations / Theses on the topic 'Representations of groups'
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Andrus, Ivan B. "Matrix Representations of Automorphism Groups of Free Groups." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd856.pdf.
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Hannesson, Sigurdur. "Representations of symmetric groups." Thesis, University of Oxford, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.442464.
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Stavis, Andreas. "Representations of finite groups." Thesis, Karlstads universitet, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-69642.
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Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces. Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups. The study focuses on character theory and how character theory can be used to extract information from a group. Prior to that, concepts needed to treat character theory, and some of their ramifications, are investigated. The study is based on existing literature, with excessive use of examples to illuminate important aspects. An example treating a class of p-groups is also discussed.
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Kasouha, Abeir Mikhail. "Symmetric representations of elements of finite groups." CSUSB ScholarWorks, 2004. https://scholarworks.lib.csusb.edu/etd-project/2605.
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This thesis demonstrates an alternative, concise but informative, method for representing group elements, which will prove particularly useful for the sporadic groups. It explains the theory behind symmetric presentations, and describes the algorithm for working with elements represented in this manner.
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Scopes, Joanna. "Representations of the symmetric groups." Thesis, University of Oxford, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279989.
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Lawrence, Ruth Jayne. "Homology representations of braid groups." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236125.
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Towers, Matthew John. "Modular representations of p-groups." Thesis, University of Oxford, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.427611.
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Can, Himmet. "Representations of complex reflection groups." Thesis, Aberystwyth University, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.289795.
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Cai, Yuanqing. "Theta representations on covering groups." Thesis, Boston College, 2017. http://hdl.handle.net/2345/bc-ir:107492.
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Thesis advisor: Solomon FriedbergKazhdan and Patterson constructed generalized theta representations on covers of general linear groups as multi-residues of the Borel Eisenstein series. For the double covers, these representations and their (degenerate-type) unique models were used by Bump and Ginzburg in the Rankin-Selberg constructions of the symmetric square L-functions for GL(r). In this thesis, we study two other types of models that the theta representations may support. We first discuss semi-Whittaker models, which generalize the models used in the work of Bump and Ginzburg. Secondly, we determine the unipotent orbits attached to theta functions, in the sense of Ginzburg. We also determine the covers for which these models are unique. We also describe briefly some applications of these unique models in Rankin-Selberg integrals for covering groups
Thesis (PhD) — Boston College, 2017
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics
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Wildon, Mark. "Modular representations of symmetric groups." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403775.
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Iano-Fletcher, Maria. "Polynomial representations of symplectic groups." Thesis, University of Warwick, 1990. http://wrap.warwick.ac.uk/89157/.
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In Chapter 1 we review some of the classical theory of reductive algebraic groups over an algebraically closed field. In Chapter 2 we summarise the work of Green on polynomial representations of GLn(k) where k is an infinite field. The irreducible polynomial representations of degree r of GLn(k) are parametrised by partitions λ of r. The irreducible polynomial GLn(k)-module F λ k can be obtained in two ways, as the quotient of a Weyl module V λ k by its unique maximal submodule, or as the unique minimal submodule of a Shur module D λ k. The two modules V λ k and D λ k are duals of one another. We describe Green's basis of D λ k consisting of one element DT for each semistandard λ-tableau T. We also describe the results of Carter and Lusztig on Weyl modules and the Carter-Lusztig basis of V λ k consisting of one element ψτ for each semistandard λ-tableau T. The aim of this thesis is to extend Green's work to polynomial representations of symplectic groups over an infinite field. The basis facts about symplectic groups are described in Chapter 3. In Chapter 4 we introduce the idea of symplectic tableaux due to R. C. King. The dimensions of the Weyl module and the Shur module is equal to the number of symplectic λ-tableaux. We shall use symplectic tableaux to parametrise basis vectors of the Weyl and Shur modules. Chapter 5 is a detailed study of the Weyl module. In Chapter 6 we consider the Shur module.
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MANARA, ELIA. "Multiplicative representations of surface groups." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2018. http://hdl.handle.net/10281/199017.
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Un gruppo di superficie è (isomorfo) al gruppo fondamentale di una superficie orientabile chiusa di genere k (maggiore o uguale a 2). È uno small cancellation group e quindi iperbolico; il suo grafo di Cayley è isomorfo a una tassellatura del piano iperbolico fatta di 2k-goni iperbolici. È possibile definire alcuni sottoinsiemi del grafo di Cayley, detti “coni”, su cui il gruppo agisce con un numero finito di orbite, chiamate “cono tipi”.
Una rappresentazione moltiplicativa di un gruppo di superficie è una rappresentazione unitaria definita sullo spazio di Hilbert delle funzioni moltiplicative.
Una funzione moltiplicativa su un gruppo di superfici ha valori vettoriali ed è definita mediante la scelta di un insieme di parametri, chiamato “sistema di matrici”. Due funzioni moltiplicative sono equivalenti se differiscono solo su un numero finito di elementi. Si può definire un prodotto interno sulle classi di equivalenza di funzioni moltiplicative. Dimostriamo che almeno per il caso di un gruppo di superficie del genere 2 ed una scelta del sistemi di matrici il prodotto interno non è identicamente nullo; dato che esso non dipende dalla scelta dei rappresentanti per le funzioni moltiplicative, è ben definito. Questa dimostrazione si basa sull'irriducibilità di una certa matrice associata alla geometria del grafo di Cayley; in particolare, un certo autovalore Perron-Frobenius deve essere semplice.
Una rappresentazione moltiplicativa agisce semplicemente per traslazione sinistra sul completamento dello spazio di Hilbert dello spazio delle funzioni moltiplicative rispetto al prodotto interno sopra menzionato.
La rappresentazione così definita è temperata: mostriamo che i coefficienti di matrice della rappresentazione regolare approssimano quelli della rappresentazione moltiplicativa.
Con il termine “rappresentazione sul bordo” intendiamo una rappresentazione di una certa C*-algebra prodotto incrociato, ottenuta dall'azione del gruppo di superficie sulla C*-algebra delle funzioni continue sul bordo - che è omeomorfo ad una circonferenza. Una rappresentazione sul bordo è data da una rappresentazione unitaria del gruppo e una rappresentazione della C*-algebra che soddisfi una condizione di covarianza.
Definiamo una famiglia di sottospazi (indicizzata da una quantità reale) di uno spazio di funzioni di quadrato integrabile con valori vettoriali sul gruppo e agiamo su questi sottospazi per traslazione a sinistra con il gruppo e per moltiplicazione con le funzioni continue sulla compattificazione del gruppo di superficie (il gruppo unito al suo bordo). Otteniamo alcune rappresentazioni del gruppo e dell'algebra che soddisfano la covarianza e mostriamo che la famiglia ha una sottosuccessione convergente. Mostriamo quindi che l'azione della C*-algebra coinvolge solo i valori delle funzioni sul bordo: otteniamo quindi una rappresentazione sul bordo. Mostriamo, inoltre, che il limite così ottenuto non dipende dalla sottosuccessione tendente a zero. Abbiamo così una rappresentazione sul bordo ben definita. Mostriamo che la parte unitaria di questa rappresentazione sul bordo è equivalente alla rappresentazione moltiplicativa: infatti, le loro funzioni di tipo positivo coincidono.
Infine, mostriamo che la rappresentazione sul bordo è irriducibile. Questo risultato si ottiene sfruttando l'unicità (a meno di prodotto per una costante positiva) dell'autovalore di Perron-Frobenius ottenuto nella dimostrazione della buona positura del prodotto interno: dimostriamo che qualsiasi proiezione che commuta sia con la rappresentazione del gruppo che con la rappresentazione dell’algebra permette di definire un autovettore della stessa matrice corrispondente all'autovalore di Perron-Frobenius. Quindi, dopo alcuni calcoli, otteniamo che la proiezione considerata deve essere banale. Da una versione del Lemma di Schur segue che la rappresentazione del prodotto incrociato è irriducibile.A surface group is (isomorphic to) the fundamental group of a closed orientable surface of genus k greater or equal than 2. It is a small cancellation group (hence hyperbolic); its Cayley graph is isomorphic to a tiling of the hyperbolic plane by 2k-gons. One can define certain subsets of the Cayley graph called cones. The group acts on the set of cones with finitely many orbits, called cone types. A multiplicative representation of a surface group is a unitary representation defined on the Hilbert space of multiplicative functions. A multiplicative function on a surface group is a vector-valued function defined through the choice of a set of parameters, called matrix system. Two multiplicative functions are equivalent if they differ only on finitely many elements. An inner product can be defined for equivalence classes of multiplicative functions. We prove that at least for the case of a surface group of genus 2 and a choice of the matrices as non-negative scalars the inner product is not identically zero; thus, since it does not depend on the representatives for the multiplicative functions, it is well posed. This proof relies on the irreducibility of a certain matrix associated with the geometry of the Cayley graph; in particular, a certain Perron-Frobenius eigenvalue must be simple. A multiplicative representation then simply acts by left translation on the Hilbert space completion of the space of multiplicative functions with respect to the inner product above mentioned. The representation thus defined is tempered: we show that the matrix coefficients of the regular representation approximate those of the multiplicative representation. By the term boundary representation, we mean a representation of a certain crossed product C*-algebra, obtained by the action of the surface group on the C*-algebra of continuous functions on its boundary – which is homeomorphic to the unit circle. Such a boundary representation is given by a unitary representation of the group and a representation of the C*-algebra satisfying a covariance condition. We define a family of subspaces (indexed by a real quantity) of a space of vector-valued square integrable functions on the group and we act on these subspaces by left translation with the group and by multiplication with continuous functions on the compactification of the surface group (the group united with its boundary). Thus, we get some representations of the group and the algebra satisfying covariance and we show that the family has a limit for a subsequence of the indexes tending to zero. We then show that the action of the C*-algebra involves only the values of the functions on the boundary. Hence, we get a boundary representation. We show, moreover, that the limit thus obtained does not depend on the subsequence tending to zero. Hence, we get a well-defined representation of the crossed product C*-algebra. We show that the unitary part of this boundary representation is equivalent to the multiplicative representation: in fact, their functions of positive type coincide. Finally, we show that the boundary representation is irreducible. This result is achieved by exploiting the uniqueness (up to scaling) of the Perron-Frobenius eigenvalue obtained in the proof of the well-posedness of the inner product: in fact, we show that any projection intertwining both the group representation and the algebra representation allows to define an eigenvector of the same matrix corresponding to the Perron-Frobenius eigenvalue. Thus, after some calculations, we get that the projection considered must be trivial. By a version of Schur’s Lemma, this yields the irreducibility of the crossed product representation.
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George, Timothy Edward. "Symmetric representation of elements of finite groups." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3105.
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The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.
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Hindeleh, Firas Y. "Tangent and Cotangent Bundles, Automorphism Groups and Representations of Lie Groups." University of Toledo / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1153933389.
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Szechtman, Fernando. "Weil representations of finite symplectic groups." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0006/NQ39598.pdf.
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Quinlan, Rachel. "Irreducible projective representations of finite groups." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ59658.pdf.
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Benjamin, Ian Francis. "Quasi-permutation representations of finite groups." Thesis, University of Liverpool, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.250561.
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Martin, Benjamin Michael Sinclair. "Varieties of representations of surface groups." Thesis, King's College London (University of London), 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.243429.
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Nabney, Ian Thomas. "Soluble minimax groups and their representations." Thesis, University of Cambridge, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.333316.
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Mason-Brown, Lucas(Lucas David). "Unipotent representations of real reductive groups." Thesis, Massachusetts Institute of Technology, 2020. https://hdl.handle.net/1721.1/126931.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 207-210).
Let G be a real reductive group and let Ĝ be the set of irreducible unitary representations of G. The determination of Ĝ (for arbitrary G) is one of the fundamental unsolved problems in representation theory. In the early 1980s, Arthur introduced a finite set Unip(G) of (conjecturally unitary) irreducible representations of G called unipotent representations. In a certain sense, these representations form the build-ing blocks of Ĝ. Hence, the determination of Ĝ requires as a crucial ingredient the determination of Unip(G). In this thesis, we prove three results on unipotent representations. First, we study unipotent representations by restriction to K [subset symbol] G, a maximal compact subgroup. We deduce a formula for this restriction in a wide range of cases, proving (in these cases) a long-standing conjecture of Vogan. Next, we study the unipotent representations attached to induced nilpotent orbits. We find that Unip(G) is 'generated' by an even smaller set Unip2(G) consisting of representations attached to rigid nilpotent orbits. Finally, we study the unipotent representations attached to the principal nilpotent orbit. We provide a complete classification of such representations, including a formula for their K-types.
by Lucas Mason-Brown.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
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Ronquillo, Rivera Javier Alfredo. "Extremely Amenable Groups and Banach Representations." Ohio University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520548085599864.
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Anwar, Muhammad F. "Representations and cohomology of algebraic groups." Thesis, University of York, 2011. http://etheses.whiterose.ac.uk/2032/.
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Let G be a semisimple simply connected linear algebraic group over an algebraically closed field k of characteristic p. In [11], Donkin gave a recursive description for the characters of cohomology of line bundles on the flag variety G/B with G = SL3. In chapter 2 of this thesis we try to give a non recursive description for these characters. In chapter 3, we give the first step of a version of formulae in [11] for G = G2. In his famous paper [7], Demazure introduced certain indecomposable modules and used them to give a short proof of the Borel-Weil-Bott theorem (characteristic zero). In chapter 5 we give the cohomology of these modules. In a recent paper [17], Doty introduces the notion of r−minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p >= 2h − 2, where h is the Coxeter number. In chapter 4, we remove the restriction on p and consider some variations involving the more general notion of (p,r)−minuscule weights.
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Chan, Ping-Shun. "Invariant representations of GSp(2)." Columbus, Ohio : Ohio State University, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1132765381.
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Sun, Binyong. "Matrix coefficients and representations of real reductive groups /." View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?MATH%202004%20SUN.
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O'Sullivan, Clodagh M. "Tolerance in intergroup relations: cognitive representations reducing ingroup projection." Thesis, University of Fort Hare, 2008. http://hdl.handle.net/10353/140.
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This study assessed the personal, career and learning skills needs of 196 psychology students (M= 56, F= 103). The preferred means of counselling assistance, preferred experience of counselling and preferred counselling provider were also assessed. The most highly endorsed needs were time management skills (83.5 pecent, n=162), learning test-taking strategies (82 percent, n=159), job search strategies (73.6 percent, n=142), increasing self-confidence (70.3 percent, n=135), increasing motivation (72.4 percent, n=134), controlling anxiety and nervousness (68.7 percent, n=134), public speaking anxiety (68 percent, n=134), understanding career interests and abilities (67.5 percent, n=131), fear of failure (68.1 percent, n=130), and improving study skills (66.5 percent, n=129). Significant sex differences were found for the following, finding a greater purpose in life, controlling weight, job search strategies, concerns about career choice, understanding career interests and abilities in the selection of major subjects and improving study skills. Males highly endorsed the need for finding a greater purpose in life, job search strategies, and concern about career choice, understanding career interest and abilities, selection of major subjects and to improve study skills, whereas females endorsed the need for controlling weight. Respondents indicated individual counselling as being their preferred means of counselling assistance, but lectures were the most prevalent means of assistance previously received by respondents. Most respondents (78.1 percent) found the assistance they had received to be helpful.
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Kielak, Dawid. "Free and linear representations of outer automorphism groups of free groups." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:f2045fba-1546-4dd3-af9f-7d02c4fc505e.
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For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(F_n) we prove that each homomorphism from Out(F_n) to GL_m(K) factors through the natural map p_n from Out(F_n) to GL(H_1(F_n,Z)) = GL_n(Z) whenever n=3, m < 7 and char(K) is not an element of {2,3}, and whenever n>5, m< n(n+1)/2 and char(K) is not an element of {2,3,...,n+1}. We also construct a new infinite family of linear representations of Out(F_n) (where n > 2), which do not factor through p_n. When n is odd these have the smallest dimension among all known representations of Out(F_n) with this property. Using the above results we establish that the image of every homomorphism from Out(F_n) to Out(F_m) is finite whenever n=3 and n < m < 6, and of cardinality at most 2 whenever n > 5 and n < m < n(n-1)/2. We further show that the image is finite when n(n-1)/2 -1 < m < n(n+1)/2. We also consider the structure of normal finite index subgroups of Out(F_n). If N is such then we prove that if the derived subgroup of the intersection of N with the Torelli subgroup T_n < Out(F_n) contains some term of the lower central series of T_n then the abelianisation of N is finite.
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Gu, Jerin. "Single-petaled K-types and Weyl group representations for classical groups." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43735.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 135-137).
In this thesis, we show that single-petaled K-types and quasi-single-petaled K-types for reductive Lie groups generalize petite K-types for split groups. First, we prove that a Weyl group algebra element represents the action of the long intertwining operator for each single-petaled K-type, and then we demonstrate that a Weyl group algebra element represents a part of the long intertwining operator for each quasi-single-petaled K-type. We classify irreducible Weyl group representations realized by quasi-single-petaled K-types for classical groups. This work proves that every irreducible Weyl group representation is realized by quasi-single-petaled K-types for SL(n;C), SL(n;R), SU(m; n), SO(m; n), and Sp(n;R).
by Jerin Gu.
Ph.D.
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Martin, Stuart. "Quivers and the modular representation theory of finite groups." Thesis, University of Oxford, 1988. http://ora.ox.ac.uk/objects/uuid:59d4dc72-60e5-4424-9e3c-650eb2b1d050.
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The purpose of this thesis is to discuss the rôle of certain types of quiver which appear in the modular representation theory of finite groups. It is our concern to study two different types of quiver. First of all we construct the ordinary quiver of certain blocks of defect 2 of the symmetric group, and then apply our results to the alternating group and to the theory of partitions. Secondly, we consider connected components of the stable Auslander-Reiten quiver of certain groups G with normal subgroup N. The main interest lies in comparing the tree class of components of N-modules, with the tree class of components of these modules induced up to G.
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Manriquez, Adam. "Symmetric Presentations, Representations, and Related Topics." CSUSB ScholarWorks, 2018. https://scholarworks.lib.csusb.edu/etd/711.
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The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J1, the Mathieu group M12, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2*60 : (2 x A5), 2*60 : A5, 2*56 : (23 : 7), and 2*28 : (PGL(2,7):2), respectively. We have also discovered the groups 24 : A5, 34 : S5, PSL(2,31), PSL(2,11), PSL(2,19), PSL(2,41), A8, 34 : S5, A52, 2• A52, 2 : A62, PSL(2,49), 28 : A5, PGL(2,19), PSL(2,71), 24 : A5, 24 : A6, PSL(2,7), 3 x PSL(3,4), 2• PSL(3,4), PSL(3,4), 2• (M12 : 2), 37:S7, 35 : S5, S6, 25 : S6, 35 : S6, 25 : S5, 24 : S6, and M12 as homomorphic images of the permutation progenitors 2*60 : (2 x A5), 2*60 : A5, 2*21 : (7: 3), 2*60 : (2 x A5), 2*120 : S5, and 2*144 : (32 : 24). We have given original proof of the 2*n Symmetric Presentation Theorem. In addition, we have also provided original proof for the Extension of the Factoring Lemma (involutory and non-involutory progenitors). We have constructed S5, PSL(2,7), and U(3,5):2 using the technique of double coset enumeration and by way of linear fractional mappings. Furthermore, we have given proofs of isomorphism types for 7 x 22, U(3,5):2, 2•(M12 : 2), and (4 x 2) :• 22.
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Hindeleh, Firas. "Tangent and cotangent bundles automorphism groups and representations of Lie groups /." See Full Text at OhioLINK ETD Center (Requires Adobe Acrobat Reader for viewing), 2006. http://www.ohiolink.edu/etd/view.cgi?acc_num=toledo1153933389.
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Thesis (Ph.D.)--University of Toledo, 2006.Typescript. "A dissertation [submitted] as partial fulfillment of the requirements of the Doctor of Philosophy degree in Mathematics." Bibliography: leaves 79-82.
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Evseeva, Elena. "Représentations du groupe pseudo-orthogonal dans les espaces des formes différentielles homogènes." Thesis, Reims, 2016. http://www.theses.fr/2016REIMS035/document.
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Dans cette thèse nous étudions des représentations du groupe de Lorentz dans les sections du fibré cotangent sur le cône isotrope. Grâce aux transformations de Fourier et de Poisson nous construisons explicitement tous les opérateurs de brisure de symétrie qui apparaissent dans les lois de branchement des produits tensoriels de telles représentationsIn this thesis we study representations of the Lorentz group acting on sectionsof the cotangent bundle over the isotropic cone. Using Fourier and Poisson transforms we construct explicitly all the symmetry breaking operators that appear in branching laws of tensor products of such representations
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Fairley, Jason Thomas. "Induced linear representations for doubly transitive groups." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.404812.
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Forrester-Barker, Magnus. "Representations of crossed modules and cat¹-groups." Thesis, Bangor University, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.401882.
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Magson, Christopher. "Real projective representations of some finite groups." Thesis, University of Liverpool, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236595.
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AFONSO, LUIS FERNANDO CROCCO. "REPRESENTATIONS OF TRIANGLE GROUPS IN COMPLEX HYPERBOLIC." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2003. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=4123@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORO principal objetivo deste trabalho é o estudo de representações que preservam tipo rho:Gamma - PU(2,1) de grupos triangulares Gamma no grupo de isometrias holomorfas do espaço hiperbólico complexo de dimensão dois H2C. O grupo triangular Gamma(p,q,r) é o grupo gerado por reflexões nos lados de um triângulo geodésico, com ângulos pi/p, pi/q e pi/r, no plano hiperbólico. Neste trabalho, nossas atenções são voltadas para os grupos Gamma (4,4,infinito) e Gamma(4,infinito,infinito). Demonstramos, entre outros resultados: Para cada caso, existe um caminho contínuo de representações rho_t que contém todas as representações que preservam tipo de Gamma em PU(2,1). Portanto, isto nos dá, em cada caso, uma descrição completa do espaço de representações de Gamma em PU(2,1). Para cada caso, existe um intervalo fechado J tal que rho_t é uma representação discreta e fiel se, e somente se, t pertence a J. Em cada caso, existe, na fronteira do espaço de deformações, uma representação com elementos parabólicos acidentais. Para demonstrar estes resultados, construímos parametrizações especiais de triângulos em H2C. Construímos poliedros fundamentais para os grupos e utilizamos uma variante do Teorema do Poliedro de Poincaré.
The main aim of this work is to study type-preserving representations p: gamma PU(2, 1) of triangle groups _ in the group of holomorphic isometries of the twodimensional complex hyperbolic space H2C. The triangle group gamma(p, q, r) is the group generated by reflections in the sides of a geodesic triangle having angles pi/p, pi/q and pi/r. We focus our attention on the groups gamma(4,4, infinit) and gamma (4,infinit, infinit). Among other results, we prove that for each case: 1. There is a continuous path of representations pt which contains all type-preserving representations of gamma in PU (2,1) up to conjugation by isometries. This gives us a complete description of the representation space of gamma in PU(2,1). 2. There is a closed interval J such that pt is a discrete and faithful representation if and only if t belongs J. 3. On the boundary of the representation space there is a representation with accidental parabolic elements. To prove these results we give special parametrizations of triangles in H2C. We also build fundamental polyhedra for the groups and use a kind of Poincares Polyhedron Theorem.
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Fayers, Matthew. "Representations of symmetric groups and Schur algebras." Thesis, University of Cambridge, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.620642.
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Almutairi, Bander Nasser. "Counting supercuspidal representations of p-adic groups." Thesis, University of East Anglia, 2012. https://ueaeprints.uea.ac.uk/48008/.
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Let F be a non-archimedean local �eld with residual characteristic p ~= 2. In this thesis we will deduce a formula for the number of irreducible supercuspidal representations of GLN(F), N C 1, with !�($F ) = 1 and level less than or equal to k. Following Blasco, we construct all irreducible supercuspidal representations of the unitary groups U(1; 1)(F~F0) and U(2; 1)(F~F0) by looking at their characteristic polynomials and then compute the number of all these representations according to their level.
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Mazhar, Siddiqua. "Composition of permutation representations of triangle groups." Thesis, University of Newcastle upon Tyne, 2017. http://hdl.handle.net/10443/3857.
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A triangle group is denoted by (p, q, r) and has finite presentation (p, q, r) = hx, y|xp = yq = (xy)r = 1i. In the 1960’s Higman conjectured that almost every triangle group has among its homomorphic images all but finitely many of the alternating groups. This was proved by Everitt in [6]. In this thesis, we combine permutation representations using the methods used in the proof of Higman’s conjecture. We do some experiments by using GAP code and then we examine the situations where the composition of a number of coset diagrams for a triangle group is imprimitive. Chapter 1 provides the introduction of the thesis. Chapter 2 contains some basic results from group theory and definitions. In Chapter 3 we describe our construction that builds compositions of coset diagrams. In Chapter 4 we describe three situations that make the composition of coset diagrams imprimitive and prove some results about the structure of the permutation groups we construct. We conduct experiments based on the theorems we proved and analyse the experiments. In Chapter 5 we prove that if a triangle group G has an alternating group as a finite quotient of degree deg > 6 containing at least one handle, then G has a quotient Cdeg−1 p o Adeg. We also prove that if, for an integer m 6= deg − 1 such that m > 4 and the alternating group Am can be generated by two product of disjoint p-cycles, and a triangle group G has a quotient Adeg containing two disjoint handles, then G also has a quotient Am o Adeg.
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Hendriksen, Michael Arent. "Minimal Permutation Representations of Classes of Semidirect Products of Groups." Thesis, The University of Sydney, 2015. http://hdl.handle.net/2123/14353.
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Given a finite group $G$, the smallest $n$ such that $G$ embeds into the symmetric group $S_n$ is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way.
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Amende, Bonnie. "G-irreducible subgroups of type A₁ /." view abstract or download file of text, 2005. http://wwwlib.umi.com/cr/uoregon/fullcit?p3190506.
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Thesis (Ph. D.)--University of Oregon, 2005.Typescript. Includes vita and abstract. Includes bibliographical references (leaf 152). Also available for download via the World Wide Web; free to University of Oregon users.
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Phillips, Aaron M. "Restricting modular spin representations of symmetric and alternating groups /." view abstract or download file of text, 2003. http://wwwlib.umi.com/cr/uoregon/fullcit?p3095271.
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Thesis (Ph. D.)--University of Oregon, 2003.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 69-71). Also available for download via the World Wide Web; free to University of Oregon users.
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Hua, Jiuzhao Mathematics & Statistics Faculty of Science UNSW. "Representations of quivers over finite fields." Awarded by:University of New South Wales. Mathematics & Statistics, 1998. http://handle.unsw.edu.au/1959.4/40405.
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The main purpose of this thesis is to obtain surprising identities by counting the representations of quivers over finite fields. A classical result states that the dimension vectors of the absolutely indecomposable representations of a quiver ?? are in one-to-one correspondence with the positive roots of a root system ??, which is infinite in general. For a given dimension vector ?? ??? ??+, the number A??(??, q), which counts the isomorphism classes of the absolutely indecomposable representations of ?? of dimension ?? over the finite field Fq, turns out to be a polynomial in q with integer coefficients, which have been conjectured to be nonnegative by Kac. The main result of this thesis is a multi-variable formal identity which expresses an infinite series as a formal product indexed by ??+ which has the coefficients of various polynomials A??(??, q) as exponents. This identity turns out to be a qanalogue of the remarkable Weyl-Macdonald-Kac denominator identity modulus a conjecture of Kac, which asserts that the multiplicity of ?? is equal to the constant term of A??(??, q). An equivalent form of this conjecture is established and a partial solution is obtained. A new proof of the integrality of A??(??, q) is given. Three Maple programs have been included which enable one to calculate the polynomials A??(??, q) for quivers with at most three nodes. All sample out-prints are consistence with Kac???s conjectures. Another result of this thesis is as follows. Let A be a finite dimensional algebra over a perfect field K, M be a finitely generated indecomposable module over A ???K ??K. Then there exists a unique indecomposable module M??? over A such that M is a direct summand of M??? ???K ??K, and there exists a positive integer s such that Ms = M ??? ?? ?? ?? ??? M (s copies) has a unique minimal field of definition which is isomorphic to the centre of End ??(M???) rad (End ??(M???)). If K is a finite field, then s can be taken to be 1.
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Steel, Allan Kenneth. "Construction of ordinary irreducible representations of finite groups." Thesis, The University of Sydney, 2012. http://hdl.handle.net/2123/20307.
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Knezevic, Marica. "Graphs of lattices in representations of finite groups." Thesis, King's College London (University of London), 2017. https://kclpure.kcl.ac.uk/portal/en/theses/graphs-of-lattices-in-representations-of-finite-groups(2f435ed5-4a11-4d45-ae2b-26e3c6264ee8).html.
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This thesis work is motivated by the Langlands program, which relates objects from number theory and representation theory. In particular it is motivated by the compatibility of the local and global cases, especially in the mod p case. Given a finite group G, E a finite extension of Qp with ring of integers OE, uniformizer E, V a finite dimensional E-vector space and : G ! AutE(V ) an absolutely irreducible representation, we associate to the following directed graph: its vertices are homothety classes of lattices in V and there is an edge from (the class of) to 0 when E 0 and the quotient =0 is irreducible. We also label the edge according to the isomorphism class of =0. The central theme of this thesis is the study of stable lattices in p-adic representations and their corresponding graphs. In particular, we show certain properties of these associated graphs, including finiteness, connectedness, a duality property and that the length of a cycle is a multiple of the number of the Jordan-Holder factors. Moreover, we restrict our attention to certain families of representations arising from admissible representations of GL2 of a p-adic field and show further properties of their graphs. In the case where is of principal series type we compute the bound, that is we find an explicit integer c and the lattice in V such that all lattices 0 in V up to homothety satisfy cE 0 . In the case where is of tame principal series type we compute the graphs and investigate their properties. We also compute graphs for certain representations of interest, where most of them are obtained using Magma. The Magma code is attached in the thesis.
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Brau, Julio. "Selmer groups of elliptic curves and Galois representations." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.708896.
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Balagović, Martina. "On representations of quantum groups and Cherednik algebras." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/68478.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 245-248).
In the first part of the thesis, we study quantum groups associated to a semisimple Lie algebra g. The classical Chevalley theorem states that for [ a Cartan subalgebra and W the Weyl group of g, the restriction of g-invariant polynomials on g to [ is an isomorphism onto the W-invariant polynomials on , Res: C[g]1 -+ C[]w. A recent generalization of [36] to the case when the target space C of the polynomial maps is replaced by a finite-dimensional representation V of g shows that the restriction map Res: (C[g] 0 V)9 -+ C[] 0 V is injective, and that the image can be described by three simple conditions. We further generalize this to the case when a semisimple Lie algebra g is replaced by a quantum group. We provide the setting for the generalization, prove that the restriction map Res: (Oq(G) 0 V)Uq(9) -+ O(H) 0 V is injective and describe the image. In the second part we study rational Cherednik algebras Hi,c(W, j) over the field of complex numbers, associated to a finite reflection group W and its reflection representation . We calculate the characters of all irreducible representations in category 0 of the rational Cherednik algebra for W the exceptional Coxeter group H3 and for W the complex reflection group G12 . In particular, we determine which of the irreducible representations are finite-dimensional, and compute their characters. In the third part, we study rational Cherednik algebras Ht,c(W, [) over the field of finite characteristic p. We first prove several general results about category 0, and then focus on rational Cherednik algebras associated to the general and special linear group over a finite field of the same characteristic as the underlying algebraically closed field. We calculate the characters of irreducible representations with trivial lowest weight of the rational Cherednik algebra associated to GL,(Fp,) and SL,(Fpr), and characters of all irreducible representations of the rational Cherednik algebra associated to GL2(F,).
by Martina Balagović.
Ph.D.
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Roth, Calvin L. (Calvin Lee). "Example of solvable quantum groups and their representations." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/28104.
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Wassink, Luke Samuel. "Split covers for certain representations of classical groups." Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1929.
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Let R(G) denote the category of smooth representations of a p-adic group. Bernstein has constructed an indexing set B(G) such that R(G) decomposes into a direct sum over s ∈ B(G) of full subcategories Rs(G) known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (τ,W) of a compact open subgroup J < G. If the functor V ↦ HomJ(W,V) is an equivalence of categories from Rs(G) → H(G,τ)mod we call (J,τ) a type.
Given a Levi subgroup L < G and a type (JL, τL) for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over” (JL, τL) by constructing an object known as a cover. In particular, a cover implements induction of H(L,τL)-modules in a manner compatible with parabolic induction of L-representations.
In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of Sp(2k) over an archimedean local field of characteristic zero. In partic- ular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.
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Fitzpatrick, Michael Colin. "Continuous families of representations of mapping class groups." Diss., University of Iowa, 2014. https://ir.uiowa.edu/etd/1316.
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The study of mapping class groups began in the 1920s with Max Dehn and Jakob Nielsen. It was about this time that topology was just being developed, so mapping class groups were of immediate interest, being invariants of topological spaces. The works of Dehn and Nielsen were continued by William Harvey in the 1960s and 70s leading to the development of the curve complex, an important construction still very relevant to mathematics today. William Thurston is another important name in this area since he was able to completely classify homeomorphisms of surfaces in 1976, leading to the famous "Nielsen-Thurston Classification Theorem".
Representations were first studied by Carl Gauss in the early 1800s and then explored more thoroughly by Ferdinand Frobenius and Richard Dedekind, among others, at the end of that century. Representation theory has since grown into an extremely important and active area of mathematics today because of its widespread applications to other areas of mathematics and even to other subject areas like physics.
Quantum group theory is the youngest area in which this thesis has its roots. This area was formalized and studied extensively for the first time in the 1980s by such mathematicians as Vladimir Drinfeld, Michio Jimbo, and Nicolai Reshetikhin, and immediately found applications in mathematics and theoretical physics. Like representation theory, the study of quantum groups is currently a highly active area of mathematics due to its widespread applications across the mathematical spectrum.
In this paper I will present two different methods of constructing projective representations of mapping class groups of surfaces. I will then prove some interesting results concerning each of these methods.
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Schoemann, Claudia. "Représentations unitaires de U(5) p-adique." Thesis, Montpellier 2, 2014. http://www.theses.fr/2014MON20101.
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Nous étudions les représentations complexes, induites par l'induction parabolique, du groupe U(5), défini sur un corps local non-archimedean de caractéristique 0. C'est Qp ou une extension finie de Qp .On parle des 'corps p-adiques'. Soit F un corps p-adique. Soit E : F une extension de corps de degré 2. Soit Gal(E : F ) = {id, σ}le groupe de Galois. On écrit σ(x) = overline{x} forall x ∈ E. Soit | |p la norme p-adique de E. Soient E* = E {0} et E 1 = {x ∈ E | xoverline{x}= 1} .U (5) a trois sous-groupes paraboliques propres. Soit P0 le sous-groupe parabolique minimal et soientP1 et P2 les deux sous-groupes paraboliques maximaux. Soient M0 , M1 et M2 les sous-groupes de Levi standards et soient N0 , N1 et N2 des sous-groupes unipotents de U (5). On a la décomposition de Levi Pi = Mi Ni , i ∈{0, 1, 2} .M0 = E* × E* × E 1 est le sous-groupe de Levi minimal, M1 = GL(2, E) × E 1 et M2 = E* × U(3) sont les sous-groupes de Levi maximaux.On considère les représentations des sous-groupes de Levi, et on les étend trivialement au sous-groupes unipotents pour obtenir des représentations des sous-groupes paraboliques. On exécute une procédure appelée 'l'induction parabolique' pour obtenir les représentations de U (5). Nous considérons les représentations de M0 , puis les représentations non-cuspidales, induites à partir de M1 et M2 . Cela veut dire que la représentation du facteur GL(2, E) de M1 est un sous-quotient propre d'une représentation induite de E* × E* à GL(2, E). La représentation du facteur U (3) de M2 est un sous-quotient propre d'une représentation induite de E* × E 1 à U(3). Un exemple pour M1 est | det |α χ(det) StGL2 * λ' , où α ∈ R, χ est un caractère unitaire de E* , StGL2 est la représentation Steinberg de GL(2, E) et λ' est un caractère de E 1 . Un exemple pour M2 est| |α χ λ (det) StU (3) , où α ∈ R, χ est un caractère unitaire de E* , λ' est un caractère unitaire de E 1et StU (3) est la représentation Steinberg de U(3). On remarque que λ' est unitaire.Ensuite on considère les représentations cuspidales de M1 .On détermine les droites et les points de réductibilité des représentations de U(5) et on détermine les sous-quotients irréductibles. Ensuite, sauf quelque cas particuliers, on détermine le dual unitaire de U(5)par rapport au quotients de Langlands. Les représentations complexes, paraboliquement induites, de U(3) sur un corps p-adique sont classifiées par Charles David Keys dans [Key84], les représentations complexes, paraboliquement induites, de U(4)sur un corps p-adique sont classifiées par Kazuko Konno dans [Kon01]We study the parabolically induced complex representations of the unitary group in 5 variables - U(5)- defined over a non-archimedean local field of characteristic 0. This is Qp or a finite extension of Qp ,where p is a prime number. We speak of a 'p-adic field'.Let F be a p-adic field. Let E : F be a field extension of degree two. Let Gal(E : F ) = {id, σ}. We write σ(x) = overline{x} forall x ∈ E. Let | |p denote the p-adic norm on E. Let E* := E {0} and let E 1 := {x ∈ E | x overline{x} = 1} .U(5) has three proper parabolic subgroups. Let P0 denote the minimal parabolic subgroup and P1 andP2 the two maximal parabolic subgroups. Let M0 , M1 and M2 denote the standard Levi subgroups and let N0 , N1and N2 denote unipotent subgroups of U(5). One has the Levi decomposition Pi = Mi Ni , i ∈ {0, 1, 2} .M0 = E* × E* × E 1 is the minimal Levi subgroup, M1 = GL(2, E) × E 1 and M2 = E* × U (3) are the two maximal parabolic subgroups.We consider representations of the Levi subgroups and extend them trivially to the unipotent subgroups toobtain representations of the parabolic groups. One now performs a procedure called 'parabolic induction'to obtain representations of U (5).We consider representations of M0 , further we consider non-cuspidal, not fully-induced representationsof M1 and M2 . For M1 this means that the representation of the GL(2, E)− part is a proper subquotientof a representation induced from E* × E* to GL(2, E). For M2 this means that the representation of theU (3)− part of M2 is a proper subquotient of a representation induced from E* × E 1 to U (3).As an example for M1 , take | det |α χ(det) StGL2 * λ' , where α ∈ R, χ is a unitary character of E* , StGL2 is the Steinberg representation of GL(2, E) and λ' is a character of E 1 . As an example forM2 , take | |α χ λ' (det) StU (3) , where α ∈ R, χ is a unitary character of E* , λ' is a character of E 1 andStU (3) is the Steinberg representation of U (3). Note that λ' is unitary.Further we consider the cuspidal representations of M1 .We determine the points and lines of reducibility of the representations of U(5), and we determinethe irreducible subquotients. Further, except several particular cases, we determine the unitary dual ofU(5) in terms of Langlands-quotients.The parabolically induced complex representations of U(3) over a p-adic field have been classied byCharles David Keys in [Key84], the parabolically induced complex representations of U(4) over a p-adicfield have been classied by Kazuko Konno in [Kon01].An aim of further study is the classication of the induced complex representations of unitary groupsof higher rank, like U (6) or U (7). The structure of the Levi subgroups of U (6) resembles the structureof the Levi subgroups of U (4), the structure of the Levi groups of U (7) resembles those of U (3) and ofU (5).Another aim is the classication of the parabolically induced complex representatioins of U (n) over ap-adic field for arbitrary n. Especially one would like to determine the irreducible unitary representations