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Du, Sautoy M. P. F. "Discrete groups, analytic groups and Poincare series." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236109.
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Assmann, Björn. "Applications of Lie methods to computations with polycyclic groups." Thesis, St Andrews, 2007. http://hdl.handle.net/10023/435.
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Silva, Jefferson dos Santos e. "Uma apresentação policíclica para o multiplicador de Schur e o quadrado tensorial não abeliano de um grupo policíclico." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4539.
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In this work, based on [9], describes an effective method for computing a consistent polycyclic presentation for the nonanbeian tensor square G G of a group G given by a consistent polycyclic presentation.
Este trabalho, baseado em [9], determina um efetivo método para calcular uma apresentação policíclica consistente para o quadrado tensorial não abeliano G G de um grupo G dado por uma apresentação policíclica consistente.
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Garibaldi, Skip. "Trialitarian algebraic groups /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1998. http://wwwlib.umi.com/cr/ucsd/fullcit?p9906492.
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Meyer, Aurel Nathan. "Essential dimension of algebraic groups." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/27091.
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We study the essential dimension of linear algebraic groups. For a group G, essential dimension is a measure for the complexity of G-torsors or, more generally, the complexity of any algebraic or geometric structure with automorphism group G. This makes essential dimension a powerful invariant with many interesting and surprising connections to problems in algebra and geometry.
We show that for various classes of groups, including finite (algebraic) groups and algebraic tori, the essential dimension is related to minimal faithful representations. In many cases this renders the exact value of the essential dimension computable and we explore several of its consequences. An important open problem is the essential dimension of the projective linear group PGLn. This topic is closely related to the structure theory of central simple algebras, which may be viewed
as twisted forms of the algebra of n x n matrices.
We study central simple algebras with additional structure such as a distinguished Galois subfield. We prove new
bounds on the essential dimension of these algebras and, as a corollary, of the group PGLn.
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Brundan, Jonathan Walter. "Double cosets in algebraic groups." Thesis, Imperial College London, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.244137.
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Craven, David Andrew. "Algebraic modules for finite groups." Thesis, University of Oxford, 2007. http://ora.ox.ac.uk/objects/uuid:7f641b33-d301-4445-8269-a5a33f4b7e5e.
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The main focus of this thesis is algebraic modules---modules that satisfy a polynomial equation with integer co-efficients in the Green ring---in various finite groups, as well as their general theory. In particular, we ask the question `when are all the simple modules for a finite group G algebraic?' We call this the (p-)SMA property. The first chapter introduces the topic and deals with preliminary results, together with the trivial first results. The second chapter provides the general theory of algebraic modules, with particular attention to the relationship between algebraic modules and the composition factors of a group, and between algebraic modules and the Heller operator and Auslander--Reiten quiver. The third chapter concerns itself with indecomposable modules for dihedral and elementary abelian groups. The study of such groups is both interesting in its own right, and can be applied to studying simple modules for simple groups, such as the sporadic groups in the final chapter. The fourth chapter analyzes the groups PSL(2,q); here we determine, in characteristic 2, which simple modules for PSL(2,q) are algebraic, for any odd q. The fifth chapter generalizes this analysis to many groups of Lie type, although most results here are in defining characteristic only. Notable exceptions include the small Ree groups, which have the 2-SMA property for all q. The sixth and final chapter focuses on the sporadic groups: for most groups we provide results on some simple modules, and some of the groups are completely analyzed in all characteristics. This is normally carried out by restricting to the Sylow p-subgroup. This thesis develops the current state of knowledge concerning algebraic modules for finite groups, and particularly for which simple groups, and for which primes, all simple modules are algebraic.
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Clarke, Matthew Charles. "Unipotent elements in algebraic groups." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/241660.
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This thesis is concerned with three distinct, but closely related, research topics focusing on the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin by obtaining a new canonical form for nilpotent matrices, up to similarity, which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi;j) -> (xn+1-j;n+1-i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group, we thus obtain a unified approach to computing representatives for nilpotent orbits for all classical groups G. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487]. We give a case-free proof of the conjectures of Lusztig from that paper. This presents a uniform picture of the unipotent elements of G, which can be viewed as an extension of the Dynkin-Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra g and the coadjoint action of G on g*. We also obtain several general results about the Hesselink stratification and Fq-rational structures on G-modules. Our third topic is concerned with generalised Gelfand-Graev representations of finite groups of Lie type. Let u be a unipotent element in such a group GF and let Γu be the associated generalised Gelfand-Graev representation of GF . Under the assumption that G has a connected centre, we show that the dimension of the endomorphism algebra of Γu is a polynomial in q (the order of the associated finite field), with degree given by dimCG(u). When the centre of G is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of q, unless one adopts a convention of considering separately various congruence classes of q. Subject to such a convention, we extend our result. We also present computational data related to the main theoretical results. In particular, tables of our canonical forms are given in the appendices, as well as tables of dimension polynomials for endomorphism algebras of generalised Gelfand-Graev representations, together with the relevant GAP source code.
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Dos, Santos João Pedro Pinto. "Fundamental groups in algebraic geometry." Thesis, University of Cambridge, 2006. https://www.repository.cam.ac.uk/handle/1810/252015.
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Sinanan, Shavak. "Algorithms for polycyclic-by-finite groups." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/49186/.
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A set of fundamental algorithms for computing with polycyclic-by-finite groups is presented here. Polycyclic-by-finite groups arise naturally in a number of contexts; for example, as automorphism groups of large finite soluble groups, as quotients of finitely presented groups, and as extensions of modules by groups. No existing mode of representation is suitable for these groups, since they will typically not have a convenient faithful permutation representation. A mixed mode is used to represent elements of such a group; utilising a polycyclic presentation or a power-conjugate presentation for the elements of the normal subgroup, and a permutation representation for the elements of the quotient.
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Stalder, Nicolas Raymond. "Algebraic monodromy groups of A-motives /." Zürich : ETH, 2007. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=17369.
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Proud, Richard. "Unipotent subgroups of reductive algebraic groups." Thesis, University of Warwick, 1997. http://wrap.warwick.ac.uk/57023/.
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Let G be a connected reductive algebraic group defined over an algebraically closed field of good characteristic p>0. Suppose uEG has order p. In [T2] it is shown that u lies in a closed reductive subgroup of G of type Al. This is the best possible group theoretic analogue of the Jacobson-Morozov theorem for simple Lie algebras. Testerman's key result is a type of `exponentiation process'. For our given element u, this process constructs a 1-dimensional connected abelian unipotent subgroup of G, hence isomorphic to Ga, containing u. This in turn yields the required Al overgroup of u. Now let 1#uEG be an arbitrary unipotent element. Such an element has order pt, for some tEN. In this thesis we extend the above result, and show that u lies in a t-dimensional closed connected abelian unipotent subgroup of G, provided p> 29 when G' contains a simple component of type E8, and that p is good for the remaining components. The structure of the resulting unipotent overgroup is also explicitly given. This is the best possible result, in terms of `minimal dimension', which we could hope for. In Chapter 1 we discuss the theory of Witt vectors, associated with a commutative ring with identity. They are closely related to the study of connected abelian unipotent algebraic groups. The unipotent overgroups are constructed using a variation of the usual exponen- tiation process. The necessary material on formal power series rings is given in 1.3. The Artin-Hasse exponentials of 1.4 play a crucial role in this construction. The connection between Witt groups and Artin-Hasse exponentials is discussed in 1.5. In Chapter 2 we apply the techniques of Chapter 1 to the various simple algebraic groups. For each type, a particular isogeny class is chosen and the required overgroup is constructed for the regular (and subregular) classes. In 2.9 we pass to the adjoint case. In Chapter 3 we extend the results of Chapter 2 to include all unipotent classes in all reductive algebraic groups (under certain restrictions). In 3.1 the Cayley Transform for the classical groups is combined with the ideas of Chapter 1 to give an explicit construction of the unipotent overgroups for every unipotent class. In 3.2 we discuss semiregular unipotent elements. Finally, in 3.3, we prove the main theorem of this thesis.
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Rainone, Raffaele. "Geometric actions of classical algebraic groups." Thesis, University of Southampton, 2014. https://eprints.soton.ac.uk/366485/.
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Let k be an algebraically closed field of arbitrary characteristic p. An affine algebraic group G is an affine algebraic variety over k with a group structure such that multiplication and inversion maps are morphisms of varieties. A special class of affine algebraic groups are the so called classical groups Cl(V), groups of isometries of a finite dimensional k-vector space V with respect to a certain form on V {e.g. a zero form, a symplectic form or a non-degenerate quadratic form. These groups are: GL(V) the general linear group, Sp(V) the symplectic group and O(V) the orthogonal group. Let G = Cl(V). Various (closed) subgroups H of G can be defined naturally in terms of the geometry of V {H may be the stabiliser of a subspace of V, or a direct sumdecomposition of V, or a non-degenerate form on V, for example. Let H be such a subgroup and let = G=H be the corresponding coset space. Then is a variety with a natural algebraic action of G. We define geometric subgroups of G to be the closed subgroups arising in this manner. Consequently, for H a geometric subgroup, we say that the natural action of G on = G=H is a geometric action. We define C (x) to be the set of points in fixed by x. Then C (x) is a subvariety, and we can show that dimC (x) = dim.
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Wright, William G. (William Glenn). "An algebraic characterization of stability groups." Thesis, University of North Texas, 1991. https://digital.library.unt.edu/ark:/67531/metadc332465/.
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The goal of this paper is to establish necessary and sufficient conditions for a subgroup of the full homeomorphism group of a manifold to be the stability group of a point in the underlying space. Such subgroups are useful in identifying the underlying space in terms of its homeomorphism group even in cases in which this space is not necessarily a manifold. Thus, stability groups are useful in classifying various spaces.
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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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Thomas, Adam Robert. "Irreducible subgroups of exceptional algebraic groups." Thesis, Imperial College London, 2014. http://hdl.handle.net/10044/1/28689.
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Let $G$ be a semisimple algebraic group over an algebraically closed field $K$, of characteristic $p \geq 0$. A closed subgroup of $G$ is said to be irreducible if it does not lie in any proper parabolic subgroup of $G$. In this thesis we address the following problem: classify the connected irreducible subgroups of $G$, up to conjugacy, where $G$ is of exceptional type. Work of Liebeck and Seitz classifies the conjugacy classes of simple, connected irreducible subgroups of rank at least 2, with a restriction on the characteristic of the underlying field ($p > 7$ is sufficient). When $G$ is of type $F_4$, Stewart has classified the conjugacy classes of simple, connected irreducible subgroups of rank at least 2 in all characteristics. We classify the conjugacy classes of simple, connected irreducible subgroups, of rank at least 2 for $E_6$, $E_7$ and $E_8$. Our approach works in all characteristics, rather than starting from the characteristics excluded in the result of Liebeck and Seitz. We use these classifications to prove corollaries concerning the representation theory of such irreducible subgroups. For example, with one exception, two simple irreducible connected subgroups of rank at least 2 are $G$-conjugate if and only if they have the same composition factors on the adjoint module of $G$. We also consider connected subgroups of rank 1. Work of Lawther and Testerman classifies conjugacy classes of rank 1 connected irreducible subgroups, with a restriction on $p$ ($p>7$ is sufficient). The connected irreducible subgroups of rank 1 were found, in arbitrary characteristic, by Amende for all but $E_8$. We give a new proof of this, finding a set of conjugacy class representatives without repetition. We prove corollaries on the overgroups of irreducible $A_1$ subgroups. For example, we prove that if $p=2$ or $3$ then any irreducible $A_1$ subgroup of $E_7$ is contained in $A_1 D_6$. Finally, consider the semisimple, non-simple connected irreducible subgroups. We classify these, up to conjugacy, for $G_2$, $F_4$ and $E_6$. So in conclusion, we classify conjugacy classes of connected irreducible subgroups of $G_2$, $F_4$ and $E_6$, all simple, connected irreducible subgroups of $E_7$ and all simple, connected irreducible subgroups of $E_8$ of rank at least 2.
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Penrod, Keith. "Infinite product groups /." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1977.pdf.
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Cook, Martin David. "Conjugacy class structure in simple algebraic groups." Thesis, Lancaster University, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.442727.
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Litterick, Alastair. "Finite simple subgroups of exceptional algebraic groups." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/14278.
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Let G = G(K) be a simple algebraic group over an algebraically closed field K of characteristic p ≥ 0. The study of subgroups of G splits naturally according to whether G is of classical or exceptional type, and according to whether the subgroups considered are finite or of positive dimension. This thesis considers finite subgroups of adjoint groups G of exceptional type. A finite subgroup of G is called Lie primitive if it lies in no proper, closed subgroup of positive dimension. This is a natural maximality condition and, when studying Lie primitive subgroups, a reduction theorem due to Borovik allows us to focus on those whose socle is a non-abelian finite simple group. The study then splits again according to whether or not this socle is a member of Lie(p), the simple groups of Lie type in characteristic p. For H = H(q) ∈ Lie(p), in [LS98b] Liebeck and Seitz prove, for all but finitely many q, that G cannot have a Lie primitive subgroup with socle H unless G and H are of the same Lie type. For H ∉ Lie(p), in [LS99] Liebeck and Seitz produce a complete (finite) list of those H which embed into an adjoint exceptional simple algebraic group, though conjugacy and Lie primitivity remain largely open. The first result of this thesis is to disprove the existence of Lie primitive embeddings of many simple groups H ∉ Lie(p). For example, for n ≥ 10 the alternating group Altn has no Lie primitive embeddings into an adjoint exceptional algebraic group, in any characteristic. This has implications for the subgroup structure of the nite groups of Lie type. In particular, it is deduced here that for n ≥ 11 the groups Altn and Symn never occur as a maximal subgroup of any nite almost-simple group of exceptional Lie type.
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Rizkallah, John. "Bounding cohomology for low rank algebraic groups." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267214.
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Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.
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Vallino, Daniele. "Algebraic and definable closure in free groups." Thesis, Lyon 1, 2012. http://www.theses.fr/2012LYO10090/document.
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Nous étudions la clôture algébrique et définissable dans les groupes libres. Les résultats principaux peuvent être résumés comme suit. Nous montrons un résultat de constructibilité des groupes hyperboliques sans torsion au-dessus de la clôture algébrique d'un sous-ensemble engendrant un groupe non abélien. Nous avons cherché à comprendre la place qu'occupe la clôture algébrique acl_G(A) dans certaines décompositions de G. Nous avons étudié la possibilité de la généralisation de la méthode de Bestvina-Paulin dans d'autres directions, en considérant les groupes de type fini qui agissent d'une manière acylindrique (au sens de Bowditch) sur les graphes hyperboliques. Enfin, nous avons étudié les relations qui existent entre les différentes notions de clôture algébrique et entre la clôture algébrique et la clôture définissableIn Chapter 1 we give basics on combinatorial group theory, starting from free groups and proceeding with the fundamental constructions: free products, amalgamated free products and HNN extensions. We outline a synthesis of Bass-Serre theory, preceded by a survey on Cayley graphs and graphs of groups. After proving the main theorem of Bass-Serre theory, we present its application to the proof of Kurosh subgroup theorem. Subsequently we recall main definitions and properties of hyperbolic spaces. In Section 1.4 we define algebraic and definable closures and recall a few other notions of model theory related to saturation and homogeneity. The last section of Chapter 1 is devoted to asymptotic cones. In Chapter 2 we prove a theorem similar to Bestvina-Paulin theorem on the limit of a sequence of actions on hyperbolic graphs. Our setting is more general: we consider Bowditch-acylindrical actions on arbitrary hyperbolic graphs. We prove that edge stabilizers are (finite bounded)-by-abelian, that tripod stabilizers are finite bounded and that unstable edge stabilizers are finite bounded. In Chapter 3 we introduce the essential notions on limit groups, shortening argument and JSJ decompositions. In Chapter 4 we present the results on constructibility of a torsion-free hyperbolic group from the algebraic closure of a subgroup. Also we discuss constructibility of a free group from the existential algebraic closure of a subgroup. We obtain a bound to the rank of the algebraic and definable closures of subgroups in torsion-free hyperbolic groups. In Section 4.2 we prove some results about the position of algebraic closures in JSJ decompositions of torsion-free hyperbolic groups and other results for free groups. Finally, in Chapter 5 we answer the question about equality between algebraic and definable closure in a free group. A positive answer has been given for a free group F of rank smaller than 3. Instead, for free groups of rank strictly greater than 3 we found some counterexample. For the free group of rank 3 we found a necessary condition on the form of a possible counterexample
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Iannone, Paola. "Automorphism groups of geometric codes." Thesis, University of East Anglia, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318091.
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Henes, Matthew Thomas. "Root subgroups of the rank two unitary groups." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2841.
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Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.
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Harkins, Andrew. "Combining lattices of soluble lie groups." Thesis, University of Newcastle Upon Tyne, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341777.
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Mohrdieck, Stephan. "Conjugacy classes of non-connected semisimple algebraic groups." [S.l. : s.n.], 2000. http://www.sub.uni-hamburg.de/disse/172/diss.pdf.
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Turner, S. M. "Hasse-Weil zeta functions for linear algebraic groups." Thesis, University of Glasgow, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318888.
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Costantini, Mauro. "On the lattice automorphisms of certain algebraic groups." Thesis, University of Warwick, 1989. http://wrap.warwick.ac.uk/101705/.
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In the first chapter we give an introduction, and a survey of known results, which we shall use throughout the dissertation. In the second chapter we first prove that every projectivity of a connected reductive non-abelian algebraic group G over K = Fp is strictly index-preserving (Theorem 2.1.6.). Then we prove that every autoprojectivity of G induces an automorphism of the building canonically associated to O. Furthermore we show how certain autoprojectivities of G act on the Weyl group of G and on the Dynkin diagram of G. In the third chapter we restrict our attention to simple algebraic groups over K. We prove that if G is a simple algebraic group over K of rank at least 2, then the problem whether every autoprojectivity of G is induced by an automorphism, is reduced to the problem whether every autoprojectivity of G fixing every parabolic subgroup of G is the identity. Namely, if we let Γ(G) – {φε Aut L(G) I Pφ = P for every parabolic subgroup P of G} , we have Aut L(G) = Γ (Aut G)*, where (Aut G)* is the group of all autoprojectivities of G induced by an automorphism (Theorem 3.4.9. and Corollary 3.4.15.). In Chapter 4 we prove that actually Γ = {1} if G has rank at least 3 and p ≠ 2 (Theorem 4.6.5.), while in Chapter 5 we prove the same result , with different arguments, for the case of rank 1 (Corollary 5.2.6.) and 2, type A2 excluded (Corollary 5.3.8.) (for groups of rank 1 we impose no restrictions on p). Finally, in Chapter 6 we show that for the groups of type A2 Theorem 4.6.5. does not hold. For this purpose we construct a non-trivial subgroup of the group Γ(SL3(F23)) (Corollary 6.4.15.).
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Hazi, Amit. "Semisimple filtrations of tilting modules for algebraic groups." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/271774.
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Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. The indecomposable tilting modules $\{T(\lambda)\}$ for $G$, which are labeled by highest weight, form an important class of self-dual representations over $k$. In this thesis we investigate semisimple filtrations of minimal length (Loewy series) of tilting modules. We first demonstrate a criterion for determining when tilting modules for arbitrary quasi-hereditary algebras are rigid, i.e. have a unique Loewy series. Our criterion involves checking that $T(\lambda)$ does not have certain subquotients whose composition factors extend more than one layer in the radical or socle series. We apply this criterion to show that the restricted tilting modules for $SL_4$ are rigid when $p \geq 5$, something beyond the scope of previous work on this topic by Andersen and Kaneda. Even when $T(\lambda)$ is not rigid, in many cases it has a particularly structured Loewy series which we call a balanced semisimple filtration, whose semisimple subquotients or "layers" are symmetric about some middle layer. Balanced semisimple filtrations also suggest a remarkably straightforward algorithm for calculating tilting characters from the irreducible characters. Applying Lusztig's character formula for the simple modules, we show that the algorithm agrees with Soergel's character formula for the regular indecomposable tilting modules for quantum groups at roots of unity. We then show that these filtrations really do exist for these tilting modules. In the modular case, high weight tilting modules exhibit self-similarity in their characters at $p$-power scales. This is due to what we call higher-order linkage, an old character-theoretic result relating modular tilting characters and quantum tilting characters at $p$-power roots of unity. To better understand this behavior we describe an explicit categorification of higher-order linkage using the language of Soergel bimodules. Along the way we also develop the algebra and combinatorics of higher-order linkage at the de-categorified level. We hope that this will provide a foundation for a tilting character formula valid for all weights in the modular case when $p$ is sufficiently large.
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Stewart, David I. "G-complete reducibility and the exceptional algebraic groups." Thesis, Imperial College London, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.528310.
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Edwards, T. "Algebraic 2-complexes over certain infinite abelian groups." Thesis, University College London (University of London), 2006. http://discovery.ucl.ac.uk/1444701/.
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Whitehead's Theorem allows the study of homotopy types of two dimensional CW complexes to be phrased in terms of chain homotopy types of algebraic complexes, arising as the cellular chains on the universal cover. It is natural to ask whether the category of algebraic complexes fully represents the category of CW complexes, in particular whether every algebraic complex is realised geometrically. The case of two dimensional complexes is of special interest, partly due to the relationship between such complexes and group presentations and partly since, as was recently proved, it relates to the question as to when cohomology is a suitable indicator of dimension. This thesis has two primary considerations. The first is the generalisation to infinite groups of F.E.A. Johnson's approach regarding problems of geometric realisation. It is shown, under certain restrictions, that the class of projective extensions containing algebraic complexes may be recognised as the unit elements of a ring, with ring elements congruence classes of extensions of the trivial module by a second homotopy module. The realisation property is shown to hold for the free abelian groups on two and three generators, and for the product of a cyclic group and a free group on a single generator. Secondly, a reinterpretation is given of the well documented relation ship between the congruence classes represented by Swan modules and the projective modules constructed via Milnor's connecting homomorphism and the relevant fibre product diagram. This relationship is shown to be typical of projective modules occurring in extensions of a two-sided ideal by a quotient ring, and we show that any two-sided ideal in a general ring results in a Mayer-Vietoris sequence which is different and complimentary to the standard excision sequence.
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Deshpande, D. V. "Topological methods in algebraic geometry : cohomology rings, algebraic cobordism and higher Chow groups." Thesis, University of Cambridge, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.598515.
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This thesis is divided into three chapters. The first chapter is about the cohomology ring of the space of rotational functions. In the second chapter, we define algebraic cobordism of classifying spaces, Ω*(BG) and G-equivariant algebraic cobordism Ω*G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted Fj(Ω*(-)); which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We compute Ω*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n + 1). We also compute Ω*(BG) when G is a finite abelian group. A finite non-abelian group for which we compute Ω*(BG) is the quaternion group of order 8. In all the above cases we check that Ω*(BG) is isomorphic to MU*(BG). The third chapter is work-in-progress on Steenrod operations on higher Chow groups. Voevodsky has defined motivic Steenrod operations on motivic cohomology and used them in his proof of the Minor Conjecture.
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Delgado, Rodríguez Jordi. "Extensions of free groups: algebraic, geometric, and algorithmic aspects." Doctoral thesis, Universitat Politècnica de Catalunya, 2017. http://hdl.handle.net/10803/458522.
Full textAbstract:
In this work we use geometric techniques in order to study certain natural extensions of free groups, and solve several algorithmic problems on them.
To this end, we consider the family of free-abelian times free groups (Zm x Fn) as a seed towards further generalization in two main directions: semidirect products, and partially commuative groups (PC-groups).
The four principal projects of this thesis are the following:
Direct products of free-abelian and free groups
We begin by studying the structure of the groups Zm x Fn , with special emphasis on their lattice of subgroups, and their endomorphisms (for which an explicit description is given, and both injectivity and surjectiveness are characterized); to then solve on them algorithmic problems involving both subgroups (the membership problem, the finite index problem, and the subgroup and coset intersection problems), and endomorphisms (the fixed points poblem, the Whitehead problems, and the twisted-conjugacy problem).
Algorithmic recognition of infinite-cyclic extensions
In the first part, we prove the algorithmic undecidability of several properties (finite generability, finite presentability, abelianity, finiteness, independence, triviality) of the base group of finitely presented cyclic extensions. In particular, we see that it is not possible to decide algorithmically if a finitely presented Z-extension admits a finitely generated base group. This last result allows us to demonstrate the undecidability of the Bieri-Neumann-Strebel (BNS) invariant.
In the second part, we prove the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for certain type of outer automorphisms, which we characterize algorithmically.
Stallings automata for free-abelian by free groups
After recreating in a purely algorithmic language the classic theory of Stallings associating an automaton to each subgroup of the free group, we extend this theory to semi-direct products of the form Zm ¿ Fn. Specifically, we associate to each subgroup of Zm ¿ Fn , an automaton ("enriched" with vectors in Zm), and we see that in the finitely generated case this construction is algorithmic and allows to solve the membership problem within this family of groups.
The geometric description obtained also shows (even in the case of direct products) not only that the intersection of finitely generated subgroups can be infinitely generated, but that even when it is finitely generated, the rank of the intersection can not be bound in terms of the ranks of the intersected subgroups. This fact is relevant because it denies any possible extension of the celebrated - and recently proven - Hanna-Neumann conjecture in this direction.
Intersection problems for Droms groups
After characterizing those partially commutative groups satisfying the Howson property, we combine the algorithmic version of the theorem of the subgroups of Kurosh given by S.V. Ivanov, with the ideas coming from our work on Zm x Fn, to prove the solvability of the subgroup and coset intersection problems within the subfamily of Droms groups (that is, those PC- groups whose subgroups are always again partially commutative).En aquest treball s'usen tècniques geomètriques per estudiar certes extensions naturals dels grups lliures, i atacar diversos problemes algorísmics sobre elles. A aquest efecte, es considera la família de grups lliure-abelians per lliure (Zm x Fn) com a punt de partida envers generalitzacions en dues direccions principals: productes semidirectes, i grups parcialment commutatius (PC-groups). Els quatre projectes principals d'aquesta tesi es descriuen a continuació. Productes directes de grups lliure-abelians per lliure. Comencem estudiant l'estructura dels grups Zm x Fn, amb especial èmfasi en el seu reticle de subgrups, i el seu monoide d'endomorfismes (per als que es dóna una descripció explícita, i es caracteritzen tant la injectivitat com l'exhaustivitat); per després resoldre sobre ells problemes algorísmics involucrant tant subgrups (el problema de la pertinença, el problema de l'índex finit, i els problemes de la intersecció de subgrups i classes laterals), com endomorfismes (el problema dels punts fixos, els problemes de Whitehead , i el problema de la "conjugació retorçada" o twisted-conjugacy problem). Reconeixement algorítmic d'extensions cícliques. A la primera part, es demostra la indecidibilitat algorísmica de diverses propietats (generabilitad finita, presentabilitad finita, abelianitat, finitud, llibertat, i trivialitat) del grup base de les extensions cícliques finitament presentades. En particular, veiem que no és possible decidir algorítmicament si una Z-extensió finitament presentada admet un grup base finitament generat. Aquest últim resultat ens permet demostrar també la indecidibilitat de l'invariant BNS (de Bieri-Neumann-Strebel). A la segona part, es demostra l'equivalència entre el problema de l'isomorfisme dins de la subclasse de Z-extensions úniques, i el problema de la semi-conjugació per a cert tipus d'automorfismes exteriors, que caracteritzem algorísmicament. Autòmats d'Stallings per a grups lliure-abelians by lliure. Després de recrear en un llenguatge purament algorísmic la teoria clàssica d'Stallings associant un autòmat a cada subgrup del grup lliure, estenem aquesta teoria a productes semidirectes de la forma Zm x Fn . Concretament associem un autòmat "enriquit" amb vectors de Zm a cada subgrup de Zm x Fn , i veiem que en el cas de subgrups finitament generats aquesta construcció és algorísmica i permet resoldre el problema de la pertinença dins d'aquesta família de grups. La descripció geomètrica obtinguda mostra a més (fins i tot en el cas de productes directes), no només que la intersecció de subgrups finitament generats pot ser infinitament generada, sinó que, fins i tot quan és finitament generada, no es pot afitar el rang de la intersecció en termes dels rangs dels subgrups intersecats. Aquest fet és rellevant perquè denega qualsevol possible extensió de la celebrada - i recentment provada - conjectura de Hanna Neumann en aquesta direcció. Problemes de la intersecció per a grups de Droms. Després de caracteritzar els grups parcialment commutatius que satisfan la propietat de Howson, combinem la versió algorísmica del teorema dels subgrups de Kurosh donada per S.V. Ivanov, amb les idees provinents del nostre treball sobre Zm x Fn, per demostrar la resolubilitat dels problemes de la intersecció de subgrups, de classes laterals (i afins) dins la subfamília de PC-grups de Droms (i.e., aquells PC-grups en que tots els subgrups son de nou parcialment commutatius).
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Haller, Sergei. "Computing Galois cohomology and forms of linear algebraic groups." Giessen Giessener Elektronische Bibliothek, 2005. http://geb.uni-giessen.de/geb/volltexte/2005/2474/index.html.
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Shin, Hyunshik. "Algebraic degrees of stretch factors in mapping class groups." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/51910.
Full textAbstract:
Given a closed surface Sg of genus g, a mapping class f in \MCG(Sg) is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number \lambda(f). The number \lambda(f) is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur.
Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on Sg with orientable foliations whose stretch factor \lambda has algebraic degree 2g. Moreover, the stretch factor \lambda is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d≤g. Our examples also give a new approach to a conjecture of Penner.
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Bowman, Christopher David. "Algebraic groups, diagram algebras, and their Schur-Weyl dualities." Thesis, University of Cambridge, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610216.
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Humphreys, Jodie John Arthur Michael. "Algebraic properties of semi-simple lattices and related groups." Thesis, University College London (University of London), 2005. http://discovery.ucl.ac.uk/1445592/.
Full textAbstract:
Two abstract theories are developed. The first concerns isomorphism in variants with the same multiplicative properties as the Euler characteristic. It is used to show that the index of a subgroup in a semi-simple lattice is deter mined by its isomorphism type when that index is finite. This is also proved to be the case for subgroups of finite index in free products of finitely many semi-simple lattices as well as certain non-trivial extensions of Z by surface groups. In addition, a criterion for the failure of this property is given which applies to a large class of central extensions. The second development concerns the syzygies of groups. The results of this theory are used to define the cohomology groups of a duality group in terms of morphisms between stable modules in the derived category. The Farrell cohomology of virtual duality groups is also considered.
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Yam, Sheung-chi Phillip. "Algebraic methods on some problems in finance /." Hong Kong : University of Hong Kong, 2001. http://sunzi.lib.hku.hk/hkuto/record.jsp?B25018723.
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Crawley-Boevey, W. W. "Polycyclic-by-finite affine group schemes and infinite soluble groups." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372868.
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Borge, I. C. "A cohomological approach to the classification of $p$-groups." Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.246353.
Full textAbstract:
In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.
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Bays, Martin. "Categoricity results for exponential maps of 1-dimensional algebraic groups." Thesis, University of Oxford, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526931.
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Dowerk, Philip. "Algebraic and Topological Properties of Unitary Groups of II_1 Factors." Doctoral thesis, Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-165242.
Full textAbstract:
The thesis is concerned with group theoretical properties of unitary groups, mainly of II_1 factors. The author gives a new and elementary proof of an result on extreme amenability, defines the bounded normal generation property and invariant automatic continuity property and proves these for various unitary groups of functional analytic types.
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Misseldine, Andrew F. "Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups." BYU ScholarsArchive, 2014. https://scholarsarchive.byu.edu/etd/5259.
Full textAbstract:
In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of Schur rings over cyclic p-groups.
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Kenneally, Darren John. "On eigenvectors for semisimple elements in actions of algebraic groups." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/224782.
Full textAbstract:
Let G be a simple simply connected algebraic group defined over an algebraically closed field K and V an irreducible module defined over K on which G acts. Let E denote the set of vectors in V which are eigenvectors for some non-central semisimple element of G and some eigenvalue in K*. We prove, with a short list of possible exceptions, that the dimension of Ē is strictly less than the dimension of V provided dim V > dim G + 2 and that there is equality otherwise. In particular, by considering only the eigenvalue 1, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of V provided dim V > dim G + 2, with a short list of possible exceptions. In the majority of cases we consider modules for which dim V > dim G + 2 where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds dim G. In more difficult cases, when dim V is only slightly larger than dim G + 2, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying dim V ≤ dim G + 2, an immediate observation yields the result for dim V < dim B where B is a Borel subgroup of G, while in other cases we argue directly.
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Uchiyama, Tomohiro. "Separability and complete reducibility of subgroups of the Weyl group of a simple algebraic group." Thesis, University of Canterbury. Mathematics and Statistics, 2012. http://hdl.handle.net/10092/7150.
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Let G be a reductive algebraic group defined over an algebraically closed field of characteristic p. A subgroup H of G is called G-complete reducible whenever H is contained in a parabolic subgroup P of G, it is contained in some Levi subgroup of P. In this thesis, we present a pair of reductive subgroups H and M of G of type E_7 such that H
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Friedrich, Nina. "Automorphism groups of quadratic modules and manifolds." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/276986.
Full textAbstract:
In this thesis we prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension $2n \geq 6$, we get an extension of their result to the case of virtually polycyclic fundamental groups. We also prove the corresponding result for manifolds equipped with tangential structures. A result on the stable homology groups of moduli spaces of manifolds by Galatius--Randal-Williams enables us to make new computations using our homological stability results. In particular, we compute the abelianisation of the mapping class groups of certain $6$-dimensional manifolds. The first computation considers a manifold built from $\mathbb{R} P^6$ which involves a partial computation of the Adams spectral sequence of the spectrum ${MT}$Pin$^{-}(6)$. For the second computation we consider Spin $6$-manifolds with $\pi_1 \cong \mathbb{Z} / 2^k \mathbb{Z}$ and $\pi_2 = 0$, where the main new ingredient is an~analysis of the Atiyah--Hirzebruch spectral sequence for $MT\mathrm{Spin}(6) \wedge \Sigma^{\infty} B\mathbb{Z}/2^k\mathbb{Z}_+$. Finally, we consider the similar manifolds with more general fundamental groups $G$, where $K_1(\mathbb{Q}[G^{\mathrm{ab}}])$ plays a role.
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Klein, Tom. "Filtered ends of pairs of groups." Diss., Online access via UMI:, 2007.
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Fitzgerald, J. G. M. "Weyl modules for groups of type B2 and G2." Thesis, University of Oxford, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.257653.
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In this thesis we determine the submodule structure of a number of Weyl modules for algebraic groups with root systems B2 and G2. We use the Jantzen sum formula to find the composition factors of Weyl modules and go on to use homomorphisms between Weyl modules, given by H.H. Andersen, and the comparison of two filtrations of tensor products of Weyl modules to establish submodule structure. A computer program in the Prolog language is given which calculates the Jantzen sum formula. In addition we find one 2-dimensional Ext group for simple modules for type G2 in characteristic greater than or equal to 7.
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Massold, Heinrich. "Labile und relative Reduktionstheorie über Zahlkörpern." Bonn : Mathematisches Institut der Universität, 2003. http://catalog.hathitrust.org/api/volumes/oclc/54890700.html.
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Burness, Timothy Charles. "Fixed point spaces in actions of finite and algebraic simple groups." Thesis, Imperial College London, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.421905.
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Lond, Daniel. "On Reductive Subgroups of Algebraic Groups and a Question of Külshammer." Thesis, University of Canterbury. Mathematics and Statistics, 2013. http://hdl.handle.net/10092/8033.
Full textAbstract:
This Thesis is motivated by two problems, each concerning representations (homomorphisms)
of groups into a connected reductive algebraic group G over an algebraically
closed field k. The first problem is due to B. Külshammer and is to do with representations
of finite groups in G:
Let Γ be a finite group and suppose k has characteristic p. Let Γp be a
Sylow p-subgroup of Γ and let ρ : Γp → G be a representation. Are there
only finitely many conjugacy classes of representations ρ' : Γ → G whose
restriction to Γp is conjugate to ρ?
The second problem follows the work of M. Liebeck and G. Seitz: describe the representations
of connected reductive algebraic H in G.
These two problems have been settled as long as the characteristic p is large enough but
not much is known in the case where the characteristic p is a so called bad prime for G,
which will be the setting for our work.
At the intersection of these two problems lies another problem which we call the algebraic
version of Külshammer's question where we no longer suppose Γ is finite. This new
variation of Külshammer's question is interesting in its own right, and a counterexample
may provide insight into Külshammer's original question.
Our approach is to convert these problems into problems in the nonabelian 1-cohomology.
Let K be a reductive algebraic group, P a parabolic subgroup of G with Levi subgroup
L < P, V the unipotent radical of P. Let ρ₀ : K → L be a representation. Then the
representations ρ : K → P that equal ρ₀ under the canonical projection P → L are
in bijective correspondence with elements of the space of 1-cocycles Z¹(K,V ) where K
acts on V by xv = ρ₀(x)vρ₀(x)⁻¹. We can then interpret P- and G-conjugacy classes
of representations in terms of the 1-cohomology H¹(K,V ).
We state and prove the conditions under which a collection of representations from K to
P is a finite union of conjugacy classes in terms of the 1-cohomology in Theorem 4.22.
Unlike other approaches, we work directly with the nonabelian 1-cohomology. Even so,
we find that the 1-cocycles in Z¹(K,V ) often take values in an abelian subgroup of V
(Lemmas 5.10 and 5.11). This is interesting, for the question "is the restriction map of
1-cohomologies H¹(H,V) → H¹(U,V) induced by the inclusion of U in K injective?"
is closely linked to the question of Külshammer, and has positive answer if V is abelian
and H = SL₂k) (Example 3.2).
We show that for G = B4 there is a family of pairwise non-conjugate embeddings of
SL₂in G, a direction provided by Stewart who proved the result for G = F4. This is
important as an example like this is first needed if one hopes to find a counterexample
to the algebraic version of Külshammer's question.