Dissertations / Theses on the topic 'Hecke'
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1
Rostam, Salim. "Algèbres de Hecke carquois et généralisations d'algèbres d'Iwahori-Hecke." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLV063/document.
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Cette thèse est consacrée à l'étude des algèbres de Hecke carquois et de certaines généralisations des algèbres d'Iwahori-Hecke. Dans un premier temps, nous montrons deux résultats concernant les algèbres de Hecke carquois, dans le cas où le carquois possède plusieurs composantes connexes puis lorsqu'il possède un automorphisme d'ordre fini. Ensuite, nous rappelons un isomorphisme de Brundan-Kleshchev et Rouquier entre algèbres d'Ariki-Koike et certaines algèbres de Hecke carquois cyclotomiques. D'une part nous en déduisons qu'une équivalence de Morita importante bien connue entre algèbres d'Ariki-Koike provient d'un isomorphisme, d'autre part nous donnons une présentation de type Hecke carquois cyclotomique pour l'algèbre de Hecke de G(r,p,n). Nous généralisons aussi l'isomorphisme de Brundan-Kleshchev pour montrer que les algèbres de Yokonuma-Hecke cyclotomiques sont des cas particuliers d'algèbres de Hecke carquois cyclotomiques. Finalement, nous nous intéressons à un problème de combinatoire algébrique, relié à la théorie des représentations des algèbres d'Ariki-Koike. En utilisant la représentation des partitions sous forme d'abaque et en résolvant, via un théorème d'existence de matrices binaires, un problème d'optimisation convexe sous contraintes à variables entières, nous montrons qu'un multi-ensemble de résidus qui est bégayant provient nécessairement d'une multi-partition bégayanteThis thesis is devoted to the study of quiver Hecke algebras and some generalisations of Iwahori-Hecke algebras. We begin with two results concerning quiver Hecke algebras, first when the quiver has several connected components and second when the quiver has an automorphism of finite order. We then recall an isomorphism of Brundan-Kleshchev and Rouquier between Ariki-Koike algebras and certain cyclotomic quiver Hecke algebras. From this, on the one hand we deduce that a well-known important Morita equivalence between Ariki--Koike algebras comes from an isomorphism, on the other hand we give a cyclotomic quiver Hecke-like presentation for the Hecke algebra of type G(r,p,n). We also generalise the isomorphism of Brundan-Kleshchev to prove that cyclotomic Yokonuma-Hecke algebras are particular cases of cyclotomic quiver Hecke algebras. Finally, we study a problem of algebraic combinatorics, related to the representation theory of Ariki-Koike algebras. Using the abacus representation of partitions and solving, via an existence theorem for binary matrices, a constrained optimisation problem with integer variables, we prove that a stuttering multiset of residues necessarily comes from a stuttering multipartition
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Heyer, Claudius. "Applications of parabolic Hecke algebras: parabolic induction and Hecke polynomials." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20137.
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Im ersten Teil wird eine neue Konstruktion der parabolischen Induktion für pro-p Iwahori-Heckemoduln gegeben. Dabei taucht eine neue Klasse von Algebren auf, die in gewisser Weise als Interpolation zwischen der pro-p Iwahori-Heckealgebra einer p-adischen reduktiven Gruppe $G$ und derjenigen einer Leviuntergruppe $M$ von $G$ gedacht werden kann. Für diese Algebren wird ein Induktionsfunktor definiert und eine Transitivitätseigenschaft bewiesen. Dies liefert einen neuen Beweis für die Transitivität der parabolischen Induktion für Moduln über der pro-p Iwahori-Heckealgebra. Ferner wird eine Funktion auf einer parabolischen Untergruppe untersucht, die als Werte nur p-Potenzen annimmt. Es wird gezeigt, dass sie eine Funktion auf der (pro-p) Iwahori-Weylgruppe von $M$ definiert, und
dass die so definierte Funktion monoton steigend bzgl. der Bruhat-Ordnung ist und einen Vergleich der Längenfunktionen zwischen der Iwahori-Weylgruppe von $M$ und derjenigen der Iwahori-Weylgruppe von $G$ erlaubt.
Im zweiten Teil wird ein allgemeiner Zerlegungssatz für Polynome über der sphärischen (parahorischen) Heckealgebra einer p-adischen reduktiven Gruppe $G$ bewiesen. Diese Zerlegung findet über einer parabolischen Heckealgebra statt, die die Heckealgebra von $G$ enthält. Für den Beweis des Zerlegungssatzes wird vorausgesetzt, dass die gewählte parabolische Untergruppe in einer nichtstumpfen enthalten ist. Des Weiteren werden die nichtstumpfen parabolischen Untergruppen von $G$ klassifiziert.The first part deals with a new construction of parabolic induction for modules over the pro-p Iwahori-Hecke algebra. This construction exhibits a new class of algebras that can be thought of as an interpolation between the pro-p Iwahori-Hecke algebra of a p-adic reductive group $G$ and the corresponding algebra of a Levi subgroup $M$ of $G$. For these algebras we define a new induction functor and prove a transitivity property. This gives a new proof of the transitivity of parabolic induction for modules over the pro-p Iwahori-Hecke algebra. Further, a function on a parabolic subgroup with p-power values is studied. We show that it induces a function on the (pro-p) Iwahori-Weyl group of $M$, that it is monotonically increasing with respect to the Bruhat order, and that it allows to compare the length function on the Iwahori-Weyl group of $M$ with the one on the Iwahori-Weyl group of $G$. In the second part a general decomposition theorem for polynomials over the spherical (parahoric) Hecke algebra of a p-adic reductive group $G$ is proved. The proof requires that the chosen parabolic subgroup is contained in a non-obtuse one. Moreover, we give a classification of non-obtuse parabolic subgroups of $G$.
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Bijakowski, Stéphane. "Classicité de formes modulaires surconvergentes sur une variété de Shimura." Thesis, Paris 13, 2014. http://www.theses.fr/2014PA132050/document.
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Nous nous intéressons aux formes modulaires surconvergentes définies sur certaines variétés de Shimura, et prouvons des théorèmes de classicité en grand poids. Dans un premier temps, nous étudions les variétés ayant bonne réduction, associées à des groupes non ramifiés en p. Nous nous intéressons aux variétés de Shimura PEL de type (A) et (C), qui sont associées respectivement à des groupes unitaires et symplectiques. Pour démontrer un théorème de classicité, nous utilisons la méthode du prolongement analytique, qui a été développée par Buzzard et Kassaei dans le cas de la courbe modulaire. Nous généralisons ensuite ce résultat de classicité à des variétés en ne supposant plus que le groupe associé est non ramifié en p. Dans le cas des formes modulaires de Hilbert, nous construisons des modèles entiers des compactifications de la variété, et démontrons un principe de Koecher. Pour des variétés de Shimura plus générales, nous travaillons avec le modèle rationnel de la variété, et utilisons un plongement vers une variété de Siegel pour définir les structures entièresWe deal with overconvergent modular forms défined on some Shimura varieties, andprove classicality results in the case of big weight. First we study the case of varieties with good reduction, associated to unramified groups in p. We deal with Shimura varieties of PEL type (A) and (C), which are associated respectively to unitary and symplectic groups. To prove a classicality theorem, we use the analytic continuation method, which has been developed by Buzzard and Kassaei in the case of the modular curve. We then generalize this classicality result for varieties without assuming that the associated group is unramified in p. In the case of Hilbert modular forms, we construct integral models of compactifications of the variety, and prove a Koecher principle. For more general Shimura varieties, we work with the rationnal model of the variety, and use an embedding to a Siegel variety to define the integral structures
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Uhl, Christine. "Quantum Drinfeld Hecke Algebras." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc862764/.
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Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.
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Parkinson, James William. "Buildings and Hecke Algebras." Thesis, The University of Sydney, 2005. http://hdl.handle.net/2123/642.
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We establish a strong connection between buildings and Hecke algebras through the study of two algebras of averaging operators on buildings. To each locally finite regular building we associate a natural algebra B of chamber set averaging operators, and when the building is affine we also define an algebra A of vertex set averaging operators. In the affine case, it is shown how the building gives rise to a combinatorial and geometric description of the Macdonald spherical functions, and of the centers of affine Hecke algebras. The algebra homomorphisms from A into the complex numbers are studied, and some associated spherical harmonic analysis is conducted. This generalises known results concerning spherical functions on groups of p-adic type. As an application of this spherical harmonic analysis we prove a local limit theorem for radial random walks on affine buildings.
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Parkinson, James William. "Buildings and Hecke Algebras." University of Sydney. Mathematics and Statistics, 2005. http://hdl.handle.net/2123/642.
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We establish a strong connection between buildings and Hecke algebras through the study of two algebras of averaging operators on buildings. To each locally finite regular building we associate a natural algebra B of chamber set averaging operators, and when the building is affine we also define an algebra A of vertex set averaging operators. In the affine case, it is shown how the building gives rise to a combinatorial and geometric description of the Macdonald spherical functions, and of the centers of affine Hecke algebras. The algebra homomorphisms from A into the complex numbers are studied, and some associated spherical harmonic analysis is conducted. This generalises known results concerning spherical functions on groups of p-adic type. As an application of this spherical harmonic analysis we prove a local limit theorem for radial random walks on affine buildings.
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Boys, Clinton. "Alternating quiver Hecke algebras." Thesis, The University of Sydney, 2014. http://hdl.handle.net/2123/12725.
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This thesis consists of a detailed study of alternating quiver Hecke algebras, which are alternating analogues of quiver Hecke algebras as defined by Khovanov-Lauda and Rouquier. The main theorem gives an isomorphism between alternating quiver Hecke algebras and alternating Hecke algebras, as introduced by Mitsuhashi, in the style of Brundan and Kleshchev, provided the quantum characteristic is odd. A proof is obtained by adapting recent methods of Hu and Mathas, which rely on seminormal forms and coefficient systems. A presentation for alternating quiver Hecke algebras by generators and relations, reminiscent of the KLR presentation for Hecke algebras, is also given. Finally, some steps are taken towards discussing the representation theoretic consequences of the results.
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Bao, Dianbin. "Identities between Hecke Eigenforms." Diss., Temple University Libraries, 2017. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/424027.
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MathematicsPh.D.
In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation $h=af^2+bfg+g^2$ is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to $\Gamma_1(N)$ of weight $k>2$ and $a,b\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\langle f^2,g\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\mathhbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $SL_2(\mathbb{Z})$ of the form $X^2+\sum_{i=1}^n \alpha_iY_i=0$ all are forced by dimension considerations, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix.
Temple University--Theses
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Jacobs, Daniel. "Slopes of compact Hecke operators." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397675.
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Canguel, Ismail Naci. "Normal subgroups of Hecke groups." Thesis, University of Southampton, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240816.
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Gunnells, Paul E. (Paul Edward). "The topology of Hecke correspondences." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/28094.
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Cocke, William Leonard. "Hecke Eigenvalues and Arithmetic Cohomology." BYU ScholarsArchive, 2014. https://scholarsarchive.byu.edu/etd/4130.
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We provide algorithms and documention to compute the cohomology of congruence subgroups of the special linear group over the integers when n=3 using the well-rounded retract and the Voronoi decomposition. We define the Sharbly complex and how one acts on a k-sharbly by the Hecke operators. Since the norm of a sharbly is not preserved by the Hecke operators we also examine the reduction techniques described by Gunnells and present our implementation of said techniques for n=3.
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Barkow, Andreas. "Die ökologische Bedeutung von Hecken für Vögel." Göttingen : [s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=966435338.
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Soriano, Solá Marcos. "Contributions to the integral representation theory of Iwahori-Hecke algebras." [S.l. : s.n.], 2002. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB9866651.
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Neshitov, Alexander. "Motivic Decompositions and Hecke-Type Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/35009.
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Let G be a split semisimple algebraic group over a field k. Our main objects of interest are twisted forms of projective homogeneous G-varieties. These varieties have been important objects of research in algebraic geometry since the 1960's.
The theory of Chow motives and their decompositions is a powerful tool for studying twisted forms of projective homogeneous varieties. Motivic decompositions were discussed in the works of Rost, Karpenko, Merkurjev, Chernousov, Calmes, Petrov, Semenov, Zainoulline, Gille and other researchers. The main goal of the present thesis is to connect motivic decompositions of twisted homogeneous varieties to decompositions of certain modules over Hecke-type algebras that allow purely combinatorial description. We work in a slightly more general situation than Chow motives, namely we consider the category of h-motives for an oriented cohomology theory h. Examples of h include Chow groups, Grothendieck K_0, algebraic cobordism of Levine-Morel, Morava K-theory and many other examples. For a group G there is the notion of a versal torsor such that any G-torsor over an infinite field can be obtained as a specialization of a versal torsor. We restrict our attention to the case of twisted homogeneous spaces of the form E/P where P is a special parabolic subgroup of G. The main result of this thesis states that there is a one-to-one correspondence between h-motivic decompositions of the variety E/P and direct sum decompositions of modules DFP* over the graded formal affine Demazure algebra DF. This algebra was defined by Hoffnung, Malagon-Lopez, Savage and Zainoulline combinatorially in terms of the character lattice, the Weyl group and the formal group law of the cohomology theory h. In the classical case h=CH the graded formal affine Demazure algebra DF coincides with the nil Hecke ring, introduced by Kostant and Kumar in 1986. So the Chow motivic decompositions of versal homogeneous spaces correspond to decompositions of certain modules over the nil Hecke ring. As an application, we give a purely combinatorial proof of the indecomposability of the Chow motive of generic Severi-Brauer varieties and the versal twisted form of HSpin8/P1.
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Wong, Michael Lennox. "Hecke modifications, wonderful compactifications and isomonodromy." Thesis, McGill University, 2011. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=104521.
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The central weight of this thesis lies in the study of the moduli space of principal bundles over a compact Riemann surface, as well as the closely related moduli spaces of Higgs bundles and local systems. Hecke modifications are used to parametrize the moduli space of principal bundles in certain cases; while extant in the literature, an attempt has been made here to systematize the exposition on Hecke modifications. One novel aspect of the thesis is the use of the "wonderful" or De Concini--Procesi compactification of a semisimple complex algebraic group of adjoint type to develop the deformation theory necessary for the parametrization. A few new facts about these compactifications are proved as were necessary for the deformation theory. The later chapters review the symplectic geometry of the moduli spaces of Higgs bundles and local systems as discovered by N.J. Hitchin and further developed by I. Biswas and S. Ramanan, F. Bottacin, and E. Markman, as well as the theory of isomonodromic deformation, which goes back to the Riemann--Hilbert problem. Finally, using the parametrizations for moduli of principal bundles obtained, we prove a result of I. Krichever, generalized by J. Hurtubise, in the principal bundle case, relating some of the Hitchin hamiltonians to differences in isomonodromic flows.Le sujet principal de cette thèse est l'étude de l'espace de modules des fibrés principaux sur une surface de Riemann compacte, et aussi des espaces de modules liés des fibrés de Higgs et des systèmes locaux. Les modifications de Hecke sont utilisées pour paramétrer l'espace de modules des fibrés principaux dans quelques instances; bien qu'elle existe dans la littérature, on a tenté de systématiser l'exposition sur des modifications de Hecke. Un aspect nouveau de cette thèse est l'usage de la compactification De Concini--Procesi d'un groupe algébraique complexe semi-simple dans la théorie des déformations nécessaire pour la paramétrisation. On a prouvé quelques nouveaux énoncés sur ces compactifications, nécessaires pour la théorie des déformations. Les derniers chapitres donnent un compte rendu de la géométrie symplectique des espaces de modules de fibrés de Higgs et des systèmes locaux découverte par N.J. Hitchin et développé plus tard par I. Biswas et S. Ramanan, F. Bottacin, et E. Markman, et aussi de la déformation isomonodromique, ce qui a des racines dans le problème de Riemann--Hilbert. Finalement, en se servant des paramétrisations pour des modules des fibrés principaux obtenues auparavant, on prouve un résultat de I. Krichever, généralisé par J. Hurtubise, dans le cas des fibrés principaux, qui donne une relation entre quelques uns des hamiltoniens de Hitchin et les différences dans les flots isomonodromiques.
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Freiberger, Marianne. "Matings between Hecke groups and polynomials." Thesis, Queen Mary, University of London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368907.
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Fakiolas, A. P. "Hecke algebras and the Lusztig isomorphism." Thesis, University of Warwick, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.379611.
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Bruno, Paul. "Rademacher Sums, Hecke Operators and Moonshine." Case Western Reserve University School of Graduate Studies / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=case1459522798.
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Anderson, Michael R. "Hecke algebras associated to Weyl groups /." The Ohio State University, 1993. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487842372897758.
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Kerschl, Alexander. "Simple modules of cyclotomic Hecke algebras." Thesis, The University of Sydney, 2019. http://hdl.handle.net/2123/20683.
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Ariki showed that the simple modules of the cyclotomic Hecke algebra are labelled by Kleshchev multipartitions. Recently, Jacon gave an alternative recursive description of Uglov multipartitions, which can be thought of as a generalisation of Kleshchev multipartitions. In this thesis we extend Jacon's combinatorics and then give a non-recursive description of Kleshchev multipartitions. We then use these combinatorial tools in the framework of the diagramatic Cherednik algebras to give a complete classification of the simple modules coming from the Webster-Bowman "many cellular bases" indexed by a loading. In particular, we recover Ariki's classification theorem in the case of Kleshchev multipartitions. As a consequence we also obtain a new lower bound for the graded dimensions of the simple modules.
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Kusilek, Jonathan. "On representations of affine Hecke algebras." Thesis, The University of Sydney, 2011. http://hdl.handle.net/2123/12074.
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We introduce a C-algebra Ht corresponding to an affine Hecke algebra H and a central character t of H, and show that the irreducible representations of Ht are precisely the irreducible representations of H with central character t. For certain choices of t we give an explicit construction of a cellular basis of Ht in terms of elementary properties of t. We thus classify, and give a construction of, the irreducible representations of Ht. While the indexing sets appear similar to those given for calibrated representations, we obtain many representations which are not calibrated.
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Nash, David A. 1982. "Graded representation theory of Hecke algebras." Thesis, University of Oregon, 2010. http://hdl.handle.net/1794/10871.
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xii, 76 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.We study the graded representation theory of the Iwahori-Hecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory. We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of two-column partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th -root of unity to those of the group algebra of the symmetric group over a field of characteristic p.
Committee in charge: Alexander Kleshchev, Chairperson, Mathematics; Jonathan Brundan, Member, Mathematics; Boris Botvinnik, Member, Mathematics; Victor Ostrik, Member, Mathematics; William Harbaugh, Outside Member, Economics
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TERRAGNI, TOMMASO. "Hecke algebras associated to coxeter groups." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2012. http://hdl.handle.net/10281/29634.
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In this thesis, we study cohomological properties of Hecke algebras $H_q(W,S)$ associated with arbitrary Coxeter groups $(W,S)$. Under mild conditions, it is possible to canonically define the Euler characteristic of such an algebra. We define an almost-canonical complex of $H$-modules that allows one to compute the Euler characteristic of $H$. It turns out that the Euler characteristic of the algebra has an interpretation as a combinatorial object attached to the Coxeter group: indeed, for suitable choices of the base ring, it is the inverse of the Poincaré series.
Some other results about Coxeter groups are proved, in particular one new characterization of minimal non-spherical, non-affine types is given.
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Pitale, Ameya. "Lifting from SL(2) to GSpin(1,4)." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1147463757.
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Schmidt, Nicolas Alexander. "Generic pro-p Hecke algebras, the Hecke algebra of PGL(2, Z), and the cohomology of root data." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/19724.
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Es wird die Theorie der generischen pro-$p$ Hecke-Algebren und ihrer Bernstein-Abbildungen entwickelt. Für eine Unterklasse diese Algebren, der \textit{affinen} pro-$p$ Hecke-Algebren wird ein Struktursatz bewiesen, nachdem diese Algebren unter anderem stets noethersch sind, wenn es der Koeffizientenring ist. Hilfsmittel ist dabei der Nachweis der Bernsteinrelationen, der in abstrakter Weise geführt wird und so die bestehende Theorie verallgemeinert.
Ferner wird der top. Raum der Orientierungen einer Coxetergruppe eingeführt und im Falle der erweiterten modularen Gruppe $\operatorname{PGL}_2(\mathds{Z})$ untersucht, und ausgenutzt um Kenntnisse über die Struktur der zugehörigen Hecke-Algebra als Modul über einer gewissen Unteralgebra, welche zur Spitze im Unendlichen zugeordnet ist, zu erlangen.
Schließlich wird die Frage des Zerfallens des Normalisators eines maximalen zerfallenden Torus innerhalb einer zerfallenden reduktiven Gruppe als Erweiterung der Weylgruppe durch die Gruppe der rationalen Punkte des Torus untersucht, und mittels zuvor erreichter Ergebnisse auf eine kohomologische Frage zurückgeführt. Zur Teilbeantwortung dieser werden dann die Kohomologiegruppen bis zur Dimension drei der Kocharaktergitter der fasteinfachen halbeinfachen Wurzeldaten einschließlich des Rangs 8 berechnet. Mittels der Theorie der $\mathbf{FI}$-Moduln wird daraus die Berechnung der Kohomologie der mod-2-Reduktion der Kowurzelgitter für den Typ $A$ in allen Rängen bewiesen.The theory of generic pro-$p$ Hecke algebras and their Bernstein maps is developed. For a certain subclass, the \textit{affine} pro-$p$ Hecke algebras, we are able to prove a structure theorem that in particular shows that the latter algebras are always noetherian if the ring of coefficients is. The crucial technical tool are the Bernstein relations, which are proven in an abstract way that generalizes the known cases. Moreover, the topological space of orientations is introduced and studied in the case of the extended modular group $\operatorname{PGL}_2(\mathds{Z})$, and used to determine the structure of its Hecke algebra as a module over a certain subalgebra, attached to the cusp at infinity. Finally, the question of the splitness of the normalizer of a maximal split torus inside a split reductive groups as an extension of the Weyl group by the group of rational points is studied. Using results obtained previously, this questioned is then reduced to a cohomological one. A partial answer to this question is obtained via computer calculations of the cohomology groups of the cocharacter lattices of all almost-simple semisimple root data of rank up to $8$. Using the theory of $\mathbf{FI}$-modules, these computations are used to determine the cohomology of the mod 2 reduction of the coroot lattices for type $A$ and all ranks.
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Solleveld, Maarten Sander. "Periodic cyclic homology of affine Hecke algebras." [S.l. : Amsterdam : s.n.] ; Universiteit van Amsterdam [Host], 2007. http://dare.uva.nl/document/45002.
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Oblomkov, Alexei. "Double affine Hecke algebras and noncommutative geometry." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/31165.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 93-96).
In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t,tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras.
by Alexei Oblomkov.
Ph.D.
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Alharbi, Badr. "Representations of Hecke algebra of type A." Thesis, University of East Anglia, 2013. https://ueaeprints.uea.ac.uk/48674/.
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We give some new results about representations of the Hecke algebra HF,q(Sn) of type A. In the first part we define the decomposition numbers dλν to be the composition multiplicity of the irreducible module Dν in the Specht module Sλ. Then we compute the decomposition numbers dλν for all partitions of the form λ = (a, c, 1b) and ν 2–regular for the Hecke algebra HC,−1(Sn). In the second part, we give some examples of decomposable Specht modules for the Hecke algebra HC,−1(Sn). These modules are indexed by partitions of the form (a, 3, 1b), where a, b are even. Finally, we find a new family of decomposable Specht modules for FSn when char(F) = 2.
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Dave, Ojas. "Irreducible Modules for Yokonuma-Type Hecke Algebras." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc862800/.
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Yokonuma-type Hecke algebras are a class of Hecke algebras built from a Type A construction. In this thesis, I construct the irreducible representations for a class of generic Yokonuma-type Hecke algebras which specialize to group algebras of the complex reflection groups and to endomorphism rings of certain permutation characters of finite general linear groups.
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Zhao, Peng. "Quantum Variance of Maass-Hecke Cusp Forms." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1243916906.
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Tzanev, Kroum. "C*-algèbres de Hecke et K-théorie." Paris 7, 2000. http://www.theses.fr/2000PA077229.
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Soit g un groupe discret et un sous-groupe de g. On dit que est presque normal dans g si pour tout g , g le sous-groupe gg 1 est d'indice fini dans. On dit egalement dans ce cas que (g,) est un couple de hecke. De tels couples se manifestent naturellement en geometrie. Par exemple en resolvant l'equation lineaire ordinaire uz + (1 u) z = 0 on voit apparaitre des revetements presque galoisiens, c'est-a-dire des revetements connexes de la forme p : x b tels que l'image de 1(x) dans 1(b) par p * soit un sous-groupe presque normal. A un couple de hecke (g,) on associe naturellement l'algebre de hecke c(g,). Cette algebre est involutive et possede plusieurs completions naturelles, comme l'algebre de banach l 1(g,), les c*-algebres c* (g,) et c* (g,), ou encore l'algebre de von neumann l(g,). On remarque a quelle point le passage du cadre des groupes discrets aux couples de hecke discrets est non trivial quand on etudie l'algebre l(g,). Alors que l(g) ne peut etre que de type i fini ou ii 1, l'algebre de von neumann l(g,) peut-etre (factorielle) de tout les types, y compris iii avec , 0,1. Au niveau des c*-algebres on montre que c* (g,) et c* (g,) coincident si et seulement si g/ est moyennable au sens d'eymard. Pour analyser de maniere plus precise la structure de ces algebres on etudie leur k-theorie en construisant un analogue de la fleche de baum et connes, ainsi qu'une k-theorie topologique k t o p *(g,). Celle-ci est calculable par des methodes isues de la topologie algebrique classique. La construction de cette fleche : k t o p *(g,) k * (c* $$(g,)) est une adaptation de la construction de n. Higson et j. Roe de l'indice co-uniforme. On espere que cette construction permettra de calculer k * (c* (g,)) = k * (c* $$(g,)) dans le cas moyennable.
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Ratliff, Leah J. "The alternating hecke algebra and its representations." Connect to full text, 2007. http://hdl.handle.net/2123/1698.
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Thesis (Ph. D.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2007.Title from title screen (viewed 13 January 2009). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.
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Ratliff, Leah Jane. "The alternating Hecke algebra and its representations." Thesis, The University of Sydney, 2007. http://hdl.handle.net/2123/1698.
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The alternating Hecke algebra is a q-analogue of the alternating subgroups of the finite Coxeter groups. Mitsuhashi has looked at the representation theory in the cases of the Coxeter groups of type A_n, and B_n, and here we provide a general approach that can be applied to any finite Coxeter group. We give various bases and a generating set for the alternating Hecke algebra. We then use Tits' deformation theorem to prove that, over a large enough field, the alternating Hecke algebra is isomorphic to the group algebra of the corresponding alternating Coxeter group. In particular, there is a bijection between the irreducible representations of the alternating Hecke algebra and the irreducible representations of the alternating subgroup. In chapter 5 we discuss the branching rules from the Iwahori-Hecke algebra to the alternating Hecke algebra and give criteria that determine these for the Iwahori-Hecke algebras of types A_n, B_n and D_n. We then look specifically at the alternating Hecke algebra associated to the symmetric group and calculate the values of the irreducible characters on a set of minimal length conjugacy class representatives.
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Ratliff, Leah Jane. "The alternating Hecke algebra and its representations." University of Sydney, 2007. http://hdl.handle.net/2123/1698.
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Doctor of PhilosophyThe alternating Hecke algebra is a q-analogue of the alternating subgroups of the finite Coxeter groups. Mitsuhashi has looked at the representation theory in the cases of the Coxeter groups of type A_n, and B_n, and here we provide a general approach that can be applied to any finite Coxeter group. We give various bases and a generating set for the alternating Hecke algebra. We then use Tits' deformation theorem to prove that, over a large enough field, the alternating Hecke algebra is isomorphic to the group algebra of the corresponding alternating Coxeter group. In particular, there is a bijection between the irreducible representations of the alternating Hecke algebra and the irreducible representations of the alternating subgroup. In chapter 5 we discuss the branching rules from the Iwahori-Hecke algebra to the alternating Hecke algebra and give criteria that determine these for the Iwahori-Hecke algebras of types A_n, B_n and D_n. We then look specifically at the alternating Hecke algebra associated to the symmetric group and calculate the values of the irreducible characters on a set of minimal length conjugacy class representatives.
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Ciappara, Joshua. "Hecke Category Actions via Smith–Treumann Theory." Thesis, The University of Sydney, 2022. https://hdl.handle.net/2123/30025.
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Let G be a connected reductive group with simply connected derived subgroup. By applying work of Riche–Williamson in the fledgling area of Smith–Treumann theory, this thesis presents a proof of a conjecture due to the former authors: there is a monoidal action of the diagrammatic Hecke category on the principal block of G. The thesis also contains background sections motivating the importance of the conjecture for modular representation theory and explaining the machinery needed for the proof (for instance, the equivariant derived category of constructible étale sheaves on a scheme). Suggestions for future research are outlined at the end.
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Brown, Keith. "Properly stratified quotients of quiver Hecke algebras." Thesis, University of East Anglia, 2017. https://ueaeprints.uea.ac.uk/63647/.
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Introduced in 2008 by Khovanov and Lauda, and independently by Rouquier, the quiver Hecke algebras are a family of infinite dimensional graded algebras which categorify the negative part of the quantum group associated to a graph. Infinite types these algebras are known to have nice homological properties, in particular they are affine quasi-hereditary. In this thesis we utilise the affine quasi-hereditary structure to create finite dimensional quotients which preserve some of the homological structure of the original algebra.
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Vankov, Kirill. "Algèbres de Hecke, séries génératices et applications." Grenoble 1, 2008. http://www.theses.fr/2008GRE10246.
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Le résultat principal dans le travail présenté est le calcul explicite de la série génératrice des opérateurs de Hecke dans l'algèbre de Hecke locale pour les groupes symplectiques de genre 3 et 4. L'algorithme est basé sur l'isomorphisme de Satake, qui permet de réaliser toutes les opérations dans l'algèbre des polynômes à plusieurs variables. C'est la première fois que cette expression est calculée pour le genre 4. Pour obtenir le résultat principal, une méthode de calcul symbolique a été développée. Cette approche algorithmique s'applique à d'autres types de séries de Hecke. En particulier, nous formulons et prouvons un analogue du Lemme de Rankin pour le genre 2. Nous avons aussi calculé les séries génératrices des carrés symétriques et des cubes symétriques. Se basant sur nos résultats nous formulons une conjecture de modularité pour les convolutions des fonctions L spineurs associées aux formes modulaires de Siegel. Nous considérons d'autres conjectures importantes liées aux formes modulaires de Siegel et à leurs fonctions L. Nous utilisons ces constructions pour calculer les facteurs algébriques rationnels aux valeurs critiques de la fonction L spineur attachée à F12 de Miyawaki. A notre connaissance c'est le premier exemple d'une fonction L-spineur de forme parabolique de Siegel de degré 3, dont certaines valeurs spéciales peuvent être calculées explicitement. Finalement, nous appliquons la théorie des algèbres de Hecke pour construire des cryptosystèmes algébriques sur ensembles finis de classes à gauches dans l'algèbre de Hecke. Nous utilisons une relation entre les classes à gauches et les points sur certains variétés algébriques projectivesThe main result in presented work consists of explicit computation of the generating power series of Hecke operators in local Hecke algebra for the symplectic groups of genus 3 and 4. The computation algorithm is based on the Satake isomorphism, which allows to carry out all operations in the algebra of polynomials in multiple variables. This is the first time when this expression was computed in genus 4. In order to obtain the main result, the method of symbolic computation was developed. This algorithmic approach is also applied to other types of Hecke series. In particular, we formulate and prove the analog of Rankin's Lemma in higher genus. We also computed the symmetric squares and symmetric cubes generating series. Based on our computational results we formulate a modularity lifting conjecture for convolutions of L-functions attached to Siegel modular forms. We review other important conjectures related to Siegel modular forms and their L-functions. We use these constructions to compute the rational algebraic factors in critical values of the spinor L-function attached to F12 of Miyawaki. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree 3, when the special values can be computed explicitly. Finally, we apply the theory of Hecke algebras to constructions of algebraic cryptosystems on some finite sets of left cosets in Hecke algebra. We use a relation between left cosets and points on certain projective algebraic varieties
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Heyer, Claudius [Verfasser], Elmar [Gutachter] Große-Klönne, Peter [Gutachter] Schneider, and Fabian [Gutachter] Januszewski. "Applications of parabolic Hecke algebras: parabolic induction and Hecke polynomials / Claudius Heyer ; Gutachter: Elmar Große-Klönne, Peter Schneider, Fabian Januszewski." Berlin : Humboldt-Universität zu Berlin, 2019. http://d-nb.info/1190641402/34.
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Couillens, Michèle. "Généralisation parabolique des polynômes de Kazhdan-Lusztig." Paris 7, 1995. http://www.theses.fr/1995PA077181.
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On considère un système de coxeter, un sous-système parabolique, et les deux algèbres de Hecke correspondantes. Pour tout caractère de degré un de l'algèbre de Hecke parabolique, on considère le module induit de l'algèbre de Hecke parabolique à la grande algèbre de Hecke. Ce module possède une base standard et deux bases de Kazhdan-Lusztig (invariantes par un automorphisme involutif du module). Il en résulte la définition, pour chaque caractère linéaire de l'algèbre de Hecke parabolique, De deux familles de polynomes de Kazhdan-Lusztig, dont on étudie les diverses propriétés : existence et unicité, symétrie, dualité, formules de récurrence, calcul en utilisant la notion de sous-expression distinguée d'une expression réduite d'un élément du groupe de Coxeter. Cela généralise une construction faite par V. Deodhar et qui correspondrait aux deux cas extrèmes des caractères indice et signe. Enfin, on établit des formules donnant l'action de l'algèbre de Hecke sur les modules induits en terme de bases de Kazhdan-Lusztig. Dans une deuxième partie, on donne une interprétation géométrique de cette situation dans le cas du groupe de Weyl d'un groupe réductif et du module induit de la représentation indice : par une méthode inspirée par les travvaux de Mars et Springer, on interprète l'action d'un élément de la base de Kazhdan-Lusztig de l'algèbre de Hecke sur un élélment de la base de Kazhdan-Lusztig de ce module comme un produit de faisceaux pervers sur le groupe réductif. Cela redonne, par une méthode directe, l'interprétation géométrique classique de la famille de polynomes associée au caractère signe
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Schroll, Sibylle. "Sur la dualité d'Alvis-Curtis et les groupes linéaires." Paris 13, 2003. http://www.theses.fr/2003PA132004.
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M. Cabanes et J. Richard ont posé la conjecture que la dualité de caractères d'Alvis-Curtis d'un groupe réductif devrait induire une équivalence homotopique que l'on obtiendrait grâce au produit tensoriel par un complexe X. Je démontre que la conjecture est vraie pour GL (2,q) si l divise q-1 et l impair en démontrant un théorème de réduction. Je la vérifie aussi pour certains blocs principaux de GL(n,q). Puis je construis un complexe H de bimodules d'algèbres de Hecke qui se spécialise au complexe de Coxeter diagonalement induit. Je montre que l'homologie de H est concentrée en un seul degré, je calcule sa structure de bimodule et je démontre que H induit une équivalence dérivée. Je considère ensuite le groupe GL(n,q),l ne divise pas q. Si on multiplie X par un idempotent ƒ, alors Xƒ est un complexe de q-algèbres de Schur et Xƒ induit une auto-équivalence dérivée de cette algèbre. Je montre que les complexes eXƒe et H sont isomorphes, en appliquant le foncteur de Schur e à Xƒ.
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Dezélée, Charlotte. "Représentations d'algèbres de Cherednik rationnelles et d'algèbres de Hecke graduées généralisées." Brest, 2003. http://www.theses.fr/2003BRES2022.
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On considère l'algèbre de Cherednik rationnelle h-(k) associée à la donnée d'un espace euclidien aR, d'un système de racines réduit R C aR* (de groupe de Weyl W) et d'une multiplicité k : R C. On utilise sa réalisation dans End(C[a]) via les opérateurs de Dunkl, qui permet de démontrer un théorème de Poincaré-Birkhoff-Witt. On étudie d'abord sa sous-algèbre sphérique (comparée à l'image de l'opérateur de composante radiale dans le cas d'une paire symétrique) et sa sous-algèbre des W-invariants (en type A2, une famille finie de générateurs est donnée). Principalement, on donne un critère d'existence de H(k)-modules de Verma de dimension finie. En rang 2, les multiplicités pour lesquelles de tels modules existent sont calculées. Enfin on généralise la notion d'algèbre de Hecke graduée pour une sous-algèbre remarquable de H(k), en types B ou D. Dans chacun de ces cas, on développe une théorie des représentations analogue (paramétrage de Langlands, modules de la série principale, critère d'irréductibilité)Let H(k) be the rational Cherednik algebra associated to an euclidean space aR*, a reduced roots system R C aR (with associated Weyl group W) and a W-invariant function k : R C. We use its realisation in End(C[a]) by Dunkl's operators. This provides a demonstration of a Poincaré-Birkhoff- Witt Theorem. We first study the spherical sub-algebra (compared to the image of the radial part operator for a symetric pair) and the W-invariants' sub-algebra H(k)w (for which we give a set of generators, in type A2). Essentially, we give a criterion for the existence of finite dirnensional H(k)-Verma modules. We determine all the parameters k for which such modules exist in rank 2. We generalize the structure of graded Hecke algebra to study a classical sub-algebra of H(k), in type B or D. In each case, we determine an analogous representation theory (Langlands' parameters, principal series representations , irreducibility criterion)
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Rassemusse, Genet Gwenaelle. "Inclusion d'algèbres de Hecke et nombres de décomposition." Phd thesis, Université Paris-Diderot - Paris VII, 2004. http://tel.archives-ouvertes.fr/tel-00006398.
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Cette thèse comporte trois parties. Dans la première, nous nous intéressons à la formule du commutateur d'un groupe admettant une BN-paire scindée. Nous montrons que sous une condition dite "condition de Lévi faible", le groupe vérifie cette formule. Dans la seconde partie, nous étudions la conservation de la forme unitriangulaire lors du passage d'une matrice de décomposition d'un module sur une algèbre graduée à la matrice de décomposition de la restriction de ce module sur l'algèbre effectuant la graduation et vice-versa. Nous verrons des applications pour des algèbres cellulaires pourvues également d'autres propriétés, notamment des algèbres de Ariki-Koike. Nous terminons par une partie traitant de la conjecture de J. Gruber et G. Hiss pour les nombres de décomposition des algèbres de Hecke de type B et D. Nous généralisons et prouvons cette conjecture dans le cas des algèbres de groupes de réflexions complexes. Puis nous observons quels sont les problèmes de la généralisation des méthodes utilisées lors du passage des algèbres de groupes aux algèbres de Hecke (de type B et D). Enfin, nous donnons une condition naturelle sur des filtrations de modules de Specht, sous-laquelle la conjecture est satisfaite.
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Wang, Yingnan, and 王英男. "Trace formulas and their applications on Hecke eigenvalues." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2012. http://hub.hku.hk/bib/B48329526.
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The objective of the thesis is to investigate the trace formulas and their applications on Hecke eigenvalues, especially on the distribution of Hecke eigenvalues. This thesis is divided into two parts..
In the first part of the thesis, a review is firstly carried out for the equidistribution of Hecke eigenvalues as primes vary and for the expected size of the error term in this equidistribution problem. Then the Kuznetsov trace formula is applied to prove a result on the size of the error term in the asymptotic distribution formula of Hecke eigenvalues. These eigenvalues become equidistributed with respect to the p-adic Plancherel measures as Hecke eigenforms vary. Next, this problem is generalized to Satake parameters of GL2 representations with prescribed supercuspidal local representations. Such a generalization is novel to the case of classical automorphic forms. To achieve this result, a trace formula of Arthur-Selberg type with a couple of key refinements is used.
In the second part of the thesis, a density theorem is proved which counts the number of exceptional nontrivial zeros of a family of symmetric power L-functions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of Elliott-Montgomery-Vaughan type for primitive Maass forms is established. The density theorem and large sieve inequality have many applications. For instance, they are used to prove statistical results on Hecke eigenvalues of primitive Maass forms and the extreme values of the symmetric power L-functions attached to primitive Maass forms.published_or_final_version
Mathematics
Doctoral
Doctor of Philosophy
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Gehles, Katrin Eva. "Properties of Cherednik algebras and graded Hecke algebras." Thesis, University of Glasgow, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.433167.
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Ivrissimtzis, Ioniis Panagioti. "Congruence subgroups of Hecke groups and regular dessins." Thesis, University of Southampton, 1998. https://eprints.soton.ac.uk/50645/.
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In this thesis we deal with dessins, that is tessellations of orientable surfaces, or from another point of view, two-cell embeddings of graphs on orientable surfaces. Our approach uses the connections of dessins with the Hecke groups Hq, and emphasizes the number-theoretic aspects of these connections. In Chapter 1 we deal with the modular group F, the simplest of the Hecke groups. We study the relations between the dessins associated with the principal congruence subgroups of F, the cosets of the special congruence subgroups of F, and the arithmetic of the finite ring Zjy. Our examples include well-known regular dessins as the icosahedron and the dessin {3, 7}s on Klein's surface of genus 3. In Chapter 2 we give some basic results on the Hecke groups and the closely related maximal real cyclotomic fields, concentrating on the factorization of the integers inside these fields. In Chapter 3 we extend the work of Chapter 1 to the other Hecke groups, especially the quadratic Hecke groups. The examples include regular dessins as the cube, the dodecahedron, the small stellated dodecahedron, and {4,5}6 on Bring's curve of genus 4. In Chapter 4 we find representations for the Hecke groups and their quotients by the principal congruence subgroups, and we use the results to do some necessary calculations. In Chapter 5, using the results of Chapter 1 as motivation, we reduce the problem of calculating the normaliser of certain subgroups of the Hecke groups into solving a system of congruences, and we solve the corresponding systems for F, H4, H6. Then, using another method we calculate the normaliser of these subgroups in PSZ^R) f°r the cases iJ4, H6, and we also calculate the corresponding quotients.
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Gonzalez-Lorca, Jorge. "Serie de drinfeld, monodromie et algebres de hecke." Paris 11, 1998. http://www.theses.fr/1998PA112155.
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Cette these est composee de quatre parties : dans la partie 1, on developpe des resultats concernant la serie de drinfeld. Cela consiste a etudier certaines solutions d'une equation differentielle lineaire du premier ordre du type de fuchs. Celles-ci s'expriment en fonction de certaines fonctions, appelees polylogarithmes. Ainsi les coefficients de cette serie sont certaines valeurs de polylogarithmes liees fortement a la fonction zeta de riemann multiple (ou nombres d'euler/zagier mixtes). La partie 2 est consacree a l'etude des systemes differentiels de knizhnik-zamolodchikov k z#n. On y introduit une certaine categorie monoidale stricte dont les morphismes t sont des arbres binaires a branches multiples. A chaque t on associe une solution f#t(z) de k z#n. Par decomposition de t on prouve qu'une solution de k z#n s'exprime comme le produit de deux fonctions qui sont solutions, l'une d'un systeme k z#n##1, deduit de k z#n, et l'autre d'une certaine equation differentielle du premier ordre. Cela etablit une compatibilite entre les solutions de k z#n et celles de k z#m (m < n) d'ordre inferieur. On fait aussi la liaison avec la monodromie universelle de k z#n en termes de cette categorie. Dans la partie 3, on etudie en detail une autre presentation, plus symetrique et directement sur les p-fonctions de riemann, des resultats classiques a propos de la fonction hypergeometrique classique. La partie 4 est consacree entierement a la construction des isomorphismes entre l'algebre de hecke du type a et l'algebre de groupe du groupe symetrique. Pour cela, on considere un systeme k z#n specialise a valeurs dans l'algebre de groupe du groupe symetrique, et ensuite on determine explicitement la monodromie correspondante. Ainsi, c'est cette monodromie, apres conjugaisons du type local, qui nous fournit un isomorphisme generique.
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Shaplin, Richard Martin III. "Spherical Elements in the Affine Yokonuma-Hecke Algebra." Thesis, Virginia Tech, 2020. http://hdl.handle.net/10919/99307.
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In Chapter 1 we introduce the Yokonuma-Hecke Algebra and a Yokonuma-Hecke Algebra-module. In Chapter 2 we determine that the possible eigenvalues of particular elements in the Yokonuma-Hecke Algebra acting on the module. In Chapter 3 we find determine module subspaces and eigenspaces that are isomorphic. In Chapter 4 we determine the structure of the q-eigenspace. In Chapter 5 we determine the spherical elements of the module.Master of Science
The Yokonuma-Hecke Algebra-module is a vector space over a particular field. Acting on vectors from the module by any element of the Yokonuma-Hecke Algebra corresponds to a linear transformation. Then, for each element we can find eigenvalues and eigenvectors. The transformations that we are considering all have the same eigenvalues. So, we consider the intersection of all the eigenspaces that correspond to the same eigenvalue. I.e. vectors that are eigenvectors of all of the elements. We find an algorithm that generates a basis for said vectors.
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Rassemusse-Genet, Gwenaëlle. "Inclusion d'algèbres de Hecke et nombres de décomposition." Paris 7, 2004. https://tel.archives-ouvertes.fr/tel-00006398.
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Johnson, Matthew Leander. "A Classification of all Hecke Eigenform Product Identities." Diss., The University of Arizona, 2012. http://hdl.handle.net/10150/228631.
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In this dissertation, we give a complete classification and list all identities of the form h = fg, where f , g and h are Hecke eigenforms of any weight with respect to Γ₁(N). This result extends the work of Ghate [Gha02] who considered this question for eigenforms with respect to Γ₁(N), with N square-free and f and g of weight 3 or greater. We remove all restrictions on the level N and the weights of f and g. For N = 1 there are only 16 eigenform identities, which are classically known. We first give a new proof of the level N = 1 case. We then give a proof which classifies all such eigenform identities for all levels N > 1. The identities fall into two categories. There are two infinite families of identities, given in Table 7.2. There are 209 other identities, listed (up to conjugacy) in Table 7.1. Thus any eigenform identity h = f g with respect to Γ₁(N) is either conjugate to an identity in Table 7.1 or takes the form of an identity described in Table 7.2.