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1
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.
4
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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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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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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Stoica, Emanuel (Emanuel I. ). "Unitary representations of rational Cherednik algebras and Hecke algebras." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/64606.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 49-50).
We begin the study of unitary representations in the lowest weight category of rational Cherednik algebras of complex reflection groups. We provide the complete classification of unitary representations for the symmetric group, the dihedral group, as well as some additional partial results. We also study the unitary representations of Hecke algebras of complex reflection groups and provide a complete classification in the case of the symmetric group. We conclude that the KZ functor defined in [16] preserves unitarity in type A. Finally, we formulate a few conjectures concerning the classification of unitary representations for other types and the preservation of unitarity by the KZ functor and the restriction functors defined in [2].
by Emanuel Stoica.
Ph.D.
8
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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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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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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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
15
Casbi, Elie. "Categorifications of cluster algebras and representations of quiver Hecke algebras." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7032.
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Cette thèse porte sur l’étude de diverses conséquences des résultats de catégorifications monoïdales d'algèbres amassées par les algèbres de Hecke carquois, établis dans les travaux de Kang-Kashiwara-Kim-Oh [69]. Nous nous intéresserons en particulier à trois aspects de cette théorie: en premier lieu celui de la combinatoire, puis de la géométrie polytopale, et enfin celui de la théorie des représentations géométrique. Nous étudierons tout d'abord certaines relations combinatoires entre objets de nature a priori différentes: d'une part, les g-vecteurs au sens de Fomin-Zelevinsky, et d'autre part les partitions de racines qui paramétrisent les représentations simples de dimension finie des algèbres de Hecke carquois de type fini. Ces relations proviennent directement de certaines compatibilités remarquables entre différents ordres partiels naturels issus respectivement de la théorie des algèbres amassées et de la théorie des représentations. Nous montrons l'existence de telles relations dans le cas d'algèbres de Hecke carquois de type A_n. Nous établissons également une expression explicite pour les partitions de racines associées aux modules déterminantaux qui catégorifient une graine standard particulière de C[N]. La deuxième partie de cette thèse est consacrée à la construction de polytopes de Newton-Okounkov en utilisant de manière naturelle la théorie des représentations des algèbres de Hecke carquois. Nous commencerons par étendre les résultats de la partie précédente au cas d'algèbres de Hecke carquois de tout type (fini) simplement lacé, et ce grâce aux récents résultats de Kashiwara-Kim [72]. Ceci joue un rôle important dans la preuve de plusieurs propriétés combinatoires et géométriques de ces polytopes. Nous montrons ainsi que les volumes de certains de ces polytopes sont reliés à des formules des équerres (colorée) issues de la théorie combinatoire des éléments complètement commutatifs des groupes de Weyl. Enfin, nous étudierons les modules déterminantaux catégorifiant les graines standard de C[N] à l’aide d'une notion géométrique a priori non reliée à la théorie des algèbres de Hecke carquois ni aux algèbres amassées et appelée multiplicité équivariante, introduite par Joseph [63], Rossmann [108] et Brion [17]. Baumann-Kamnitzer-Knutson [6] ont récemment défini un morphisme d'algèbre D sur C[N] relié aux multiplicités équivariantes des cycles de Mirkovic-Vilonen via la correspondance de Satake géométrique. Nous montrons qu'en types A_n et D_4, l’évaluation de D sur les mineurs drapeaux de C[N] prend une forme distinguée, semblable aux valeurs prises par D sur les éléments de la base canonique duale correspondant aux modules fortement homogènes des algèbres de Hecke carquois selon la construction de Kleshchev-Ram [78]. Ceci soulève également la question de certaines propriétés de lissité des cycles MV correspondant aux mineurs drapeaux de C[N]. Nous mettons également en évidence certaines relations entre les images par D des mineurs drapeaux d'une même graine standard et nous montrons qu'en tous types ADE ces relations sont préservées par mutation d'une graine standard à une autreThe purpose of this thesis is to investigate various consequences of Kang-Kashiwara-Kim-Oh's monoidal categorifications of cluster algebras via quiver Hecke algebras [69]. We are interested in three different aspects of this theory: combinatorics, polytopal geometry, and geometric representation theory. We begin by studying some combinatorial relationships between objects of different natures: the g-vectors in the sense of Fomin-Zelevinsky on the one hand, and the root partitions parametrizing irreducible finite-dimensional representations of finite type quiver Hecke algebras on the other hand. These relationships arise from certain compatibilities between various natural partial orderings respectively coming from cluster theory and representation theory. We prove the existence of such relationships in the case of quiver Hecke algebras of type A_n. We also provide an explicit description of the root partitions associated to the determinantial modules categorifying a particular standard seed in C[N]. The second part of this thesis is devoted to constructing Newton-Okounkov polytopes in a natural way using the representation theory of quiver Hecke algebras. We begin by extending the results of the previous part to any (finite) simply-laced type using recent results of Kashiwara-Kim [72]. This plays a key role for proving several combinatorial and geometric properties of these polytopes. In particular, we show that the volumes of certain of these polytopes are related to (colored) hook formulae coming from the combinatorics of fully-commutative elements of Weyl groups. Finally, we study the determinantial modules categorifying the standard seeds of C[N] using certain a priori unrelated geometric tools, called equivariant multiplicities, introduced by Joseph [63], Rossmann [108] and Brion [17]. Baumann-Kamnitzer-Knutson [6] recently defined an algebra morphism on C[N] related to the equivariant multiplicities of Mirkovic-Vilonen cycles via the geometric Satake correspondence. We show that in types A_n and D_4, the evaluation of D on the flag minors of C[N] takes a distinguished form, similar to the values of D on the elements of the dual canonical basis corresponding to Kleshchev-Ram's [78] strongly homogeneous modules over quiver Hecke algebras. This also raises the question of certain smoothness properties of the MV cycles corresponding to the flag minors of C[N]. We also exhibit certain identities relating the images under D of the flag minors belonging to the same standard seed and we show that in any ADE type these relations are preserved under cluster mutation from one standard seed to another
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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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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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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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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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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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Black, Samson 1979. "Representations of Hecke algebras and the Alexander polynomial." Thesis, University of Oregon, 2010. http://hdl.handle.net/1794/10847.
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viii, 50 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 a certain quotient of the Iwahori-Hecke algebra of the symmetric group Sd , called the super Temperley-Lieb algebra STLd. The Alexander polynomial of a braid can be computed via a certain specialization of the Markov trace which descends to STLd. Combining this point of view with Ocneanu's formula for the Markov trace and Young's seminormal form, we deduce a new state-sum formula for the Alexander polynomial. We also give a direct combinatorial proof of this result.
Committee in charge: Arkady Vaintrob, Co-Chairperson, Mathematics Jonathan Brundan, Co-Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Dev Sinha, Member, Mathematics; Paul van Donkelaar, Outside Member, Human Physiology
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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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Speyer, Liron. "Representation theory of Khovanov-Lauda-Rouquier algebras." Thesis, Queen Mary, University of London, 2015. http://qmro.qmul.ac.uk/xmlui/handle/123456789/9114.
Full textAbstract:
This thesis concerns representation theory of the symmetric groups and related algebras. In recent years, the study of the “quiver Hecke algebras”, constructed independently by Khovanov and Lauda and by Rouquier, has become extremely popular. In this thesis, our motivation for studying these graded algebras largely stems from a result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which generalise the Hecke algebras of type A, and thus the group algebras of the symmetric groups. This has allowed the study of the graded representation theory of these algebras. In particular, the Specht modules for the Ariki–Koike algebras can be graded; in this thesis we investigate graded Specht modules in the KLR setting. First, we conduct a lengthy investigation of the (graded) homomorphism spaces between Specht modules. We generalise the rowand column removal results of Lyle and Mathas, producing graded analogues which apply to KLR algebras of arbitrary level. These results are obtained by studying a class of homomorphisms we call dominated. Our study provides us with a new result regarding the indecomposability of Specht modules for the Ariki–Koike algebras. Next, we use homomorphisms to produce some decomposability results pertaining to the Hecke algebra of type A in quantum characteristic two. In the remainder of the thesis, we use homogeneous homomorphisms to study some graded decomposition numbers for the Hecke algebra of type A. We investigate graded decomposition numbers for Specht modules corresponding to two-part partitions. Our investigation also leads to the discovery of some exact sequences of homomorphisms between Specht modules.
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Ruff, Oliver. "Completely splittable representations of symmetric groups and affine Hecke algebras /." view abstract or download file of text, 2005. http://wwwlib.umi.com/cr/uoregon/fullcit?p3190545.
Full textAbstract:
Thesis (Ph. D.)--University of Oregon, 2005.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 44-45). Also available for download via the World Wide Web; free to University of Oregon users.
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Miemietz, Vanessa. "On representations of affine Hecke algebras of type B." [S.l.] : [s.n.], 2005. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB12103647.
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Chlouveraki, Maria. "On the cyclotomic Hecke algebras of complex reflection groups." Paris 7, 2007. http://www.theses.fr/2007PA077083.
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Krawzik, Naomi. "Graded Hecke Algebras for the Symmetric Group in Positive Characteristic." Thesis, University of North Texas, 2020. https://digital.library.unt.edu/ark:/67531/metadc1707315/.
Full textAbstract:
Graded Hecke algebras are deformations of skew group algebras which arise from a group acting on a polynomial ring. Over fields of characteristic zero, these deformations have been studied in depth and include both symplectic reflection algebras and rational Cherednik algebras as examples. In Lusztig's graded affine Hecke algebras, the action of the group is deformed, but not the commutativity of the vectors. In Drinfeld's Hecke algebras, the commutativity of the vectors is deformed, but not the action of the group. Lusztig's algebras are all isomorphic to Drinfeld's algebras in the nonmodular setting. We find new deformations in the modular setting, i.e., when the characteristic of the underlying field divides the order of the group. We use Poincare-Birkhoff-Witt conditions to classify these deformations arising from the symmetric group acting on a polynomial ring in arbitrary characteristic, including the modular case.
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Alhaddad, Shemsi I. "Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups." Thesis, University of North Texas, 2006. https://digital.library.unt.edu/ark:/67531/metadc5235/.
Full textAbstract:
The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.
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Abubakar, Ahmed Bello. "The structure of symmetric group algebras at arbitrary characteristic." Thesis, University of East London, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.300326.
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Aiston, Anna Katherine. "Skein theoretic idempotents of Hecke algebras and quantum group invariants." Thesis, University of Liverpool, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.307662.
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Spencer, Matthew. "The representation theory of Iwahori-Hecke algebras with unequal parameters." Thesis, Queen Mary, University of London, 2014. http://qmro.qmul.ac.uk/xmlui/handle/123456789/8644.
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The Iwahori-Hecke algebras of finite Coxeter groups play an important role in many areas of mathematics. In this thesis we study the representation theory of the Iwahori-Hecke algebras of the Coxeter groups of type Bn and F4, in the unequal parameter case. We denote these algebras HQ and KQ respectively. This follows on from work done by Dipper, James, Murphy and Norton. We are interested in the Iwahori-Hecke algebras of type Bn and F4 since these are the only cases, apart from the dihedral groups, where the Coxeter generators lie in different conjugacy classes, and therefore the Iwahori-Hecke algebra can have unequal parameters. There are two parameters associated with these algebras, Q and q. Norton dealt with the case Q = q = 0, whilst Dipper, James and Murphy addressed the case q 6= 0 in type Bn. In this thesis we look at the case Q 6= 0; q = 0. We begin by constructing the simple modules for HQ, then compute the Ext quiver and find the blocks of HQ. We continue by observing that there is a natural embedding of the algebra of type n 1 in the algebra of type n, and this gives rise to the notion of an induced module. We look at the structure of the induced module associated with a given simple HQ-module. Here we are able to construct a composition series for the induced module and show that in a particular case the induced modules are self-dual. Finally, we look at KQ and find that the representation theory is related to representation theory of the Iwahori-Hecke algebra of type B3. Using this relationship we are able to construct the simple modules for KQ and begin the analysis of the Ext quiver.
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Graber, John Eric Goodman Frederick M. "Cellularity and Jones basic construction." Iowa City : University of Iowa, 2009. http://ir.uiowa.edu/etd/292.
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Li, Ge Jr. "Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier Algebras of type A." Thesis, The University of Sydney, 2012. http://hdl.handle.net/2123/8844.
Full textAbstract:
The main purpose of this thesis is to prove that the cyclotomic Khovanov-Lauda-Rouquier algebras of type A over Z are free by giving a graded cellular basis of the cyclotomic KLR algebra. We then extend it to obtain a graded cellular basis of the affine KLR algebra, which indicates that the affine KLR algebra is an affine graded cellular algebra. Finally we work with the Jucys-Murphy elements of the cyclotomic Hecke algebras of type A and proved a periodic property of these elements.
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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.
Full textAbstract:
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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Livesey, Daria. "High performance computations with Hecke algebras : bilinear forms and Jantzen filtrations." Thesis, University of Aberdeen, 2014. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=214835.
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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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Stoll, Friederike. "On the action of Ariki-Koike algebras on tensor space." [S.l. : s.n.], 2004. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB12168104.
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Dodd, Christopher Stephen. "Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/67788.
Full textAbstract:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 97-100).
In this thesis, we examine three different versions of "categorification" of the affine Hecke algebra and its periodic module: the first is by equivariant coherent sheaves on the Grothendieck resolution (and related objects), the second is by certain classes on bimodules over polynomial rings, called Soergel bimodules, and the third is by certain categories of constructible sheaves on the affine flag manifold (for the Langlands dual group). We prove results relating all three of these categorifications, and use them to deduce nontrivial equivalences of categories. In addition, our main theorem allows us to deduce the existence of a strict braid group action on all of the categories involved; which strengthens a theorem of Bezrukavnikov-Riche.
by Christopher Stephen Dodd.
Ph.D.
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Mak, Chi Kin School of Mathematics UNSW. "On complex reflection groups G(m, 1, r) and their Hecke algebras." Awarded by:University of New South Wales. School of Mathematics, 2003. http://handle.unsw.edu.au/1959.4/20777.
Full textAbstract:
We construct an algorithm for getting a reduced expression for any element in a complex reflection group G(m, 1, r) by sorting the element, which is in the form of a sequence of complex numbers, to the identity. Thus, the algorithm provides us a set of reduced expressions, one for each element. We establish a one-one correspondence between the set of all reduced expressions for an element and a set of certain sorting sequences which turn the element to the identity. In particular, this provides us with a combinatorial method to check whether an expression is reduced. We also prove analogues of the exchange condition and the strong exchange condition for elements in a G(m, 1, r). A Bruhat order on the groups is also defined and investigated. We generalize the Geck-Pfeiffer reducibility theorem for finite Coxeter groups to the groups G(m, 1, r). Based on this, we prove that a character value of any element in an Ariki-Koike algebra (the Hecke algebra of a G(m, 1, r)) can be determined by the character values of some special elements in the algebra. These special elements correspond to the reduced expressions, which are constructed by the algorithm, for some special conjugacy class representatives of minimal length, one in each class. Quasi-parabolic subgroups are introduced for investigating representations of Ariki- Koike algebras. We use n x n arrays of non-negative integer sequences to characterize double cosets of quasi-parabolic subgroups. We define an analogue of permutation modules, for Ariki-Koike algebras, corresponding to certain subgroups indexed by multicompositions. These subgroups are naturally corresponding, not necessarily one-one, to quasi-parabolic subgroups. We prove that each of these modules is free and has a basis indexed by right cosets of the corresponding quasi-parabolic subgroup. We also construct Murphy type bases, Specht series for these modules, and establish a Young's rule in this case.
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Lipp, Johannes. "Representations of Hecke algebras of Weyl groups of type A and B." [S.l. : s.n.], 2001. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB9600259.
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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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Hill, David Edward. "The Jantzen-Shapovalov form and Cartan invariants of symmetric groups and Hecke algebras /." view abstract or download file of text, 2007. http://proquest.umi.com/pqdweb?did=1400959351&sid=1&Fmt=2&clientId=11238&RQT=309&VName=PQD.
Full textAbstract:
Thesis (Ph. D.)--University of Oregon, 2007.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 107-108). Also available for download via the World Wide Web; free to University of Oregon users.
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Bernhardt, Karen 1977. "The generalized Harish-Chandra homomorphism for Hecke algebras of real reductive Lie groups." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/28922.
Full textAbstract:
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 73-74).
For complex reductive Lie algebras g, the classical Harish-Chandra homomorphism allows to link irreducible finite dimensional representations of g to those of certain subalgebras l. The Casselman-Osborne theorem establishes an extension of this link to infinite dimensional irreducible representations. In this paper we present a generalized Harish-Chandra homomorphism construction for Hecke algebras, and establish the corresponding generalized Casselman-Osborne theorem. This homomorphism can be used to link representations of (g, L n K)-pairs to those of (g, L n K)-pairs, where is a certain subalgebra of g as in the classical case. Since representations of such pairs are closely related to those of the underlying Lie group G, this construction is a good first approximation to lifting the Harish-Chandra homomorphism from the Lie algebra to the Lie group level.
by Karen Bernhardt.
S.M.
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Yu, Shona Huimin. "The Cyclotomic Birman-Murakami-Wenzl Algebras." Thesis, The University of Sydney, 2007. http://hdl.handle.net/2123/3560.
Full textAbstract:
This thesis presents a study of the cyclotomic BMW algebras, introduced by Haring-Oldenburg as a generalization of the BMW (Birman-Murakami-Wenzl) algebras related to the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. The motivation behind the definition of the BMW algebras may be traced back to an important problem in knot theory; namely, that of classifying knots (and links) up to isotopy. The algebraic definition of the BMW algebras uses generators and relations originally inspired by the Kauffman link invariant. They are intimately connected with the Artin braid group of type A, Iwahori-Hecke algebras of type A, and with many diagram algebras, such as the Brauer and Temperley-Lieb algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics, and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, Haring-Oldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the Ariki-Koike algebra h_{n,k} is a quotient of B_n^k, in the same way the Iwahori-Hecke algebra of type A is a quotient of the BMW algebra. In this thesis, we investigate the structure of these algebras and show they have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. In particular, they are shown to be R-free of rank k^n (2n-1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and Ariki-Koike algebras, one must impose extra so-called "admissibility conditions" on the parameters of the ground ring in order for these results to hold. This is due to potential torsion caused by the polynomial relation of order k imposed on one of the generators of B_n^k. It turns out that the representation theory of B_2^k is crucial in determining these conditions precisely. The representation theory of B_2^k is analysed in detail in a joint preprint with Wilcox in [45] (http://arxiv.org/abs/math/0611518). The admissibility conditions and a universal ground ring with admissible parameters are given explicitly in Chapter 3. The admissibility conditions are also closely related to the existence of a non-degenerate Markov trace function of B_n^k which is then used together with the cyclotomic Brauer algebras in the linear independency arguments contained in Chapter 4. Furthermore, in Chapter 5, we prove the cyclotomic BMW algebras are cellular, in the sense of Graham and Lehrer. The proof uses the cellularity of the Ariki-Koike algebras (Graham-Lehrer [16] and Dipper-James-Mathas [8]) and an appropriate "lifting" of a cellular basis of the Ariki-Koike algebras into B_n^k, which is compatible with a certain anti-involution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by Morton-Wasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and Hauschild-Mosley also attempt similar arguments to establish the freeness and diagram algebra results mentioned above. However, they withdrew their preprints ([14] and [15]), due to issues with their generic ground ring crucial to their linear independence arguments. A similar strategy to that proposed in [14], together with different trace maps and the study of rings with admissible parameters in Chapter 3, is used in establishing linear independency of our basis in Chapter 4. Since the submission of this thesis, new versions of these preprints have been released in which Goodman and Hauschild-Mosley use alternative topological and Jones basic construction theory type arguments to establish freeness of B_n^k and an isomorphism with the cyclotomic Kauffman Tangle algebra. However, they require their ground rings to be an integral domain with parameters satisfying the (slightly stronger) admissibility conditions introduced by Wilcox and the author in [45]. Also, under these conditions, Goodman has obtained cellularity results. Rui and Xu have also obtained freeness and cellularity results when k is odd, and later Rui and Si for general k, under the assumption that \delta is invertible and using another stronger condition called "u-admissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic Nazarov-Wenzl algebras and involve the construction of seminormal representations; their preprints have recently been released on the arXiv. It should also be noted there are slight differences between the definitions of cyclotomic BMW algebras and ground rings used, as explained partly above. Furthermore, Goodman and Rui-Si-Xu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the anti-involution axiom of the original Graham-Lehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.
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Yu, Shona Huimin. "The Cyclotomic Birman-Murakami-Wenzl Algebras." School of Mathematics and Statistics, 2007. http://hdl.handle.net/2123/3560.
Full textAbstract:
Doctor of PhilosophyThis thesis presents a study of the cyclotomic BMW algebras, introduced by Haring-Oldenburg as a generalization of the BMW (Birman-Murakami-Wenzl) algebras related to the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. The motivation behind the definition of the BMW algebras may be traced back to an important problem in knot theory; namely, that of classifying knots (and links) up to isotopy. The algebraic definition of the BMW algebras uses generators and relations originally inspired by the Kauffman link invariant. They are intimately connected with the Artin braid group of type A, Iwahori-Hecke algebras of type A, and with many diagram algebras, such as the Brauer and Temperley-Lieb algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics, and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, Haring-Oldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the Ariki-Koike algebra h_{n,k} is a quotient of B_n^k, in the same way the Iwahori-Hecke algebra of type A is a quotient of the BMW algebra. In this thesis, we investigate the structure of these algebras and show they have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. In particular, they are shown to be R-free of rank k^n (2n-1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and Ariki-Koike algebras, one must impose extra so-called "admissibility conditions" on the parameters of the ground ring in order for these results to hold. This is due to potential torsion caused by the polynomial relation of order k imposed on one of the generators of B_n^k. It turns out that the representation theory of B_2^k is crucial in determining these conditions precisely. The representation theory of B_2^k is analysed in detail in a joint preprint with Wilcox in [45] (http://arxiv.org/abs/math/0611518). The admissibility conditions and a universal ground ring with admissible parameters are given explicitly in Chapter 3. The admissibility conditions are also closely related to the existence of a non-degenerate Markov trace function of B_n^k which is then used together with the cyclotomic Brauer algebras in the linear independency arguments contained in Chapter 4. Furthermore, in Chapter 5, we prove the cyclotomic BMW algebras are cellular, in the sense of Graham and Lehrer. The proof uses the cellularity of the Ariki-Koike algebras (Graham-Lehrer [16] and Dipper-James-Mathas [8]) and an appropriate "lifting" of a cellular basis of the Ariki-Koike algebras into B_n^k, which is compatible with a certain anti-involution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by Morton-Wasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and Hauschild-Mosley also attempt similar arguments to establish the freeness and diagram algebra results mentioned above. However, they withdrew their preprints ([14] and [15]), due to issues with their generic ground ring crucial to their linear independence arguments. A similar strategy to that proposed in [14], together with different trace maps and the study of rings with admissible parameters in Chapter 3, is used in establishing linear independency of our basis in Chapter 4. Since the submission of this thesis, new versions of these preprints have been released in which Goodman and Hauschild-Mosley use alternative topological and Jones basic construction theory type arguments to establish freeness of B_n^k and an isomorphism with the cyclotomic Kauffman Tangle algebra. However, they require their ground rings to be an integral domain with parameters satisfying the (slightly stronger) admissibility conditions introduced by Wilcox and the author in [45]. Also, under these conditions, Goodman has obtained cellularity results. Rui and Xu have also obtained freeness and cellularity results when k is odd, and later Rui and Si for general k, under the assumption that \delta is invertible and using another stronger condition called "u-admissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic Nazarov-Wenzl algebras and involve the construction of seminormal representations; their preprints have recently been released on the arXiv. It should also be noted there are slight differences between the definitions of cyclotomic BMW algebras and ground rings used, as explained partly above. Furthermore, Goodman and Rui-Si-Xu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the anti-involution axiom of the original Graham-Lehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.
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Liu, Wille. "Double affine Hecke algebra of general parameters : perverse sheaves and Knizhnik--Zamolodchikov functor." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7144.
Full textAbstract:
Le présent travail de thèse porte sur l'étude de la catégorie O des algèbres de Hecke doublement affines dégénérées (dDAHA) au point de vue de la théorie de Springer et celle des faisceaux pervers. Dans les premiers deux chapitres nous étudions de manière algébrique les dDAHA et leurs généralisations, algèbres de Hecke doubles carquois (QDHA). Nous introduisons le foncteur de Knizhnik--Zamolodchikov (KZ) pour les QDHA et démontrons qu'ils vérifient la propriété bicommutante dans chapitre 2. Les chapitres 3 et 4 sont consacrés à l'étude des faisceaux pervers sur les algèbres de Lie munies de graduations cycliques et la théorie de Springer pour les dDAHA avec certaines familles de paramètres. Dans le chapitre 5, nous expliquons comment le foncteur KZ se réalise en termes de faisceaux pervers et nous montrons comment des structures plus fines sur la catégorie O se déduisent de l'analyse faisceautique sur les algèbres de Lie cycliquement graduéesThe present thesis work focuses on the study of the category O of degenerate double affine Hecke algebras (dDAHA) with the point of view of Springer theory and perverse sheaves. In the first two chapiters we study algebraically the dDAHAs and their generalisations, quiver double Hecke algebras (QDHA). We in introduce the Knizhnik--Zamolodchikov (KZ) functor for the QDHA and prove that it satisfies the double centraliser property in chapter 2. Chapters 3 and 4 are devoted to the study of perverse sheaves on a Lie algebra equipped with a cyclic grading and the Springer theory for the dDAHAs with certain families of parameters. In chapter 5, we explain how the KZ functor can be realised in terms of perverse sheaves and we show how finer structures on the category O can be deduced from the sheaf-theoretic analysis on cyclically graded Lie algebras
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Schmidt, Nicolas Alexander [Verfasser], Elmar Gutachter] Große-Klönne, Yuval Z. [Gutachter] [Flicker, and Ulrich [Gutachter] Görtz. "Generic pro-p Hecke algebras, the Hecke algebra of PGL(2, Z), and the cohomology of root data / Nicolas Alexander Schmidt ; Gutachter: Elmar Große-Klönne, Yuval Zvi Flicker, Ulrich Görtz." Berlin : Humboldt-Universität zu Berlin, 2019. http://d-nb.info/1197159886/34.
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Schmidt, Nicolas Alexander Verfasser], Elmar [Gutachter] Große-Klönne, Yuval Z. [Gutachter] [Flicker, and Ulrich [Gutachter] Görtz. "Generic pro-p Hecke algebras, the Hecke algebra of PGL(2, Z), and the cohomology of root data / Nicolas Alexander Schmidt ; Gutachter: Elmar Große-Klönne, Yuval Zvi Flicker, Ulrich Görtz." Berlin : Humboldt-Universität zu Berlin, 2019. http://d-nb.info/1197159886/34.
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