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Dissertations / Theses on the topic 'Hecke algebra'
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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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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.<br>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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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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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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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.<br>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 Philosophy<br>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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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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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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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.<br>Master of Science<br>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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Forsberg, Love. "Effective representations of Hecke-Kiselman monoids of type A." Thesis, Uppsala universitet, Algebra och geometri, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-174648.
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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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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/.
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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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Leclerc, Marc-Antoine. "The Hyperbolic Formal Affine Demazure Algebra." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/35218.
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In this thesis, we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malagón-López, Savage and Zainoulline in two directions. First, we introduce and study the notion of formal Demazure lattices of a Kac-Moody root system and show that the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra.
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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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Roberts, Jeremiah. "Complex and p-adic Hecke Algebra with Applications to SL(2)." OpenSIUC, 2020. https://opensiuc.lib.siu.edu/theses/2754.
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We discuss two versions of the Hecke algebra of a locally profinite group G, one that is complex valued and one that is p-adic valued. We reproduce several results which are well known for the complex valued Hecke algebra for the p-adic valued Hecke algebra. Specifically we show the equivalence of smooth representations of G and smooth modules of the Hecke algebra of G. We specialize to the group G=GLn(F) for F an extension of Qp, and show that the spherical Hecke algebra of G is finitely generated, and exhibit its generators. This is a standard fact for the complex valued Hecke algebra that we reproduce for the p-adic valued case. We then show that the spherical Hecke algebra of SLnF is isomorphic to a subalgebra of the spherical Hecke algebra of GLnF. Then a character of the spherical Hecke algebra ofGLn(F) can also be viewed as a character of the spherical Hecke algebra of SLn(F). Therefore such a character has two induced modules, one for the Hecke algebra of GLn(F) and another for the Hecke algebra of SLn(F). Theorem 3.4.3 and corollary 3.4.4give a condition under which the coinduced and induced modules of such a character areisomorphic as vector spaces.
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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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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.
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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 graduées<br>The 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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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.
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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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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/.
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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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Lawrence, Ruth Jayne. "Homology representations of braid groups." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236125.
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Fan, C. Kenneth (Chenteh Kenneth). "A Hecke algebra quotient and properties of commutative elements of a Weyl group." Thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/11578.
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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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Orellana, Rosa C. "The Hecke algebra of type B at roots of unity, Markov traces and subfactors /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1999. http://wwwlib.umi.com/cr/ucsd/fullcit?p9938595.
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Lin, Qiang Flach Matthias. "Bloch-Kato conjecture for the adjoint of H1(X0(N)) with integral Hecke algebra /." Diss., Pasadena, Calif. : California Institute of Technology, 2004. http://resolver.caltech.edu/CaltechETD:etd-11182003-084742.
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Schüler, Axel. "Klassifikation von bikovarianten Differentialkalkülen auf Quantengruppen." Doctoral thesis, Universitätsbibliothek Leipzig, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-218907.
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Unter der Voraussetzung, dass q keine Einheitswurzel ist und dass die Differentiale duij der Fundamentalmatrix den Linksmodul der 1-Formen erzeugen, werden die bikovarianten Differentialkalküle auf den Quantengruppen SLq(N), Oq(N) und Spq(N) klassifiziert. Es wird gezeigt, dass es auf den Quantengruppen SLq(N), N ≥ 3, abgesehen von eindimensionalen Kalkülen und endlich vielen Werten von q genau 2N bikovariante Differentialkalküle gibt. Diese Kalküle haben die Dimension N². Für die Quantengruppen Oq(N) und Spq(N), N ≥ 3, gibt es unter den genannten Voraussetzungen bis auf endlich viele Werte von q genau zwei bikovariante Differentialkalküle der Dimension N². Die Bimodulstruktur der Kalküle sowie die zugeordneten ad-invarianten Rechtsideale werden explizit angegeben. Für die Quantengruppen SLq(N), N ≥ 3, wird gezeigt, dass es, sofern q keine Einheitwurzel ist, genau 2N² + 2N bikovariante Bimoduln vom Typ (u^c u; f) gibt<br>If q is not a root of unity and under the assumption that the differentials duij of the fundamental matrix (uij) generate the left module of 1-forms, all bicovariant differential calculi on quantum groups SLq(N), Oq(N) and Spq(N) are classified. It is shown that on quantum groups SLq(N), N ≥ 3, except of 1-dimensional calculi and finitely many values of q, thre are exactly 2N bicovariant differential calculi. The space of invariant forms has dimension N². For quantum groups Oq(N) and Spq(N), N ≥ 3, under the same assumptions and up to finitely many values of q, there are exactly two bicovariant differential calculi of dimension N². The bimodule structure of the calculi as well as the corresponding ad-invariant right ideals are explicitely described. For quantum groups SLq(N), N ≥ 3, there are exactly 2N² + 2N bicovariant bimodules of type (u^c u; f) provided q is not a root of unity
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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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Graber, John Eric. "Cellularity and Jones basic construction." Diss., University of Iowa, 2009. https://ir.uiowa.edu/etd/292.
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This thesis establishes a framework for cellularity of algebras related to the Jones basic construction. The framework allows a uniform proof of cellularity of Brauer algebras, BMW algebras, walled Brauer algebras, partition algebras, and others. In this setting, the cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for this class of algebras, the cellular structures are compatible with restriction and induction of modules.
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Fujita, Ryo. "A geometric study of Dynkin quiver type quantum affine Schur-Weyl duality." Kyoto University, 2019. http://hdl.handle.net/2433/242573.
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Hebert, Auguste. "Études des masures et de leurs applications en arithmétique." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSES027/document.
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Les masures ont été introduites en 2008 par Gaussent et Rousseau afin d’étudier les groupes de Kac-Moody sur les corps locaux. Elles généralisent les immeubles de Bruhat-Tits. Dans cette thèse, j’étudie d’une part les propriétés des masures et d’autre part leurs applications en arithmétique et en théorie des représentations. Rousseau a donné une définition axiomatique des masures, inspirée par la définition de Tits des immeubles de Bruhat-Tits. Je propose une axiomatique plus simple et plus agréable à manipuler et je montre que mon axiomatique est équivalente à celle de Rousseau.Nous étudions (en collaboration avec Ramla Abdellatif) les algèbres de Hecke sphériques et d’Iwahori-Hecke introduites par Bardy-Panse, Gaussent et Rousseau. Nous démontrons que contrairement au cas réductif, le centre de leur algèbre d’Iwahori-Hecke est quasiment trivial, et n’est en particulier pas isomorphe à l’algèbre de Hecke sphérique. Nous introduisons donc une algèbre d’Iwahori-Hecke complétée, dont le centre est isomorphe à l’algèbre de Hecke sphérique. Nous associons aussi des algèbres de Hecke à des faces sphériques comprises entre 0 et l’alcôve fondamentale de la masure,généralisant la construction de Bardy-Panse, Gaussent et Rousseau de l’algèbre d’Iwahori-Hecke.La formule de Gindikin-Karpelevich est une formule importante dans la théorie des groupes réductifs sur les corps locaux. Récemment, Braverman,Garland, Kazhdan, et Patnaik ont généralisé cette formule au cas des groupes de Kac-Moody affines. Une partie importante de leur preuve consiste à montrer que cette formule est bien définie, c’est à dire que les nombres intervenants dans cette formule, qui sont les cardinaux de certains sous groupes de quotients du groupe étudié sont bien finis. Je démontre cette finitude dans le cas des groupes de Kac-Moody généraux. J’étudie aussi les distances sur une masure. Je montre qu’on ne peux pas avoir de distance ayant les mêmes propriétés que dans le cas réductif. Je construis des distances ayant des propriétés moins forte mais qui semblent intéressantes<br>Masures were introduced in 2008 by Gaussent and Rousseau in order to study Kac-Moody groups over local fields. They generalize Bruhat-Tits buildings. In this thesis, I study the properties of masures and the application of the theory of masures in arithmetic and representation theory. Rousseau gave an axiomatic of masures, inspired by the definition by Tits of Bruhat-Tits buildings. I propose an axiomatic, which is simpler and easyer to handle and I prove that my axiomatic is equivalent to the one of Rousseau. We study (in collaboration with Ramla Abdellatif) the spherical and Iwahori-Hecke algebras introduced by Bardy-Panse, Gaussent and Rousseau. We prove that on the contrary to the reductive case, the center of the Iwahori-Hecke algebra is almost trivial and is in particular not isomorphic to the spherical Hecke algebra. We thus introduce a completed Iwahori-Hecke algebra, whose center is isomorphic to the spherical Hecke algebra. We also associate Hecke algebras to spherical faces between 0 and the fundamental alcove of the masure, generalizing the construction of Bardy-Panse, Gaussent and Rousseau of the Iwahori-Hecke algebra.The Gindikin-Karpelevich formula is an important formula in the theory of reductive groups over local fields. Recently, Braverman, Garland, Kazhdanand Patnaik generalized this formula to the case of affine Kac-Moody groups. An important par of their prove consists in proving that this formula iswell-defined, which means that the numbers involved in this formula, which are the cardinals of certain subgroup of quotients of the studied subgroupare finite. I prove this finiteness in the case of general Kac-Moody groups.I also study distances on a masure. I prove that there is no distance having the same properties as in the reductive case. I construct distances having weaker properties, but which seem interesting
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Poulain, D'Andecy Loïc. "Algèbres de Hecke cyclotomiques : représentations, fusion et limite classique." Phd thesis, Aix-Marseille Université, 2012. http://tel.archives-ouvertes.fr/tel-00748920.
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Une approche inductive est développée pour la théorie des représentations de la chaîne des algèbres de Hecke cyclotomiques de type G(m,1,n). Cette approche repose sur l'étude du spectre d'une famille commutative maximale, formée par les analogues des éléments de Jucys-Murphy. Les représentations irréductibles, paramétrées par les multi-partitions, sont construites avec l'aide d'une nouvelle algèbre associative, dont l'espace vectoriel sous-jacent est le produit tensoriel de l'algèbre de Hecke cyclotomique avec l'algèbre associative libre engendrée par les multi-tableaux standards. L'analogue de cette approche est présentée pour la limite classique, c'est-à-dire la chaîne des groupes de réflexions complexes de type G(m,1,n). Dans une seconde partie, une base des algèbres de Hecke cyclotomiques est donnée et la platitude de la déformation est montrée sans utiliser la théorie des représentations. Ces résultats sont généralisés aux algèbres de Hecke affines de type A. Ensuite, une procédure de fusion est présentée pour les groupes de réflexions complexes et les algèbres de Hecke cyclotomiques de type G(m,1,n). Dans les deux cas, un ensemble complet d'idempotents primitifs orthogonaux est obtenu par évaluation consécutive d'une fonction rationnelle. Dans une troisième partie, une nouvelle présentation est obtenue pour les sous-groupes alternés de tous les groupes de Coxeter. Les générateurs sont reliés aux arêtes orientées du graphe de Coxeter. Cette présentation est ensuite étendue, pour tous les types, aux extensions spinorielles des groupes alternés, aux algèbres de Hecke alternées et aux sous-groupes alternés des groupes de tresses.
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Martino, Marcelo Gonçalves de. "On the unramified spherical automorphic spectrum." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4017/document.
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Cette thèse a deux résultats d'analyse harmonique sur des groupes réductifs. Soit G connexe et défini sur un corps de nombres F, A les adèles et K un sous-groupe compact maximal de G(A). On a étudié la décomposition de l'espace des fonctions de carré intégrable sur le l'espace quotient G(F)\G(A)/K, en tant que module sur une algèbre de Hecke global. Des résultats similaires que ceux obtenus ici ont été établies par divers auteurs pour de nombreux cas particuliers. La caractéristique principale de la présente approche réside dans le fait qu'il est uniforme. Cette approche a été inspirée par des résultats de G. Heckman et E. Opdam dans les problèmes spectraux pour les algèbre de Hecke graduée. Dans la démonstration, nous avons besoin d'un résultat par M. Reeder sur les espaces de poids des représentations (anti)sphériques de la série discrète de l’algèbre de Hecke affine, aussi, nous sommes confrontés au problème du calcul de certains constantes rationnelles dans le spectre global mesurer en termes de mesures de Plancherel locales.Pour le second résultat, nous montrons qu'un complexe de Coxeter et un immeuble euclidienne peuvent être dotés de fonctions de Morse PL qui permet d'écrire des contractions explicites des complexes cellulaires sous-jacents. Cette approche par la théorie de Morse pour étudier les immeubles de Bruhat-Tits a été inspiré par les idées de G. Savin et M. Bestvina dans le cas de l’immeuble de SL(n). Nous conjecturer que ces contractions ont de bonnes bornes sur leurs coefficients et peuvent donc être utilisés pour calculer les groupes Ext entre les représentations tempérée d'une manière analogue à celle qui a été fait par M. Solleveld et E. Opdam<br>This thesis contains two results on harmonic analysis of reductive groups. First, let G be connected and defined over a number field F, A be the ring of adèles and K be a maximal compact subgroup of G(A). We studied the decomposition of the space of square-integrable functions on the quotient G(F)\G(A)/K, as a module for a global Hecke algebra. Similar results than the ones obtained here have been established by various authors for many special cases of reductive groups. The main feature of the present approach is the fact that it is uniform. Such approach was greatly inspired by results of G. Heckman and E. Opdam in treating spectral problems for graded affine Hecke algebras. In the proof, we need a result by M. Reeder on the weight spaces of the (anti)spherical discrete series representations of affine Hecke algebras, as well as we are faced with the problem of computing certain rational constants factors involved in the global spectral measure in terms of local Plancherel measures which are known only in the affine Hecke algebra context. As for the second result, we show that a Coxeter complex and a Euclidean building can be endowed with piecewise linear Morse functions that allows one to write down explicit contractions of the underlying cell complexes. Such approach via PL Morse theory to study buildings was heavily inspired by ideas from G. Savin and M. Bestvina in the specific case of the building of SL(n). We conjecture that these contractions have nice bounds on their coefficients and thus can be used to compute Ext groups between tempered representations in an analogous way as was done by M. Solleveld and E. Opdam
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Laugerotte, Eric. "Combinatoire et calcul symbolique en théorie des représentations." Rouen, 1997. http://www.theses.fr/1997ROUES069.
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Ce mémoire concerne le traitement algorithmique des représentations matricielles. Les techniques y sont illustrées sur deux exemples, les algèbres de Hecke et les automates à multiplicités. Les algèbres de Hecke interviennent dans plusieurs domaines (dont l'algèbre ou la physique statistique) qui demandent de pouvoir y calculer efficacement. Ici sont rassemblés des algorithmes implémentés en Maple constituant la bibliothèque SHRI. Par l'action d'opérateurs de symétrisation sur des Q-Vandermonde, on détermine un système complet de représentations polynomiales. En calculant les polynômes minimaux de chaque bloc de la représentation régulière, on en déduit l'inverse d'un élément s'il existe. On construit une famille complète d'idempotents minimaux orthogonaux en évaluant les gz-polynômes en les q-analogues des éléments de Jucys-Murphy. Ces polynômes sont de degré minimal en la variable d'index maximal (la plus coûteuse). On en déduit un calcul des bases de Gelfand-Zetlin des modules irréductibles. Le caractère d'un élément de l'algèbre de Hecke est, grâce à un algorithme de conjugaison, une combinaison linéaire d'évaluations sur des produits de cycles calculées efficacement par une formule de J. Desarmenien généralisant la formule de Murnaghan-Nakayama. Une forme bilinéaire invariante permet d'expliciter les idempotents centraux via la formule de Kilmoyer. Le phénomène de compression spectrale observé lors de tests sur le package réside en la compression des deux paramètres formels de l'algèbre de Hecke générique en un seul par l'implémentation des isomorphismes semi-linéaires entre les algèbres de Hecke. Les calculs dans l'algèbre de Hecke générique sont alors plus efficaces. Dans une dernière partie, on établit, pour le cas non-commutatif, l'algorithme classique de minimisation des représentations linéaires des séries rationnelles dû à M. P. Schutzenberger. On montre comment calculer les isomorphismes d'automates minimaux.
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Amorós, Carafí Laia. "Images of Galois representations and p-adic models of Shimura curves." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/471452.
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The Langlands program is a vast and unifying network of conjectures that connect the world of automorphic representations of reductive algebraic groups and the world of Galois representations.
These conjectures associate an automorphic representation of a reductive algebraic group to every n-dimensional representation of a Galois group, and the other way around: they attach a Galois representation to any automorphic representation of a reductive algebraic group. Moreover, these correspondences are done in such a way that the automorphic L-functions attached to the two objects coincide.
The theory of modular forms is a field of complex analysis whose main importance lies on its connections and applications to number theory. We will make use, on the one hand, of the arithmetic properties of modular forms to study certain Galois representations and their number theoretic meaning. On the other hand, we will use the geometric meaning of these complex analytic functions to study a natural generalization of modular curves. A modular curve is a geometric object that parametrizes isomorphism classes of elliptic curves together with some additional structure depending on some modular subgroup. The generalization that we will be interested in are the so called Shimura curves. We will be particularly interested in their p-adic models.
In this thesis, we treat two different topics, one in each side of the Langlands program. In the Galois representations' side, we are interested in Galois representations that take values in local Hecke algebras attached to modular forms over finite fields. In the automorphic forms' side, we are interested in Shimura curves: we develop some arithmetic results in definite quaternion algebras and give some results about Mumford curves covering p-adic Shimura curves.
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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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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 bégayante<br>This 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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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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Borie, Nicolas. "Calcul des invariants de groupes de permutations par transformée de Fourier." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112294/document.
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Cette thèse porte sur trois problèmes en combinatoire algébrique effective et algorithmique.Les premières parties proposent une approche alternative aux bases de Gröbner pour le calcul des invariants secondaires des groupes de permutations, par évaluation en des points choisis de manière appropriée. Cette méthode permet de tirer parti des symétries du problème pour confiner les calculs dans un quotient de petite dimension, et ainsi d'obtenir un meilleur contrôle de la complexité algorithmique, en particulier pour les groupes de grande taille. L'étude théorique est illustrée par de nombreux bancs d'essais utilisant une implantation fine des algorithmes. Un prérequis important est la génération efficace de vecteurs d'entiers modulo l'action d'un groupe de permutation, dont l'algorithmique fait l'objet d'une partie préliminaire.La quatrième partie cherche à déterminer, pour un certain quotient naturel d'une algèbre de Hecke affine, quelles spécialisations des paramètres aux racines de l'unité donne un comportement non générique.Finalement, la dernière partie présente une conjecture sur la structure d'une certaine $q$-déformation des polynômes harmoniques diagonaux en plusieurs paquets de variables pour la famille infinie de groupes de réflexions complexes.Tous ces chapitres s'appuient fortement sur l'exploration informatique, et font l'objet de multiples contributions au logiciel Sage<br>This thesis concerns algorithmic approaches to three challenging problems in computational algebraic combinatorics.The firsts parts propose a Gröbner basis free approach for calculating the secondary invariants of a finite permutation group, proceeding by using evaluation at appropriately chosen points. This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups. The theoretical study is illustrated by extensive benchmarks using a fine implementation of algorithms. An important prerequisite is the generation of integer vectors modulo the action of a permutation group, whose algorithmic constitute a preliminary part of the thesis.The fourth part of this thesis is determining for a certain interesting quotient of an affine Hecke algebra exactly which root-of-unity specialization of its parameter lead to non-generic behavior.Finally, the last part presents a conjecture on the structure of certain q-deformed diagonal harmonics in many sets of variables for the infinite family of complex reflection groups.All chapters proceed widely by computer exploration, and most of established algorithms constitute contributions of the software Sage
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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.<br>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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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.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 49-50).<br>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].<br>by Emanuel Stoica.<br>Ph.D.
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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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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.<br>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.<br>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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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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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 autre<br>The 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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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.<br>Includes bibliographical references (p. 93-96).<br>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.<br>by Alexei Oblomkov.<br>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.