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Journal articles on the topic 'Hecke algebras'

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Published: 4 June 2021

Last updated: 11 March 2023

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1

Rouquier, Raphaël. "Quiver Hecke Algebras and 2-Lie Algebras." Algebra Colloquium 19, no. 02 (May 3, 2012): 359–410. http://dx.doi.org/10.1142/s1005386712000247.

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We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

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2

Yang, Guiyu, and Yanbo Li. "Standardly based algebras and 0-Hecke algebras." Journal of Algebra and Its Applications 14, no. 10 (September 2015): 1550141. http://dx.doi.org/10.1142/s0219498815501418.

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In this paper we prove that standardly based algebras are invariant under Morita equivalences. As an application, we prove 0-Hecke algebras and 0-Schur algebras are standardly based algebras. From this point of view, we give a new way to construct the simple modules of 0-Hecke algebras, and prove that the dimension of the center of a symmetric 0-Hecke algebra is not less than the number of its simple modules.

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3

Krieg, Aloys. "Hecke algebras." Memoirs of the American Mathematical Society 87, no. 435 (1990): 0. http://dx.doi.org/10.1090/memo/0435.

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4

Takebayashi, Tadayoshi. "Double affine Hecke algebras and elliptic Hecke algebras." Journal of Algebra 253, no. 2 (July 2002): 314–49. http://dx.doi.org/10.1016/s0021-8693(02)00055-8.

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5

Savage, Alistair. "Affine Wreath Product Algebras." International Mathematics Research Notices 2020, no. 10 (May 24, 2018): 2977–3041. http://dx.doi.org/10.1093/imrn/rny092.

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Abstract We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke–Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth.

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6

Liu, Wille. "Knizhnik–Zamolodchikov functor for degenerate double affine Hecke algebras: algebraic theory." Representation Theory of the American Mathematical Society 26, no. 30 (August 30, 2022): 906–61. http://dx.doi.org/10.1090/ert/614.

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In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.

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7

Opdam, Eric, and Maarten Solleveld. "Homological algebra for affine Hecke algebras." Advances in Mathematics 220, no. 5 (March 2009): 1549–601. http://dx.doi.org/10.1016/j.aim.2008.11.002.

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8

Rostam, Salim. "Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras." Advances in Mathematics 311 (April 2017): 662–729. http://dx.doi.org/10.1016/j.aim.2017.03.004.

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9

Poulain d'Andecy, L., and R. Walker. "Affine Hecke algebras and generalizations of quiver Hecke algebras of type B." Proceedings of the Edinburgh Mathematical Society 63, no. 2 (March 9, 2020): 531–78. http://dx.doi.org/10.1017/s0013091519000294.

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AbstractWe define and study cyclotomic quotients of affine Hecke algebras of type B. We establish an isomorphism between direct sums of blocks of these algebras and a generalization, for type B, of cyclotomic quiver Hecke algebras, which are a family of graded algebras closely related to algebras introduced by Varagnolo and Vasserot. Inspired by the work of Brundan and Kleshchev, we first give a family of isomorphisms for the corresponding result in type A which includes their original isomorphism. We then select a particular isomorphism from this family and use it to prove our result.

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10

Putcha, Mohan S. "Monoid Hecke algebras." Transactions of the American Mathematical Society 349, no. 9 (1997): 3517–34. http://dx.doi.org/10.1090/s0002-9947-97-01823-0.

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11

Marin, Ivan. "Infinitesimal Hecke algebras." Comptes Rendus Mathematique 337, no. 5 (September 2003): 297–302. http://dx.doi.org/10.1016/s1631-073x(03)00369-8.

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12

Berenstein, Arkady, and David Kazhdan. "Hecke-Hopf algebras." Advances in Mathematics 353 (September 2019): 312–95. http://dx.doi.org/10.1016/j.aim.2019.06.018.

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13

Etingof, Pavel, Wee Liang Gan, and Victor Ginzburg. "Continuous Hecke algebras." Transformation Groups 10, no. 3-4 (December 2005): 423–47. http://dx.doi.org/10.1007/s00031-005-0404-2.

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14

Wang, Kun, Shanshan Qi, Haitao Ma, and Zhujun Zheng. "Stability for Representations of Hecke Algebras of Type A." Mathematics 10, no. 1 (December 22, 2021): 32. http://dx.doi.org/10.3390/math10010032.

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In this paper, we introduce the concept of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence of the representations of Hecke algebras is representation stable.

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15

Orellana, R. C. "Weights of Markov traces on Hecke algebras." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 508 (March 12, 1999): 157–78. http://dx.doi.org/10.1515/crll.1999.508.157.

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Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

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16

Aubert, Anne-Marie, Paul Baum, Roger Plymen, and Maarten Solleveld. "HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS." Journal of the Institute of Mathematics of Jussieu 16, no. 2 (May 5, 2015): 351–419. http://dx.doi.org/10.1017/s1474748015000079.

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Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

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17

OGIEVETSKY, O. V., and L. POULAIN D'ANDECY. "ON REPRESENTATIONS OF CYCLOTOMIC HECKE ALGEBRAS." Modern Physics Letters A 26, no. 11 (April 10, 2011): 795–803. http://dx.doi.org/10.1142/s0217732311035377.

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An approach, based on Jucys–Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys–Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without using the representation theory.

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18

Hu, Jun, and Fang Li. "Modified affine Hecke algebras and quiver Hecke algebras of type A." Journal of Algebra 541 (January 2020): 219–69. http://dx.doi.org/10.1016/j.jalgebra.2019.10.001.

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19

Khongsap, Ta. "Hecke–Clifford algebras and spin Hecke algebras III: The trigonometric type." Journal of Algebra 322, no. 8 (October 2009): 2731–50. http://dx.doi.org/10.1016/j.jalgebra.2009.07.028.

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20

Palma, Rui. "On envelopingC⁎-algebras of Hecke algebras." Journal of Functional Analysis 264, no. 12 (June 2013): 2704–31. http://dx.doi.org/10.1016/j.jfa.2013.03.011.

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21

Levandovskyy, Viktor, and Anne V. Shepler. "Quantum Drinfeld Hecke Algebras." Canadian Journal of Mathematics 66, no. 4 (August 1, 2014): 874–901. http://dx.doi.org/10.4153/cjm-2013-012-2.

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AbstractWe consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

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22

Cho, Ilwoo. "Free probability on Hecke algebras and certain group C^{*}-algebras induced by Hecke algebras." Opuscula Mathematica 36, no. 2 (2016): 153. http://dx.doi.org/10.7494/opmath.2016.36.2.153.

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23

Rostam, Salim. "Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$." Transactions of the American Mathematical Society 371, no. 6 (November 16, 2018): 3877–916. http://dx.doi.org/10.1090/tran/7485.

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24

LANDSTAD, MAGNUS B., and NADIA S. LARSEN. "GENERALIZED HECKE ALGEBRAS AND C*-COMPLETIONS." International Journal of Mathematics 20, no. 01 (January 2009): 45–76. http://dx.doi.org/10.1142/s0129167x09005169.

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For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra [Formula: see text], which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection [Formula: see text] such that [Formula: see text] is isomorphic to [Formula: see text]. We study the structure and properties of C*-completions of the generalized Hecke algebra arising from this corner realisation, and via Morita–Fell–Rieffel equivalence, we identify, in some cases explicitly, the resulting proper ideals of [Formula: see text]. By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + b-groups and the Heisenberg group.

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25

Murin, Max, and Seth Shelley-Abrahamson. "Parameters for generalized Hecke algebras in type B." Journal of Algebra and Its Applications 18, no. 09 (July 17, 2019): 1950173. http://dx.doi.org/10.1142/s0219498819501731.

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The irreducible representations of full support in the rational Cherednik category [Formula: see text] attached to a Coxeter group [Formula: see text] are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in [Formula: see text] of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup [Formula: see text] in [Formula: see text] for [Formula: see text] and [Formula: see text].

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26

Savage, Alistair, and John Stuart. "Frobenius nil-Hecke algebras." Pacific Journal of Mathematics 311, no. 2 (July 31, 2021): 455–73. http://dx.doi.org/10.2140/pjm.2021.311.455.

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27

Gutkin, Eugene. "Representations of Hecke algebras." Transactions of the American Mathematical Society 309, no. 1 (January 1, 1988): 269. http://dx.doi.org/10.1090/s0002-9947-1988-0957070-0.

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28

James, Gordon. "Hecke algebras and immanants." Linear Algebra and its Applications 197-198 (January 1994): 659–70. http://dx.doi.org/10.1016/0024-3795(94)90508-8.

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29

Benson, David, Karin Erdmann, and Aram Mikaelian. "Cohomology of Hecke algebras." Homology, Homotopy and Applications 12, no. 2 (2010): 353–70. http://dx.doi.org/10.4310/hha.2010.v12.n2.a12.

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30

Du, Jie, Brian Parshall, and Leonard Scott. "Extending Hecke endomorphism algebras." Pacific Journal of Mathematics 279, no. 1-2 (December 21, 2015): 229–54. http://dx.doi.org/10.2140/pjm.2015.279.229.

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31

Malle, Gunter, and Andrew Mathas. "Symmetric Cyclotomic Hecke Algebras." Journal of Algebra 205, no. 1 (July 1998): 275–93. http://dx.doi.org/10.1006/jabr.1997.7339.

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32

Alvis, Dean. "Units in Hecke Algebras." Journal of Algebra 216, no. 2 (June 1999): 417–30. http://dx.doi.org/10.1006/jabr.1998.7801.

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33

Parkinson, James. "Buildings and Hecke algebras." Journal of Algebra 297, no. 1 (March 2006): 1–49. http://dx.doi.org/10.1016/j.jalgebra.2005.08.036.

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34

Witherspoon, Sarah. "Twisted graded Hecke algebras." Journal of Algebra 317, no. 1 (November 2007): 30–42. http://dx.doi.org/10.1016/j.jalgebra.2007.05.025.

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35

Boys, Clinton. "Alternating quiver Hecke algebras." Journal of Algebra 449 (March 2016): 246–63. http://dx.doi.org/10.1016/j.jalgebra.2015.10.020.

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36

Poulain d'Andecy, L., and R. Walker. "Affine Hecke algebras of type D and generalisations of quiver Hecke algebras." Journal of Algebra 552 (June 2020): 1–37. http://dx.doi.org/10.1016/j.jalgebra.2019.11.039.

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37

Khongsap, T. A., and Weiqiang Wang. "Hecke-Clifford Algebras and Spin Hecke Algebras I: The Classical Affine Type." Transformation Groups 13, no. 2 (June 2008): 389–412. http://dx.doi.org/10.1007/s00031-008-9012-2.

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38

Hivert, Florent, Anne Schilling, and Nicolas M. Thiéry. "Hecke group algebras as quotients of affine Hecke algebras at level 0." Journal of Combinatorial Theory, Series A 116, no. 4 (May 2009): 844–63. http://dx.doi.org/10.1016/j.jcta.2008.11.010.

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39

RAINBOLT, JULIANNE G. "WEYL GROUPS AND BASIS ELEMENTS OF HECKE ALGEBRAS OF GELFAND–GRAEV REPRESENTATIONS." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 849–64. http://dx.doi.org/10.1142/s0219498811005002.

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The initial section of this article provides illustrative examples on two ways to construct the Weyl group of a finite group of Lie type. These examples provide the background for a comparison of the elements in the Weyl groups of GL(n, q) and U(n, q) that are used in the construction of the standard bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q). Using a theorem of Steinberg, a connection between a theoretic description of bases of these Hecke algebras and a combinatorial description of these bases is provided. This leads to an algorithmic method for generating bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q).

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40

Lee, Kyu-Hwan. "Iwahori-Hecke Algebras of SL2 over 2-Dimensional Local Fields." Canadian Journal of Mathematics 62, no. 6 (December 14, 2010): 1310–24. http://dx.doi.org/10.4153/cjm-2010-072-x.

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AbstractIn this paper we construct an analogue of Iwahori–Hecke algebras of SL2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL2, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori–Matsumoto type relations.

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41

GECK, MEINOLF, and NICOLAS JACON. "OCNEANU'S TRACE AND STARKEY'S RULE." Journal of Knot Theory and Its Ramifications 12, no. 07 (November 2003): 899–904. http://dx.doi.org/10.1142/s0218216503002834.

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We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

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42

Doty, Stephen, Karin Erdmann, and Anne Henke. "A generic algebra associated to certain Hecke algebras." Journal of Algebra 278, no. 2 (August 2004): 502–31. http://dx.doi.org/10.1016/j.jalgebra.2004.04.007.

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43

Bowman, Chris, Anton Cox, Amit Hazi, and Dimitris Michailidis. "Path combinatorics and light leaves for quiver Hecke algebras." Mathematische Zeitschrift 300, no. 3 (September 29, 2021): 2167–203. http://dx.doi.org/10.1007/s00209-021-02829-0.

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AbstractWe recast the classical notion of “standard tableaux" in an alcove-geometric setting and extend these classical ideas to all “reduced paths" in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias–Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the “Bott–Samelson truncation" of the quiver Hecke algebra.

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44

Kaliszewski, S., Magnus B. Landstad, and John Quigg. "Hecke C*-Algebras, Schlichting Completions and Morita Equivalence." Proceedings of the Edinburgh Mathematical Society 51, no. 3 (October 2008): 657–95. http://dx.doi.org/10.1017/s0013091506001419.

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AbstractThe Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

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45

Reeks, Michael. "Cocenters of Hecke–Clifford and spin Hecke algebras." Journal of Algebra 476 (April 2017): 85–112. http://dx.doi.org/10.1016/j.jalgebra.2016.11.039.

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46

Hohnen, Winfried. "Period polynomials and congruences for Hecke algebras." Proceedings of the Edinburgh Mathematical Society 42, no. 2 (June 1999): 217–24. http://dx.doi.org/10.1017/s0013091500020204.

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Using the Eichler-Shimura isomorphism and the action of the Hecke operator T2 on period polynomials, we shall give a simple and new proof of the following result (implicitly contained in the literature): let f be a normalized Hecke eigenform of weight k with respect to the full modular group with eigenvalues λp under the usual Hecke operators Tp (p a prime). Let Kf be the field generated over Q by the λp for all p. Let p be a prime of Kf lying above 5. Then

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47

Chlouveraki, Maria. "Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)." Nagoya Mathematical Journal 197 (March 2010): 175–212. http://dx.doi.org/10.1215/00277630-2009-004.

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The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series G(de, e, r), thus completing their calculation for all complex reflection groups.

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48

Chlouveraki, Maria. "Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)." Nagoya Mathematical Journal 197 (March 2010): 175–212. http://dx.doi.org/10.1017/s0027763000009880.

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The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

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49

Takebayashi, Tadayoshi. "Elliptic Hecke algebras and modified Cherednik algebras." Proceedings of the Japan Academy, Series A, Mathematical Sciences 79, no. 4 (April 2003): 80–84. http://dx.doi.org/10.3792/pjaa.79.80.

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50

Chari, Vyjayanthi, and Andrew Pressley. "Quantum affine algebras and affine Hecke algebras." Pacific Journal of Mathematics 174, no. 2 (June 1, 1996): 295–326. http://dx.doi.org/10.2140/pjm.1996.174.295.

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