Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory (original) (raw)
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Affine Hecke algebras and generalized standard Young tableaux
Journal of Algebra, 2003
This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine Hecke algebra on a specific basis in the representation space. The indexing sets for weight spaces are generalizations of standard Young tableaux and the construction of the irreducible calibrated affine Hecke algebra modules is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type.
Representations of A-type Hecke algebras
arXiv: Quantum Algebra, 2009
We review some facts about the representation theory of the Hecke algebra. We adapt for the Hecke algebra case the approach of Okounkov and Vershik which was developed for the representation theory of symmetric groups. We justify an explicit construction of the idempotents in the Hecke algebra in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented.
Classification of graded Hecke algebras for complex reflection groups
Commentarii Mathematici Helvetici, 2003
The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups.
Calibrated representations of affine Hecke algebras
2004
This paper introduces the notion of calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The main results are that (1) irreducible calibrated representations are indexed by placed skew shapes, (2) the dimension of an irreducible calibrated representation is the number of standard Young tableaux corresponding to the placed skew shape and (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of each generator of the affine Hecke algebra on a specific basis in the representation space. This construction is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type.
A generic algebra associated to certain Hecke algebras
2004
We initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain generic Hecke algebra, infinite-dimensional in general, coming from the universal enveloping algebra of gl n (or sl n). The endomorphism algebras and the generic algebras are cellular (in the latter case, of profinite type in the sense of R.M. Green). We give several equivalent descriptions of these algebras, find a number of explicit bases, and describe indexing sets for their irreducible representations.
Lie algebras and degenerate Affine Hecke Algebras of type A
Eprint Arxiv Q Alg 9710037, 1997
We construct a family of exact functors from the Bernstein-Gelfand-Gelfand category O of sl n-modules to the category of finite-dimensional representations of the degenerate affine Hecke algebra H β of GL β. These functors transform Verma modules to standard modules or zero, and simple modules to simple modules or zero. Any simple H β-module can be thus obtained.
Local and global methods in representations of Hecke algebras
Science China-mathematics, 2017
This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. The authors first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. They prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells. Contents 1. Introduction. 1 2. Localization of integral quasi-hereditary algebras (QHAs). 3 3. Stratified algebras and their localizations. 10 4. Some Morita equivalences. 15 5. The Hecke algebras at good primes. 19 6. Bad primes and standardly stratified algebras.
Hecke algebras, π_{π}π π_{π}, and the Donald-Flanigan conjecture for π_{π}
Transactions of the American Mathematical Society, 1997
The DonaldβFlanigan conjecture asserts that the integral group ring Z G \mathbb {Z}G of a finite group G G can be deformed to an algebra A A over the power series ring Z [ [ t ] ] \mathbb {Z}[[t]] with underlying module Z G [ [ t ] ] \mathbb {Z}G[[t]] such that if p p is any prime dividing # G \#G then A β Z [ [ t ] ] F p ( ( t ) ) Β― A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))} is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of C G . \mathbb {C}G. We prove this for G = S n G = S_{n} using the natural representation of its Hecke algebra H \mathcal {H} by quantum Yang-Baxter matrices to show that over Z [ q ] \mathbb {Z}[q] localized at the multiplicatively closed set generated by q q and all i q 2 = 1 + q 2 + q 4 + β― + q 2 ( i β 1 ) , i = 1 , 2 , β¦ , n i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n , the Hecke algebra becomes a direct sum of total matric algebras. The correspo...
Modular Representations and Branching Rules for Wreath Hecke Algebras
International Mathematics Research Notices, 2010
We introduce a generalization of degenerate affine Hecke algebra, called wreath Hecke algebra, associated to an arbitrary finite group G. The simple modules of the wreath Hecke algebra and of its associated cyclotomic algebras are classified over an algebraically closed field of any characteristic p β₯ 0. The modular branching rules for these algebras are obtained, and when p does not divide the order of G, they are further identified with crystal graphs of integrable modules for quantum affine algebras. The key is to establish an equivalence between a module category of the (cyclotomic) wreath Hecke algebra and its suitable counterpart for the degenerate affine Hecke algebra. Contents 1. Introduction 1 2. Definition and properties of the wreath Hecke algebra 3 3. An equivalence of module categories 8 4. Classification of simple modules and modular branching rules 12 5. Cyclotomic wreath Hecke algebras and crystals 16 References 22 Partially supported by NSF grant DMS-0800280.