Journal ArticleDOI
Representations of Coxeter Groups and Hecke Algebras.
David Kazhdan,George Lusztig +1 more
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In this article, the problem of decomposing this space of functions into irreducible representations of a finite Chevalley group G(Fq) is equivalent to decomposing the regular representation o f ~ | | (12) of a Coxeter group.Abstract:
here l(w) is the length of w In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functions on the flag manifold of the corresponding finite Chevalley group G(Fq) (see [loc cit, Ex 24]) Therefore, the problem of decomposing this space of functions into irreducible representations of G(Fq) is equivalent to the problem of decomposing the regular representation o f ~ | (12 It is known that, in this case, | is isomorphic to the group algebra of W; however, in general, this isomorphism cannot be defined without introducing a square root of q (see [1]) It is therefore, natural to extend the ground ring of ~ as follows For any Coxeter group (W, S) we define the Hecke algebra ~ to be J{' | A, where A is the ring of Laurent polynomials with integral coefficients in the indeterminate ql/2 Our purpose is to construct representations oL,Uf endowed with a special basis They will be defined in terms of certain graphs We define a W-graph to be a set of vertices X, with a set Y of edges (an edge is a subset of X consisting of two elements) together with two additional data: for each vertex xeX , we are given a subset I x of S and, for each ordered pair of vertices y, x such that {y, x} e Y, we are given an integer p(y, x) +0 These data are subject to the requirements (10a), (10b) below Let E beread more
Citations
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Journal ArticleDOI
Canonical bases arising from quantized enveloping algebras
TL;DR: In this paper, the problem of constructing bases of U+ as a Q(v) vector space has been studied, and a class of bases of PBW type has been given.
Journal ArticleDOI
Koszul Duality Patterns in Representation Theory
TL;DR: In this paper, Soergel et al. showed that the block of the Bernstein-Gelfand-gelfand category O that corresponds to any fixed central character is a Koszul ring and the dual of that ring governs a certain subcategory of the category O again.
Book
The geometry and topology of Coxeter groups
TL;DR: The authors introduced the theory of Coxeter groups in the context of the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua University on May 10, 2013.
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The Art of Computer Programming
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
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Moduln mit einem höchsten Gewicht
TL;DR: In this article, the wichtigsten Eigenschaften der Moduln with einem hochsten Gewicht fur unsere halbeinfache, zerfallende Lie-Algebra g referiert and allgemein die Kategorie O betrachtet.
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A class of irreducible representations of a Weyl group
TL;DR: In this paper, the authors describe a class f/w of irreducible representations of W. This class arises naturally in the study of representations of a reductive group over a finite field.
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Hecke algebras and characters of parabolic type of finite group with $(B, N)$-pairs
TL;DR: In this paper, the authors present a set of conditions générales d'utilisation of commercial or impression systématique, i.e., the copie ou impression de ce fichier doit contenir la présente mention de copyright.
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On the degrees and rationality of certain characters of finite Chevalley groups
C. T. Benson,C. W. Curtis +1 more
TL;DR: In this article, the authors consider a system of finite groups with (5, N)-pairs, with Coxeter system (W, R) and set of characteristic powers {q} (see [4]).