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189
Gaussian groups and Garside groups, two generalisations of Artin groups
 Proc. London Math. Soc
, 1998
"... It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. ..."
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Cited by 85 (19 self)
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It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups.
Coxeter arrangements
 Proceedings of Symposia in Pure Mathematics 40
, 1983
"... Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the ..."
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Cited by 61 (6 self)
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Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D (m) (A) = {θ ∈ DerS  θ(αH) ∈ Sα m H}. The module is known to be a free Smodule of rank ℓ by K. Saito (1975) for m = 1 and L. SolomonH. Terao (1998) for m = 2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D (m) (A). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m − 1)h/2) + mi(1 ≤ i ≤ ℓ) (when m is odd). Here m1 ≤ · · · ≤ mℓ are the exponents of G and h = mℓ + 1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G.) Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.
Quantum affine algebras and their representations, preprint
, 1994
"... Abstract. We prove a highest weight classification of the finitedimensional irreducible representations of a quantum affine algebra, in the spirit of Cartan’s classification of the finitedimensional irreducible representations of complex simple Lie algebras in terms of dominant integral weights. W ..."
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Cited by 56 (10 self)
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Abstract. We prove a highest weight classification of the finitedimensional irreducible representations of a quantum affine algebra, in the spirit of Cartan’s classification of the finitedimensional irreducible representations of complex simple Lie algebras in terms of dominant integral weights. We also survey what is currently known about the structure of these representations. 1.
Affine Weyl groups, discrete dynamical systems and Painlevé equations
"... Abstract. A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlevé type are discussed. ..."
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Cited by 51 (12 self)
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Abstract. A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlevé type are discussed.
Flat pencils of metrics and Frobenius manifolds., ArXiv: math.DG/9803106
 In: Proceedings of 1997 Taniguchi Symposium ”Integrable Systems and Algebraic Geometry”, editors M.H.Saito, Y.Shimizu and K.Ueno
, 1998
"... Abstracts This paper is based on the author’s talk at 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumpt ..."
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Cited by 35 (6 self)
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Abstracts This paper is based on the author’s talk at 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold M appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space L(M). This elucidates the relations between Frobenius manifolds and integrable hierarchies. 1
Degenerations Of Flag And Schubert Varieties To Toric Varieties
 Transformation Groups
, 1996
"... . In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In ..."
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Cited by 32 (4 self)
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. In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In this paper, we carry out the proof of the results announced in [21]. Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k. Fix a maximal torus T in G, a Borel subgroup B oe T . Let W be the Weyl group of G relative to T . Let Q ' B be a parabolic subgroup of classical type, say Q = T r i=1 P k i , where P k i , 1 i r, is a maximal parabolic subgroup of classical type (see [26] for the definition of a parabolic subgroup of classical type). Let W (Q) be the Weyl group of Q. For w 2 W=W (Q), let X(w)(= BwQ (mod Q) with the canonical reduced structure of a scheme) denote the Schubert variety in G=Q, associated to w. Given m = (m 1 ; : : : ; m ...
A decomposition of the descent algebra of a finite Coxeter group
 J. Algebraic Combin
, 1992
"... The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [5], [4] and [10]) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent ..."
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Cited by 30 (3 self)
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The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [5], [4] and [10]) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/WJ, where the WJ are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra. 1.
Extended affine Weyl groups and Frobenius manifolds
 Compositio Math
, 1998
"... Abstract. For the root system of type Bl and Cl, we generalize the result of [5] by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of [5]. 1. ..."
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Cited by 30 (7 self)
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Abstract. For the root system of type Bl and Cl, we generalize the result of [5] by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of [5]. 1.
Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory
"... We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, IwahoriHecke algebras of types A, B, and D, the complex reflection groups G(r; p; n) and the corresponding cyclotomic Hecke algebras H r;p;n , can be obtained, in all c ..."
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Cited by 29 (2 self)
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We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, IwahoriHecke algebras of types A, B, and D, the complex reflection groups G(r; p; n) and the corresponding cyclotomic Hecke algebras H r;p;n , can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to ane Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.