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156
Branching rules for modular representations of symmetric groups III: Some . . .
 J. LONDON MATH. SOC
, 1996
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The affine permutation groups of rank three
 Proc. London Math. Soc
, 1987
"... Introduction and statement of results Finite primitive permutation groups of rank 3 have been the subject of much study in the past twenty years, leading on the one hand to interesting new groups (for instance, some sporadic groups) and on the other to new techniques in the theory of permutation gro ..."
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Cited by 28 (0 self)
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Introduction and statement of results Finite primitive permutation groups of rank 3 have been the subject of much study in the past twenty years, leading on the one hand to interesting new groups (for instance, some sporadic groups) and on the other to new techniques in the theory of permutation groups. It is readily seen that if G is a primitive rank 3
Nilpotent orbits in good characteristic and the Kempf–Rousseau theory
 J. Algebra
, 2003
"... Abstract. Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p> 0, g = LieG, and suppose that p is a good prime for the root system of G. In this paper, we give a fairly short conceptual proof of Pommerening’s theorem [27, 28] which states that a ..."
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Cited by 24 (5 self)
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Abstract. Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p> 0, g = LieG, and suppose that p is a good prime for the root system of G. In this paper, we give a fairly short conceptual proof of Pommerening’s theorem [27, 28] which states that any nilpotent element in g is Richardson in a distinguished parabolic subalgebra of the Lie algebra of a Levi subgroup of G. As a byproduct, we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent Gorbits in g (for earlier proofs of this theorem see [15, 36, 24]). We extend recent results of Sommers [35] to reductive Lie algebras of good characteristics thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in g and unipotent elements in G. Earlier computations of these groups in positive characteristics relied, mostly, on work of Mizuno [20, 21]. Our approach is based on the theory of optimal parabolic subgroups for Gunstable vectors, also known as the Kempf–Rousseau theory, which provides a good substitute for the sl(2)theory prominent in the characteristic zero case. 1.
KostkaFoulkes polynomials and Macdonald spherical functions
 in Surveys in Combinatorics 2003, C. Wensley ed., London Math. Soc. Lect. Notes 307 Camb
, 2003
"... Abstract. Generalized HallLittlewood polynomials (Macdonald spherical functions) and generalized KostkaFoulkes polynomials (qweight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different defin ..."
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Cited by 22 (4 self)
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Abstract. Generalized HallLittlewood polynomials (Macdonald spherical functions) and generalized KostkaFoulkes polynomials (qweight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from “scratch”, in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Schützenberger’s positive formula for the KostkaFoulkes poynomials in the type A case. The classical theory of HallLittlewood polynomials and the KostkaFoulkes polynomials appears in the monograph of I.G. Macdonald [Mac]. The HallLittlewood polynomials form a basis of the ring of symmetric functions and the KostkaFoulkes polynomials are the entries of the transition matrix between the HallLittlewood polynomials and the Schur functions.
Schubert cells and cohomology of the space G/P
 31 Fomin S.V. and Kirillov A.N., Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in Geometry, 147182, Progress in Math. 172
, 1973
"... We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and Ρ a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the oth ..."
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Cited by 22 (0 self)
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We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and Ρ a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the other consists in identifying the cohomology of G/P with certain polynomials on the Lie algebra of the Cartan subgroup Η of G. The results obtained are used to describe the algebraic action of the Weyl group W of G on the cohomology of G/P. Contents
Group representations and lattices
 J. Amer. Math. Soc
, 1990
"... This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Sh I, Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irred ..."
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Cited by 20 (1 self)
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This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Sh I, Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irreducibility hypotheses on a rational representation V of a finite group G, then the Gstable Euclidean lattices in V are severely restricted. Unfortunately, these hypotheses are rarely satisfied when V is absolutely irreducible over Q. But there are many examples where the ring End G ( V) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the MordellWeillattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves. In § I we discuss lattices and Hermitian forms on T/, and in §§24 the strong irreducibility hypotheses we wish to make. In § 5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean Z[G]lattices
A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical
 TRANSFORM. GROUPS
, 1997
"... Let G be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroup P of G acts on its unipotent radical Pu, or on pu, the Lie algebra of Pu, with only a finite number of orbits. The proof proceeds in two parts. First we obtain a red ..."
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Cited by 20 (11 self)
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Let G be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroup P of G acts on its unipotent radical Pu, or on pu, the Lie algebra of Pu, with only a finite number of orbits. The proof proceeds in two parts. First we obtain a reduction to the case of general linear groups. In a second step, a solution for these is achieved by studying the representation theory of a particular quiver with certain relations. Furthermore, for the general linear groups we obtain a combinatorial formula for the number of orbits in the finite cases.
On Decomposition Numbers And Branching Coefficients For Symmetric And Special Linear Groups
 Proc. London Math. Soc.(3) 75
, 1997
"... this paper describes the multiplicities of some composition factors in ..."
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Cited by 20 (6 self)
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this paper describes the multiplicities of some composition factors in
Generic transfer for general spin groups
 Duke Math. J
"... Abstract. We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be selfdual. Together with cases of classical groups, t ..."
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Cited by 19 (9 self)
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Abstract. We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be selfdual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose Lgroups have classical derived groups. The important transfer from GSp 4 to GL4 follows from our result as a special case. 1.
Parametrizations of flag varieties
"... Abstract. For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (settheoretical) crosssection φ: G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w0 in the Weyl group W. It assigns to any gB areprese ..."
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Cited by 18 (1 self)
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Abstract. For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (settheoretical) crosssection φ: G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w0 in the Weyl group W. It assigns to any gB arepresentative g ∈ G together with a factorization into simple root subgroups and simple reflections. The crosssection φ is continuous along the components of Deodhar’s decomposition of G/B. We introduce a generalization of the Chamber Ansatz and give formulas for the factors of g = φ(gB). These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety (G/B)≥0 defined by Lusztig, giving a new proof of Lusztig’s conjectured cell decomposition of (G/B)≥0. We also give minimal sets of inequalities describing these cells. 1.