Results 1  10
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125
Branching rules for modular representations of symmetric groups
 J. London Math. Soc
, 1995
"... Let K be a field of characteristic p> 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first « — 1 letters, and D k the irreducible ATnmodule corresponding to a (/^regular) partition X of n. In [9] we described the socle of the restriction D x ..."
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Cited by 64 (14 self)
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Let K be a field of characteristic p> 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first « — 1 letters, and D k the irreducible ATnmodule corresponding to a (/^regular) partition X of n. In [9] we described the socle of the restriction D x [T and obtained a number of other results
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 23 (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
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 21 (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.
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 19 (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
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.
Minimal representations: spherical vectors and automorphic functionals, in Algebraic groups and arithmetic
, 2004
"... Abstract. In the first part of this paper we study minimal representations of simply connected simple split groups G of type Dk or Ek over local nonarchimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of groups over R and C, such ..."
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Cited by 17 (0 self)
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Abstract. In the first part of this paper we study minimal representations of simply connected simple split groups G of type Dk or Ek over local nonarchimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of groups over R and C, such a formula was obtained recently in [8]. We also use our techniques to study the structure of the space of smooth vectors in the minimal representation. In the second part we consider groups G as above defined over a global field K. In this situation we describe the form of the automorphic functional on the minimal representation of the corresponding adelic group.
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 17 (6 self)
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this paper describes the multiplicities of some composition factors in
Existence of lattices in KacMoody groups over finite fields
 Commun. Contemp. Math
, 1999
"... Let g be a Kac–Moody Lie algebra. We give an interpretation of Tits ’ associated group functor using representation theory of g and we construct a locally compact “Kac–Moody group ” G over a finite field k. Using (twin) BNpairs (G, B, N) and(G, B −,N)forG we show that if k is “sufficiently large”, ..."
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Cited by 17 (11 self)
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Let g be a Kac–Moody Lie algebra. We give an interpretation of Tits ’ associated group functor using representation theory of g and we construct a locally compact “Kac–Moody group ” G over a finite field k. Using (twin) BNpairs (G, B, N) and(G, B −,N)forG we show that if k is “sufficiently large”, then the subgroup B − is a nonuniform lattice in G. We have also constructed an uncountably infinite family of both uniform and nonuniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of nonuniform lattices in rank 2 is a spherical Tits system for G which we also construct. Keywords: 2000 Mathematics Subject Classification: Primary 20F32; Secondary 22F50
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 17 (10 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.