Results 1  10
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13
A path model for geodesics in Euclidean buildings and its applications to representation theory
, 2004
"... In this paper we give a combinatorial characterization of projections of geodesics in Euclidean buildings to Weyl chambers. We apply these results to the representation theory of complex reductive Lie groups and to spherical Hecke rings associated with split nonarchimedean reductive Lie groups. Our ..."
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Cited by 22 (6 self)
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In this paper we give a combinatorial characterization of projections of geodesics in Euclidean buildings to Weyl chambers. We apply these results to the representation theory of complex reductive Lie groups and to spherical Hecke rings associated with split nonarchimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SLn to other complex semisimple Lie groups. 1
Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity
 J. Differential Geometry
"... In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homogeneous linear inequalities, which we call the generalized trian ..."
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Cited by 18 (9 self)
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In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homogeneous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G/P for maximal parabolic subgroups P. The side lengths of polygons in Euclidean buildings are studied in the related paper [KLM2]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM3]. 1.
Structure of the tensor product semigroup
 Asian J. of Math
"... Abstract. We study the structure of semigroup T ens(G) consisting of triples of dominant weights (λ, µ, ν) of a complex reductive Lie group G such that (Vλ ⊗ Vµ ⊗ Vν) G � = 0. We prove two general structural results for T ens(G) and give an explicit computation of T ens(G) for G = Sp(4, C) and G = G ..."
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Cited by 7 (4 self)
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Abstract. We study the structure of semigroup T ens(G) consisting of triples of dominant weights (λ, µ, ν) of a complex reductive Lie group G such that (Vλ ⊗ Vµ ⊗ Vν) G � = 0. We prove two general structural results for T ens(G) and give an explicit computation of T ens(G) for G = Sp(4, C) and G = G2. 1.
Polygons with prescribed Gauss map in Hadamard spaces and Euclidean
"... We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration if the configuration is stable. Moreover, the same result holds for semistable configurations on arbitra ..."
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Cited by 3 (0 self)
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We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration if the configuration is stable. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings. 1 In the first section, we recall some background material on Hadamard spaces and Euclidean buildings, and we introduce the concepts needed to state and prove our Theorems. In particular, we define stability for weighted configurations on the boundary at infinity of a Hadamard space. In the second section, we introduce ultralimits and the special cases ultraproducts and asymptotic tubes which we use in our proofs. In the third section, we prove our results: Main Theorem. Let X be a Euclidean building and c a semistable weighted configuration on its boundary at infinity, or let X be a locally compact Hadamard space and c a stable weighted configuration on its boundary at infinity. Then the associated weak contraction Φc has a fixed point. In particular, there exists a polygon p in X such that c is a Gauss map for p. For a slightly more general statement in the case of a Hadamard space, see Corollary 3.9. As an immediate consequence, we can formulate the following classification of configurations which can occur as Gauss maps on Euclidean buildings and symmetric spaces: Corollary. Let X be a symmetric space of noncompact type or a Euclidean building, and let c be a weighted configuration on its boundary at infinity. Then there exists a polygon having this configuration as a Gauss map if and only if the configuration is semistable in the building case and nice semistable in the case of a symmetric space. Necessity of semistability, as well as the Theorem and the Corollary in the case where X is a symmetric space or a locally compact Euclidean building were shown in [KLM1],
SATURATION AND IRREDUNDANCY FOR SPIN(8)
"... Dedicated to F. Hirzebruch on the occasion of his eightieth birthday Abstract. We explicitly calculate the triangle inequalities for the group P SO(8), thereby explicitly solving the eigenvalues of a sum problem for this group (equivalently describing the sidelengths of geodesic triangles in the co ..."
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Cited by 2 (0 self)
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Dedicated to F. Hirzebruch on the occasion of his eightieth birthday Abstract. We explicitly calculate the triangle inequalities for the group P SO(8), thereby explicitly solving the eigenvalues of a sum problem for this group (equivalently describing the sidelengths of geodesic triangles in the corresponding symmetric space for the metric d ∆ with values in the Weyl chamber ∆). We then apply some computer programs to verify two basic questions/conjectures. First, we verify that the above system of inequalities is irredundant. Then, we verify the “saturation conjecture ” for the decomposition of tensor products of finitedimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights (λ, µ, ν) such that λ + µ + ν is in the root lattice, and any positive integer N, if and only if
The Eigencone and Saturation for Spin(8)
, 2006
"... Dedicated to F. Hirzebruch on the occasion of his eightieth birthday Abstract: We explicitly calculate the system of restricted triangle inequalities for the group P SO(8) given by BelkaleKumar, thereby explicitly solving the eigenvalues of a sum problem for this group (equivalently describing the ..."
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Cited by 2 (1 self)
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Dedicated to F. Hirzebruch on the occasion of his eightieth birthday Abstract: We explicitly calculate the system of restricted triangle inequalities for the group P SO(8) given by BelkaleKumar, thereby explicitly solving the eigenvalues of a sum problem for this group (equivalently describing the sidelengths of geodesic triangles in the corresponding symmetric space for the metric d ∆ with values in the Weyl chamber ∆). We then apply some computer programs to verify the saturation conjecture for the decomposition of tensor products of finitedimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights (λ, µ, ν) such that λ + µ + ν is in the root lattice, and any positive integer N, if and only if
Convex rank 1 subsets of Euclidean Buildings (of type A2)
, 2008
"... For a Euclidean building X of type A2, we classify the 0dimensional subbuildings A of ∂TX that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of A) is (essentially) sufficient. To prove this, we construct new ..."
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For a Euclidean building X of type A2, we classify the 0dimensional subbuildings A of ∂TX that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of A) is (essentially) sufficient. To prove this, we construct new convex subsets as the union of convex sets. 1 1
unknown title
, 2005
"... Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity ..."
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Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity
IDEAL TRIANGLES IN EUCLIDEAN BUILDINGS AND BRANCHING TO LEVI SUBGROUPS
"... Abstract. Let G denote a connected reductive group, defined and split over Z, and let M ⊂ G denote a Levi subgroup. In this paper we study varieties of geodesic triangles with fixed vectorvalued sidelengths α, β, γ in the BruhatTits buildings associated to G, along with varieties of ideal triangl ..."
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Abstract. Let G denote a connected reductive group, defined and split over Z, and let M ⊂ G denote a Levi subgroup. In this paper we study varieties of geodesic triangles with fixed vectorvalued sidelengths α, β, γ in the BruhatTits buildings associated to G, along with varieties of ideal triangles associated to the pair M ⊂ G. The ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex ξ such that other infinite side containing ξ has fixed “ideal length ” λ and the remaining finite side has fixed length µ. We establish an isomorphism between varieties in the second family and certain varieties in the first family (the pair (µ, λ) and the triple (α, β, γ) satisfy a certain relation). We apply these results to the study of the Hecke ring of G and the restriction homomorphism R ( G) → R ( M) between representation rings. We deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism. 1.