The Experts below are selected from a list of 42015 Experts worldwide ranked by ideXlab platform
Dongho Moon - One of the best experts on this subject based on the ideXlab platform.
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Reduction formulae of Littlewood–Richardson coefficients
Advances in Applied Mathematics, 2011Co-Authors: Soojin Cho, Dongho MoonAbstract:AbstractThere are two well-known reduction formulae by Griffiths–Harris for Littlewood–Richardson coefficients. Our observation is that some special cases of the factorization theorem of Littlewood–Richardson coefficients by King, Tollu and Toumazet give reduction formulae including the Griffiths–Harris formulae. We provide explicit statements of those reduction formulae in more general forms, and extend them to their conjugated forms also. Eight useful reduction formulae deleting one or two rows (columns) of each partition are listed up as results. As an application, we prove that if the Littlewood–Richardson coefficient is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the Littlewood–Richardson coefficient is 1. Furthermore, our conjecture is that one of four types of our reduction formulae is always applicable to all triples of partitions if the corresponding Littlewood–Richardson coefficient is 1
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Reduction formulae of Littlewood-Richardson coefficients
Advances in Applied Mathematics, 2010Co-Authors: Soojin Cho, Dongho MoonAbstract:There are two well-known reduction formulae by Griffiths-Harris for Littlewood-Richardson coefficients. Our observation is that some special cases of the factorization theorem of Littlewood-Richardson coefficients by King, Tollu and Toumazet give reduction formulae including the Griffiths-Harris formulae. We provide explicit statements of those reduction formulae in more general forms, and extend them to their conjugated forms also. Eight useful reduction formulae deleting one or two rows (columns) of each partition are listed up as results. As an application, we prove that if the Littlewood-Richardson coefficient is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the Littlewood-Richardson coefficient is 1. Furthermore, our conjecture is that one of four types of our reduction formulae is always applicable to all triples of partitions if the corresponding Littlewood-Richardson coefficient is 1.
Soojin Cho - One of the best experts on this subject based on the ideXlab platform.
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Reduction formulae of Littlewood–Richardson coefficients
Advances in Applied Mathematics, 2011Co-Authors: Soojin Cho, Dongho MoonAbstract:AbstractThere are two well-known reduction formulae by Griffiths–Harris for Littlewood–Richardson coefficients. Our observation is that some special cases of the factorization theorem of Littlewood–Richardson coefficients by King, Tollu and Toumazet give reduction formulae including the Griffiths–Harris formulae. We provide explicit statements of those reduction formulae in more general forms, and extend them to their conjugated forms also. Eight useful reduction formulae deleting one or two rows (columns) of each partition are listed up as results. As an application, we prove that if the Littlewood–Richardson coefficient is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the Littlewood–Richardson coefficient is 1. Furthermore, our conjecture is that one of four types of our reduction formulae is always applicable to all triples of partitions if the corresponding Littlewood–Richardson coefficient is 1
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Reduction formulae of Littlewood-Richardson coefficients
Advances in Applied Mathematics, 2010Co-Authors: Soojin Cho, Dongho MoonAbstract:There are two well-known reduction formulae by Griffiths-Harris for Littlewood-Richardson coefficients. Our observation is that some special cases of the factorization theorem of Littlewood-Richardson coefficients by King, Tollu and Toumazet give reduction formulae including the Griffiths-Harris formulae. We provide explicit statements of those reduction formulae in more general forms, and extend them to their conjugated forms also. Eight useful reduction formulae deleting one or two rows (columns) of each partition are listed up as results. As an application, we prove that if the Littlewood-Richardson coefficient is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the Littlewood-Richardson coefficient is 1. Furthermore, our conjecture is that one of four types of our reduction formulae is always applicable to all triples of partitions if the corresponding Littlewood-Richardson coefficient is 1.
Shuichi Sato - One of the best experts on this subject based on the ideXlab platform.
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Littlewood–Paley operators and Sobolev spaces
Illinois Journal of Mathematics, 2014Co-Authors: Shuichi SatoAbstract:We prove some weighted estimates for two kinds of Littlewood–Paley operators related to the Riesz potentials, which can be used to characterize the weighted Sobolev spaces. Also, we show the boundedness from the weighted Hardy space $H^{1}_{w}$ to the weighted weak $L^{1}$ space of a Littlewood–Paley operator arising from spherical means.
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Estimates for Littlewood-Paley Functions and Extrapolation
Integral Equations and Operator Theory, 2008Co-Authors: Shuichi SatoAbstract:We prove certain Lp-estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove Lp-boundedness of the Littlewood-Paley functions under a sharp kernel condition.
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DeLeeuw's theorem on Littlewood-Paley functions
Nagoya Mathematical Journal, 2002Co-Authors: Chang-pao Chen, Dashan Fan, Shuichi SatoAbstract:We establish certain deLeeuw type theorems for Littlewood-Paley functions. By these theorems, we know that the boundedness of a Littlewood- Paley function on n is equivalent to the boundedness of its corresponding Littlewood-Paley function on the torus n . x1. Introduction Let R n be the n-dimensional Euclidean space and T n be the n-dimen- sional torus. T n can be identied with R n =, where is the unit lattice which is the additive group of points in R n having integral coordinates. For an L 1 (R n ) function we dene t(x) = 2 tn ( x=2 t ), t 2 R. Then the Fourier transform of t is ^ t( ) = ^ (2 t ). The Littlewood-Paley g-function
Ravi Vakil - One of the best experts on this subject based on the ideXlab platform.
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Solving Schubert Problems with Littlewood-Richardson Homotopies
arXiv: Numerical Analysis, 2010Co-Authors: Frank Sottile, Ravi Vakil, Jan VerscheldeAbstract:We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.
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A geometric Littlewood-Richardson rule
Annals of Mathematics, 2006Co-Authors: Ravi VakilAbstract:We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base eld, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri’s rule to arbitrary Schubert classes, by way of explicit homotopies. It has straightforward bijections to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao’s puzzles. This gives the rst geometric proof and interpretation of the Littlewood-Richardson rule. Geometric consequences are described here and in [V2], [KV1], [KV2], [V3]. For example, the rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory.
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A geometric Littlewood-Richardson rule
arXiv: Algebraic Geometry, 2003Co-Authors: Ravi VakilAbstract:We describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties so that they break into Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. It has a host of geometric consequences, described in the companion paper "Schubert induction". The rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory. The rule suggests a natural approach to the open question of finding a Littlewood-Richardson rule for the flag variety, leading to a conjecture, shown to be true up to dimension 5. Finally, the rule suggests approaches to similar open problems, such as Littlewood-Richardson rules for the symplectic Grassmannian and two-flag varieties.
Frank Sottile - One of the best experts on this subject based on the ideXlab platform.
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Solving Schubert Problems with Littlewood-Richardson Homotopies
arXiv: Numerical Analysis, 2010Co-Authors: Frank Sottile, Ravi Vakil, Jan VerscheldeAbstract:We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.
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Skew Littlewood―Richardson rules from Hopf algebras
Discrete Mathematics and Theoretical Computer Science, 2010Co-Authors: Thomas Lam, Aaron Lauve, Frank SottileAbstract:We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.
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skew Littlewood richardson rules from hopf algebras
arXiv: Combinatorics, 2009Co-Authors: Thomas Lam, Aaron Lauve, Frank SottileAbstract:We use Hopf algebras to prove a version of the Littlewood-Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood-Richardson rules for Schur P- and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group.
