Schubert's method for solving systems of sparse equations has achieved a great deal of computational success. In this paper, Schubert's method was extended to multiple version, and the compact representation o...Schubert's method for solving systems of sparse equations has achieved a great deal of computational success. In this paper, Schubert's method was extended to multiple version, and the compact representation of multple Schubert's updating matrix was derived. The compact representation could be used to efficiently implement limited memory methods for large problems.展开更多
In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coo...In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coordinates,and it is proved that such equations generate the defining ideal of variety of type C in those of type A.As applications of this result,the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed,and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections.Finally,the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed,filling gaps in the study of algebraic varieties of the same type.展开更多
Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard ...Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional F_4- homogeneous manifold associated to a short simple root.展开更多
We develop two parallel algorithms progressively based on C＋＋ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of ...We develop two parallel algorithms progressively based on C＋＋ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.展开更多
By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schube...By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.展开更多
We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety F?_(n1,...,nk;n) via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimens...We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety F?_(n1,...,nk;n) via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases,including the complex Grassmannian Gr(2, n) and the complete flag variety F?_(1,2,3;4).展开更多
Let w be a permutation of{1,2,...,n},and let D(w)be the Rothe diagram of w.The Schubert polynomial■w_(x)can be realized as the dual character of the flagged Weyl module associated with D(w).This implies the following...Let w be a permutation of{1,2,...,n},and let D(w)be the Rothe diagram of w.The Schubert polynomial■w_(x)can be realized as the dual character of the flagged Weyl module associated with D(w).This implies the following coefficient-wise inequality:Min_(x)≤■_(w)(x)≤Max_(w)xwhere both Min_(w)(x)and Max_(w)(x)are polynomials determined by D(w).Fink et al.(2018)found that■w_(x)equals the lower bound Min_(w)(x)if and only if w avoids twelve permutation patterns.In this paper,we show that■w_(x)reaches the upper bound Max_(w)(x)if and only if w avoids two permutation patterns 1432 and 1423.Similarly,for any given compositionα∈Z^(n)≥0,one can define a lower bound Min_(α)(x)and an upper bound Max_(α)(x)for the key polynomialκ_(α)(x).Hodges and Yong(2020)established thatκ_(α)(x)equals Min_(α)(x)if and only ifαavoids five composition patterns.We show thatκ_(α)(x)equals Max_(α)(x)if and only ifαavoids a single composition pattern(0,2).As an application,we obtain that whenαavoids(0,2),the key polynomialκ_(α)(x)is Lorentzian,partially verifying a conjecture of Huh et al.(2019).展开更多
文摘Schubert's method for solving systems of sparse equations has achieved a great deal of computational success. In this paper, Schubert's method was extended to multiple version, and the compact representation of multple Schubert's updating matrix was derived. The compact representation could be used to efficiently implement limited memory methods for large problems.
文摘In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coordinates,and it is proved that such equations generate the defining ideal of variety of type C in those of type A.As applications of this result,the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed,and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections.Finally,the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed,filling gaps in the study of algebraic varieties of the same type.
基金supported by the National Researcher Program 2010-0020413 of NRFGA17-19437S of Czech Science Foundation(GACR)+3 种基金partially supported by the Simons-Foundation grant 346300the Polish Government MNi SW 2015-2019 matching fundsupported by BK21 PLUS SNU Mathematical Sciences DivisionIBS-R003-Y1
文摘Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional F_4- homogeneous manifold associated to a short simple root.
基金The authors sincerely appreciate the referees for acknowledging the manuscript and providing valuable comments and suggestions that benefit their manuscript. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11131008, 11271157, 11201453, 11471141）, the 973 Program （2011CB302400）, the Open Project Program of the State Key Lab of CAD＆CG （A1302） of Zhejiang University, and the Scientific Research Foundation for Returned Scholars, Ministry of Education of China. They also wish to thank the High Performance Computing Center of Jilin University and Computing Center of Jilin Province for essential computing support.
文摘We develop two parallel algorithms progressively based on C＋＋ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.
文摘By applying a Grobner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk＇s formula.
基金supported by the Samsung Science and Technology Foundation(Grant No. SSTF-BA1602-03)supported by the National Research Foundation of Korea (Grant No. NRF-2019R1F1A1058962)+1 种基金supported by National Natural Science Foundation of China (Grant Nos. 11771455, 11822113 and 11831017)Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No. 2017ZT07X355)。
文摘We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety F?_(n1,...,nk;n) via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases,including the complex Grassmannian Gr(2, n) and the complete flag variety F?_(1,2,3;4).
基金supported by National Natural Science Foundation of China(Grant Nos.11971250 and 12071320)Sichuan Science and Technology Program(Grant No.2020YJ0006)。
文摘Let w be a permutation of{1,2,...,n},and let D(w)be the Rothe diagram of w.The Schubert polynomial■w_(x)can be realized as the dual character of the flagged Weyl module associated with D(w).This implies the following coefficient-wise inequality:Min_(x)≤■_(w)(x)≤Max_(w)xwhere both Min_(w)(x)and Max_(w)(x)are polynomials determined by D(w).Fink et al.(2018)found that■w_(x)equals the lower bound Min_(w)(x)if and only if w avoids twelve permutation patterns.In this paper,we show that■w_(x)reaches the upper bound Max_(w)(x)if and only if w avoids two permutation patterns 1432 and 1423.Similarly,for any given compositionα∈Z^(n)≥0,one can define a lower bound Min_(α)(x)and an upper bound Max_(α)(x)for the key polynomialκ_(α)(x).Hodges and Yong(2020)established thatκ_(α)(x)equals Min_(α)(x)if and only ifαavoids five composition patterns.We show thatκ_(α)(x)equals Max_(α)(x)if and only ifαavoids a single composition pattern(0,2).As an application,we obtain that whenαavoids(0,2),the key polynomialκ_(α)(x)is Lorentzian,partially verifying a conjecture of Huh et al.(2019).
