Effective algorithms for computing triangular operator in Schubert calculus

doi:10.1007/s11464-014-0417-z

Front. Math. China

2015, Vol. 10

Issue (1) : 221-237 https://doi.org/10.1007/s11464-014-0417-z

RESEARCH ARTICLE

Effective algorithms for computing triangular operator in Schubert calculus

Kai ZHANG^1,^*(

),Jiachuan ZHANG¹,Haibao DUAN²,Jingzhi LI^3,^*(

)

1. Department of Mathematics, Jilin University, Changchun 130023, China
2. Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, China
3. Faculty of Science, South University of Science and Technology of China, Shenzhen 518055, China

Download: PDF(239 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

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.

Keywords Triangular operator Schubert calculus parallel algorithm central binomial coefficient

Corresponding Author(s): Kai ZHANG

Issue Date: 30 December 2014

Cite this article:

Haibao DUAN,Jingzhi LI,Kai ZHANG, et al. Effective algorithms for computing triangular operator in Schubert calculus[J]. Front. Math. China, 2015, 10(1): 221-237.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0417-z
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I1/221

1	Bott R, Samelson H. The cohomology ring of G/T. Nat Acad Sci, 1955, 41: 490-492 https://doi.org/10.1073/pnas.41.7.490
2	Donaghey R, Shapiro L W. Motzkin numbers. J Combin Theory Ser A, 1977, 23: 291-301 https://doi.org/10.1016/0097-3165(77)90020-6
3	Duan H B. Multiplicative rule of Schubert classes. Invent Math, 2005, 159(2): 407-436 https://doi.org/10.1007/s00222-004-0394-z
4	Duan H B. The degree of a Schubert variety. Adv Math, 2003, 180: 112-133 https://doi.org/10.1016/S0001-8708(02)00098-1
5	Duan H B, Zhao X Z. Algorithm for multiplying Schubert classes. Inter J Algebra Comput, 2006, 16: 1197-1210 https://doi.org/10.1142/S021819670600344X
6	Duan H B, Zhao X Z. A unified formula for Steenrod operations in flag manifolds. Compos Math, 2007, 143(1): 257-270 https://doi.org/10.1112/S0010437X06002405
7	Duan H B, Zhao X Z. Schubert presentation of the integral cohomology ring of the flag manifolds G/T. arXiv: 0801.2444
8	Duan H B, Zhao X Z. The Chow rings of generalized Grassmannians. Found Comput Math, 2010, 10(3): 245-274 https://doi.org/10.1007/s10208-010-9058-0
9	Elizaldea S, Mansour T. Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials. Discrete Math, 2005, 305(3): 170-189 https://doi.org/10.1016/j.disc.2005.10.010
10	Huber B, Sottile F, Sturmfels B. Numerical Schubert calculus. J Symbolic Comput, 1998, 26: 767-788 https://doi.org/10.1006/jsco.1998.0239
11	Kleiman S, Laksov D. Schubert calculus. Amer Math Monthly, 1974, 79: 1061-1082 https://doi.org/10.2307/2317421
12	Knuth D. The Art of Computer Programming, Vol 3. Sorting and Searching. Boston: Addison-Wesley, 1973, 506-542
13	Koshy T. Catalan Numbers with Applications. Oxford: Oxford University Press, 2008 https://doi.org/10.1093/acprof:oso/9780195334548.001.0001
14	Li T Y, Wang X S, Wu M N. Numerical Schubert calculus by the Pieri homotopy algorithm. SIAM J Numer Anal, 2002, 40: 578-600 https://doi.org/10.1137/S003614290139175X
15	Pitman J. Probability. Berlin: Springer-Verlag, 1993
16	Sottile F. Four entries for Kluwer Encyclopaedia of Mathematics. arXiv: math/0102047

[1]	Yunnan ZHANG, Huaijie ZHONG. Strongly irreducible operators and Cowen-Douglas operators on c₀, l_p (1≤p<∞)[J]. Front Math Chin, 2011, 6(5): 987-1001.

Viewed

Full text

Abstract

Cited

Shared

Discussed