The Chow Ring of flag manifolds - Open Research Data - Bridge of Knowledge

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The Chow Ring of flag manifolds

Description

Schubert calculus is the intersection theory of 19th century. Justifying this calculus is the content of the 15th problem of Hilbert. In the course to establish the foundation of algebraic geometry, Van der Vaerden and A. Weil attributed the problem to the determination of the chow ring of flag manifolds G/P, where G is a compact Lie group and P is a parabolic subgroup. This problem has been solved by Borel for the classical Lie groups, and by Duan and Zhao for all compact Lie groups see [H. Duan, Xuezhi Zhao, On Schubert's Problem of Characteristics, to appear in Springer proceedings in Mathematics & Statistics, arXiv:1912.10745].

This databset is created to record the preliminary data to formulate the Chow rings of the flag manifolds associated to the exceptional Lie groups G=G2, F4, E6, E7, which are generated by a general algorithm (i.e. applicable to all flag manifolds) illustrated in the paper of Duan and Zhao mentioned above.

Each filename begins with the name an exceptional Lie group G, followed by an integer k, indicating that P is the parabolic subgroup of G corresponding to the k-th root. Each file has two part: two lists in the format of Mathematica. In the first part of each file we present the Schubert cells on G/P in term of its minimal word representation in the set W(P;G) of left cosets of the Weyl group of G by the Weyl group of P, where s(i,j) denotes the j-th Schubert cell of dimension 2i, and all of them form an additive basis of the Chow ring of G/P. In the second part of each file, we list all the L-R coefficients, a(i,j,k), required to express the product s(1,1) s(i,j) as a linear combination in the Schubert basis, that is s(1,1) s(i,j) = a(i,j,1) s(i+1,1) + a(i,j,2) s(i+1,2)+.... .

Dataset file

efg.zip
435.0 kB, S3 ETag 5f858c239593b91bad00381467e01885-1, downloads: 73
The file hash is calculated from the formula
hexmd5(md5(part1)+md5(part2)+...)-{parts_count} where a single part of the file is 512 MB in size.

Example script for calculation:
https://github.com/antespi/s3md5
download file efg.zip

File details

License:
Creative Commons: by 4.0 open in new tab
CC BY
Attribution

Details

Year of publication:
2020
Verification date:
2020-12-17
Dataset language:
English
Fields of science:
  • mathematics (Natural sciences)
DOI:
DOI ID 10.34808/ffc2-6s76 open in new tab
Verified by:
Gdańsk University of Technology

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