schubertpy

Overview

schubertpy is a powerful Python package designed for performing advanced mathematical operations on the Grassmannian, a key concept in algebraic geometry and representation theory. This module facilitates operations such as quantum Pieri rules, quantum Giambelli formulae, and the manipulation of Schubert classes. It is a Python implementation based on the comprehensive maple library available at https://sites.math.rutgers.edu/~asbuch/qcalc/.

References:

• https://ar5iv.labs.arxiv.org/html/0809.4966
• https://sites.math.rutgers.edu/~asbuch/notes/grass.pdf
• https://sites.math.rutgers.edu/~asbuch/notes/schurfcns.pdf
• https://sites.math.rutgers.edu/~asbuch/papers/qschub.pdf
• https://sites.math.rutgers.edu/~asbuch/papers/isogiam.pdf
• https://sites.math.rutgers.edu/~asbuch/papers/qgig.pdf
• https://sites.math.rutgers.edu/~asbuch/papers/oggiam.pdf
• https://sites.math.rutgers.edu/~asbuch/talks/cirm2020.pdf
• https://sites.math.rutgers.edu/~asbuch/papers/

Features

• Quantum Pieri Rule Calculations: Efficient computation of quantum Pieri rules applied to Schubert classes.
• Quantum Giambelli Formulae: Expression of products of Schubert classes in alternative forms using quantum Giambelli formulae.
• Schubert Class Operations: Perform actions and multiplications on Schubert classes, in both classical and quantum contexts.
• Dualization and Conversion: Dualize Schubert classes and convert between different Schubert class representations.

Installation

To install the schubertpy module, run the following command:

bash pip install schubertpy

If you wanna use with sagemath, run the following command:

bash sage -pip install schubertpy

Usage

Example on Google Colab

Example usage demonstrating the capabilities of schubertpy:

```python from schubertpy import Grassmannian, OrthogonalGrassmannian, IsotropicGrassmannian

def main(): # Initialize the Grassmannian object with dimensions gr = Grassmannian(2, 5) print(gr.qpieri(1, 'S[2,1] - 7S[3,2]')) print(gr.qact('S[1]+S[2]S[3]', 'S[2,1]+S[3,2]')) print(gr.qgiambelli('S[2,1]S[2,1]')) print(gr.qmult('S[2,1]', 'S[2,1]+S[3,2]')) print(gr.qtoS('S[2,1]S[2,1]S[2,1]')) print(gr.pieri(1, 'S[2,1] - 7S[3,2]')) print(gr.act('S[1]+S[2]S[3]', 'S[2,1]+S[3,2]')) print(gr.giambelli('S[2,1]S[2,1]')) print(gr.mult('S[2,1]', 'S[2,1]+S[3,2]')) print(gr.toS('S[2,1]S[2,1]S[2,1]')) print(gr.dualize('S[1]+S[2]'))

ig = IsotropicGrassmannian(2, 6)
print(ig.qpieri(1, 'S[2,1] - 7*S[3,2]'))
print(ig.qact('S[1]+S[2]*S[3]', 'S[2,1]+S[3,2]'))
print(ig.qgiambelli('S[2,1]*S[2,1]'))
print(ig.qmult('S[2,1]', 'S[2,1]+S[3,2]'))
print(ig.qtoS('S[2,1]*S[2,1]*S[2,1]'))
print(ig.pieri(1, 'S[2,1] - 7*S[3,2]'))
print(ig.act('S[1]+S[2]*S[3]', 'S[2,1]+S[3,2]'))
print(ig.giambelli('S[2,1]*S[2,1]'))
print(ig.mult('S[2,1]', 'S[2,1]+S[3,2]'))
print(ig.toS('S[2,1]*S[2,1]*S[2,1]'))
print(ig.dualize('S[1]+S[2]'))

og = OrthogonalGrassmannian(2, 7)
print(og.qpieri(1, 'S[2,1] - 7*S[3,2]'))
print(og.qact('S[1]+S[2]*S[3]', 'S[2,1]+S[3,2]'))
print(og.qgiambelli('S[2,1]*S[2,1]'))
print(og.qmult('S[2,1]', 'S[2,1]+S[3,2]'))
print(og.qtoS('S[2,1]*S[2,1]*S[2,1]'))
print(og.pieri(1, 'S[2,1] - 7*S[3,2]'))
print(og.act('S[1]+S[2]*S[3]', 'S[2,1]+S[3,2]'))
print(og.giambelli('S[2,1]*S[2,1]'))
print(og.mult('S[2,1]', 'S[2,1]+S[3,2]'))
print(og.toS('S[2,1]*S[2,1]*S[2,1]'))
print(og.dualize('S[1]+S[2]'))

og = OrthogonalGrassmannian(2, 6)
print(og.qpieri(1, 'S[2,1] - 7*S[3,2]'))
print(og.qact('S[1]+S[2]*S[3]', 'S[2,1]+S[3,2]'))
print(og.qgiambelli('S[2,1]*S[2,1]'))
print(og.qmult('S[2,1]', 'S[2,1]+S[3,2]'))
print(og.qtoS('S[2,1]*S[2,1]*S[2,1]'))
print(og.pieri(1, 'S[2,1] - 7*S[3,2]'))
print(og.act('S[1]+S[2]*S[3]', 'S[2,1]+S[3,2]'))
print(og.giambelli('S[2,1]*S[2,1]'))
print(og.mult('S[2,1]', 'S[2,1]+S[3,2]'))
print(og.toS('S[2,1]*S[2,1]*S[2,1]'))
print(og.dualize('S[1]+S[2]'))

if name == "main": main() ```

You wanna use with sagemath? You can save above example to main.py and then run:

bash sage -python main.py

For detailed examples and more operations, refer to the test cases provided within the module's documentation.

Running Tests

To verify the module's functionality, you can run the included tests with either of the following commands:

bash make test

Or directly with Python:

bash python3 -m unittest schubertpy/testcases/*.py

Authors

• Dang Tuan Hiep 🇻🇳

  • 
  • Email: hiepdt@dlu.edu.vn (Đặng Tuấn Hiệp)

• Trần Duy Thanh 🇻🇳

  • 
  • Email: fbtranduythanh@gmail.com
  • Email: 2015830@dlu.edu.vn

• Hoàng Minh Đức 🇻🇳

  • 
    
  • Email: 2113423@dlu.edu.vn

• Nguyễn Trương Thiên Ân 🇻🇳

  • 
  • Email: 2113421@dlu.edu.vn

Contributing

We highly encourage contributions to schubertpy. Whether you are looking to expand functionality, enhance performance, or fix bugs, your input is valuable. To get started:

• Report Issues: If you encounter issues or have suggestions, please report them by creating an issue on our GitHub page.
• Submit Pull Requests: Feel free to fork the repository and submit pull requests. Whether it's adding new features, optimizing existing code, or correcting bugs, your contributions are welcome.

Please ensure your pull requests are well-documented and include any necessary tests. For more details on contributing, refer to our contribution guidelines on GitHub.

License

schubertpy is open source software (under the GNU General Public License).

Citing

We encourage you to cite our work if you have used our package. See "Cite this repository" on this page.

Documentation

• Pieri Algorithms
• Pieri Type A Algorithms
• Pieri Type B Algorithms
• Pieri Type C Algorithms
• Pieri Type D Algorithms
• Giambelli Algorithms
• Littlewood-Richardson Algorithms