schubertpy 0.3.16

This Python module facilitates operations such as quantum Pieri rules, quantum Giambelli formulae, action and multiplication of Schubert classes, and conversion between different representations of Schubert classes

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:

pip install schubertpy

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

sage -pip install schubertpy

Usage

Example usage demonstrating the capabilities of schubertpy:

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] - 7*S[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] - 7*S[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 = Grassmannian(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 = Grassmannian(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]'))

    og = Grassmannian(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]'))


if __name__ == "__main__":
    main()

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

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:

make test

Or directly with Python:

python3 -m unittest schubertpy/testcases/*.py

Contributing

Contributions to schubertpy are highly encouraged, whether they involve extending functionality, enhancing performance, or fixing bugs. Please feel free to submit issues or pull requests on GitHub to suggest changes or improvements.

License

schubertpy is made available under the MIT License. For more details, see the LICENSE file included with the source code.