pySymmPol: Symmetric Polynomials

This Python package is designed for the manipulation of various symmetric polynomials or functions. It includes the following types:

1. Complete homogeneous Symmetric Functions
2. Elementary symmetric polynomials
3. Monomial symmetric polynomials 
4. Schur polynomials
5. Hall-Littlewood polynomials

Additionally, the package contains a module with basic functionalities for manipulating integer partitions and Young diagrams.

Read our Statement of need here.

Tutorials can be found in the documentation page.

Dependencies

This package has been tested with the following versions: - Python >= 3.11 - SymPy >= 1.11 - NumPy >= 1.26.2

Installation

The package can be installed with pip: bash $ pip install pysymmpol

Basic Usage

The PySymmPol package has seven main classes for manipulating various symmetric polynomials.

YoungDiagram and ConjugacyClass

For the construction and manipulation of Young diagrams, we need to import the YoungDiagram and the ConjugacyClass classes. python from pysymmpol import YoungDiagram, ConjugacyClass The distinction between these two classes lies in the representations of the diagrams they handle. The YoungDiagram class represents diagrams using a monotonic decreasing sequence, while the ConjugacyClass class represents them as a sequence representing the cycle of the symmetric group SnSn​. For example, let's consider the partition (3,2,1), which is represented as a tuple in the YoungDiagram class and as a dictionary in the ConjugacyClass class {1: 1, 2: 1, 3: 1}, respectively. python young = YoungDiagram((3,2,1)) conjugacy = ConjugacyClass({1: 1, 2: 1, 3: 1}) Both objects describe the same mathematical entity, the partition 6=3+2+1. In fact, we have the usual pictorial representation ```python young.draw_diagram(4)

conjugacy.draw_diagram(4) ``` that give the same output (the argument 4 means that we draw the octothorpe, and there are 4 other symbols available).

```python

#

#

``` Further details on the other functionalities can be found in the tutorial.

Homogeneous and Elementary Polynomials

These classes can be initialized as python from pysymmpol import HomogeneousPolynomial, ElementaryPolynomial from pysymmpol.utils import create_miwa We also imported the function create_miwa from the utils module for convenience. Now, let's create the polynomials at level n=3. We can instantiate them and find their explicit expressions using the explicit(t) method, where t represents the Miwa coordinates. The block ```python n = 3 t = create_miwa(n)

homogeneous = HomogeneousPolynomial(n) elementary = ElementaryPolynomial(n) print(f"homogeneous: {homogeneous.explicit(t)}") print(f"elementary: {elementary.explicit(t)}") gives the output homogeneous: t13/6 + t1*t2 + t3 elementary: t13/6 - t1*t2 + t3 ```

Schur Polynomials

To create Schur polynomials, we first need to instantiate a partition before defining the polynomial itself. Let's use the Young diagram we considered a few lines above, then python from pysymmpol import YoungDiagram from pysymmpol import SchurPolynomial from pysymmpol.utils import create_miwa The YoungDiagram class includes a getter method for retrieving the number of boxes in the diagram, which we utilize to construct the Miwa coordinates. Subsequently, the SchurPolynomial class is instantiated using the Young diagram. Then ```python young = YoungDiagram((3,2,1)) t = create_miwa(young.boxes)

schur = SchurPolynomial(young)

print(f"schur: {schur.explicit(t)}") gives schur: t16/45 - t13t3/3 + t1t5 - t3**2 ``` The documentation and tutorial contain examples demonstrating how to find skew-Schur polynomials.

Monomial Symmetric Polynomials

For Monomial symmetric polynomials, we have a similar structure. python from pysymmpol import YoungDiagram from pysymmpol import MonomialPolynomial from pysymmpol.utils import create_x_coord The only difference is the function create_x_coord from the utils module. Therefore, ```python young = YoungDiagram((3,2,1))

n = 3 x = createxcoord(n)

monomial = MonomialPolynomial(young)

print(f"monomial: {monomial.explicit(x)}") gives the output monomial: x1x2x3(x12x2 + x12x3 + x1x22 + x1x32 + x22x3 + x2x3*2) ```

Hall-Littlewood Polynomials

In addition to partitions, for the Hall-Littlewood polynomials, we also require the deformation parameter Q (as t has been used to denote the Miwa coordinates). python from sympy import Symbol from pysymmpol import YoungDiagram from pysymmpol import HallLittlewoodPolynomial from pysymmpol.utils import create_x_coord The method explicit(x, Q) needs another argument. Finally, the code ```python Q = Symbol('Q') young = YoungDiagram((3,2,1))

n = 3 x = createxcoord(n)

hall_littlewood = HallLittlewoodPolynomial(young)

print(f"hall-littlewood: {hall_littlewood.explicit(x, Q)}") gives hall-littlewood: x1x2x3(-Q2x1x2x3 - Qx1x2x3 + x12x2 + x12x3 + x1x22 + 2x1x2x3 + x1x32 + x22x3 + x2x3**2) ```

References

Here are some recommended resources covering symmetric polynomials, combinatorics, and their significance in theoretical physics:

• Symmetric Functions and Hall Polynomials - Ian Macdonald
• An Introduction to Symmetric Functions and Their Combinatorics - Eric S. Egge
• Counting with Symmetric Functions - Mendes and Remmel
• Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras - Miwa, Jimbo and Date
• The uses of random partitions - Andrei Okounkov

Citation & Contributing

If you found this package useful in your research, please consider citing the companion paper available here: arxiv.org/abs/2403.13580. @article{Araujo:2024piv, author = "Araujo, Thiago", title = "{PySymmPol: Symmetric Polynomials in Python}", eprint = "2403.13580", archivePrefix = "arXiv", primaryClass = "math.CO", doi = "10.21105/joss.06724", journal = "J. Open Source Softw.", volume = "9", pages = "6724", year = "2024" } Feeling like contributing? Fork the project and open a pull request with your modifications. Found a bug? Just open a GitHub issue.