pysymmpol 0.1.0

A python package for manipulation of symmetric polynomials.

pySymmPol: Symmetric Polynomials

This is a python package for manipulation of some symmetric polynomials (or functions). Among them, we have

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

Moreover, we have a module with basic functionalities for integer partitions and Young Diagrams manipulation.

Documentation

The documentation is [here], where I also included some [tutorials].

Dependencies

This package was tested on:

• Python 3.12
• Sympy 1.12
• Numpy 1.26.4

Installation

The package can be installed with pip:

$ pip install pysymmpol

Basic Usage

PySymmPol has seven main classes for manipulation of different symmetric polynomials.

YoungDiagram and ConjugacyClass

For the construction and manipulation of Young diagrams, we need to import the YoungDiagram and the ConjugacyClass classes.

from pysymmpol import YoungDiagram, ConjugacyClass

The difference between these two classes are the different representations of the diagrams. YoungDiagram represents them using a monotonic decreasing sequence; and ConjugacyClass as a sequence representing the cycle of the symmetric group S_n. Let us create the partition (3,2,1) represented as a tuple in the YoungDiagram class, and as a dictionary in the ConjugacyClass {1: 1, 2: 1, 3: 1},

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

young.draw_diagram(4)

conjugacy.draw_diagram(4)

that gives the output (the argument 4 means that we draw the octothorpe, there are 4 other symbols).

#
# #
# # #

#
# #
# # #

Description of the other functionalities can be seen in the tutorial.

Homogeneous and Elementary Polynomials

These classes can be initialized as

from pysymmpol import HomogeneousPolynomial, ElementaryPolynomial
from pysymmpol.utils import create_miwa

We also imported the function create_miwa in the utils module, for convenience. Let us create the polynomials at the level n=3. We can instanciate them, and find the explicit expression using the method explicit(t), there t are the Miwa coordinates.

n = 3
t = create_miwa(n)

homogeneous = HomogeneousPolynomial(n)
elementary = ElementaryPolynomial(n)
print(f"homogeneous: {homogeneous.explicit(t)}")
print(f"elementary: {elementary.explicit(t)})

that gives the output

homogeneous: t1**3/6 + t1*t2 + t3
elementary: t1**3/6 - t1*t2 + t3

Schur Polynomials

For Schur polynomials, we need to instanciate a partition before the polynomial itself. Let us use the Young diagram we considered a few lines above,

from pysymmpol import YoungDiagram
from pysymmpol import SchurPolynomial
from pysymmpol.utils import create_miwa

The class Young diagram has a getter for the number of boxes in the diagram. We use it to built the Miwa coordinates. The class SchurPolynomial is instanciated using the young diagram.

young = YoungDiagram((3,2,1))
t = create_miwa(young.boxes)

schur = SchurPolynomial(young)

print(f"schur: {schur.explicit(t)}")

that gives the output

schur: t1**6/45 - t1**3*t3/3 + t1*t5 - t3**2

In the documentation and tutorial, you can find some examples to find skew-Schur polynomials.

Monomial Symmetric Polynomials

For Monomial symmetric polynomials, we have a similar structure.

from pysymmpol import YoungDiagram
from pysymmpol import MonomialPolynomial
from pysymmpol.utils import create_x_coord

The only difference is that we import the function create_x_coord from the utils module.

young = YoungDiagram((3,2,1))

n = 3
x = create_x_coord(n)

monomial = MonomialPolynomial(young)

print(f"monomial: {monomial.explicit(x)}")

that gives the output

monomial: x1*x2*x3*(x1**2*x2 + x1**2*x3 + x1*x2**2 + x1*x3**2 + x2**2*x3 + x2*x3**2)

Hall-Littlewood Polynomials

Finally, for the Hall-Littlewood polynomials, besides the partitions, we also need the deformation parameter Q (because t has been used to denote the Miwa coordinates).

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.

Q = Symbol('Q')
young = YoungDiagram((3,2,1))

n = 3
x = create_x_coord(n)

hall_littlewood = HallLittlewoodPolynomial(young)

print(f"hall-littlewood: {hall_littlewood.explicit(x, Q)}")

that gives the output

hall-littlewood: x1*x2*x3*(-Q**2*x1*x2*x3 - Q*x1*x2*x3 + x1**2*x2 + x1**2*x3 + x1*x2**2 + 2*x1*x2*x3 + x1*x3**2 + x2**2*x3 + x2*x3**2)