Getting Started

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:

$ 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.

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.

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

#
# #
# # #

#
# #
# # #

Further details on the other functionalities can be found 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 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

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

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

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

schur = SchurPolynomial(young)

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

gives

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

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,

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

n = 3
x = create_x_coord(n)

monomial = MonomialPolynomial(young)

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

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

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).

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

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)}")

gives

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)