haarpy 0.0.5

Symbolic calculation of Weingarten functions

Haarpy is a Python library for the symbolic calculation of Weingarten functions and related averages of unitary matrices $U(d)$ sampled uniformly at random from the Haar measure.

The original Mathematica version of this code, for the calculation of Weingarten functions of the unitary group, can be found here.

Haarpy in action

The main functions of Haarpy are weingarten_class, weingarten_element and haar_integral allowing for the calculation of Weingarten functions and integrals over unitaries sampled at random from the Haar measure. We recommend importing the following when working with Haarpy:

from sympy import Symbol
from sympy.combinatorics import Permutation

d = Symbol("d")

weingarten_class

Takes a partition, labeling a conjugacy class of $S_p$, and a dimension $d$ as arguments. For the conjugacy class labeled by partition $\lbrace 3,1\rbrace$, the function returns

from haarpy import weingarten_class
weingarten_class((3,1),d)

(2*d**2 - 3)/(d**2*(d - 3)*(d - 2)*(d - 1)*(d + 1)*(d + 2)*(d + 3))

The previous can also be called with integer values as such

weingarten_class((3,1),4)

29/20160

weingarten_element

Takes an element and the degree $p$ of the symmetric group $S_p,$ and a dimension $d$ as arguments, the conjugacy class being obtained from the first two.

from haarpy import weingarten_element
weingarten_element(Permutation(0,1,2), 4, d)

(2*d**2 - 3)/(d**2*(d - 3)*(d - 2)*(d - 1)*(d + 1)*(d + 2)*(d + 3))

Which yields the same result as before since $\lbrace 3,1\rbrace$ is the class of permutation $(0,1,2)$ in $S_4$.

haar_integral

Takes in a tuple of sequences $((i_1,\dots,i_p),\ (j_1,\dots,j_p),\ (i\prime_1,\dots, i\prime_p),\ (j\prime_1,\dots,j\prime_p))$, and the dimension $d$ of the unitary group. Returns the value of the integral $\int dU \ U_{i_1j_1}\dots U_{i_pj_p}U^\ast_{i\prime_1 j\prime_1}\dots U^\ast_{i\prime_p j\prime_p}$.

from haarpy import haar_integral
haar_integral(((1,2), (1,2), (1,2), (1,2)), d)

1/((d-1)*(d+1))

Auxiliary functions include, but are not limited to, the following. For a comprehensive list of functionalities, please refer to the documentation.

murn_naka_rule

Implementation of the Murnaghan-Nakayama rule for the characters irreducible representations of the symmetric group $S_p$. Takes a partition characterizing an irrep of $S_p$ and a conjugacy class and yields the associate character.

from haarpy import murn_naka_rule
murn_naka_rule((3,1), (1,1,1,1))

3

get_conjugacy_class

Returns the class of a given element of $S_p$ when given the order and the element.

from haarpy import get_conjugacy_class
get_conjugacy_class(Permutation(0,1,2), 4)

(3,1)

irrep_dimension

Takes a partition labeling an irrep of $S_p$ and returns the dimension of this irrep.

from haarpy import irrep_dimension
irrep_dimension((5,4,2,1,1,1))

63063

representation_dimension

Takes a partition labeling a representation of the unitary group $U(d)$, as well as the dimension $d$, and returns the dimension of the representation.

from haarpy import representation_dimension
representation_dimension((5, 4, 2, 1, 1, 1),d)

d**2*(d - 5)*(d - 4)*(d - 3)*(d - 2)*(d - 1)**2*(d + 1)**2*(d + 2)**2*(d + 3)*(d + 4)/1382400

Which can also be done numerically.

ud_dimension((5, 4, 2, 1, 1, 1),8)

873180

Tables of Weingarten functions for $n \le 5$

The following have been retrieved using the weingarten_class function. Weingarten functions of symmetric groups of higher degrees can just as easily be obtained.

Symmetric group $S_2$

Class	Weingarten
$\lbrace2\rbrace$	$-\displaystyle\frac{1}{(d-1) d (d+1)}$
$\lbrace1,1\rbrace$	$\displaystyle\frac{1}{(d-1)(d+1)}$

Symmetric group $S_3$

Class	Weingarten
$\lbrace 3\rbrace$	$\displaystyle\frac{2}{(d-2) (d-1) d (d+1) (d+2)}$
$\lbrace 2,1\rbrace$	$-\displaystyle\frac{1}{(d-2) (d-1) (d+1) (d+2)}$
$\lbrace 1,1,1\rbrace$	$\displaystyle\frac{d^2-2}{(d-2) (d-1) d (d+1) (d+2)}$

Symmetric group $S_4$

Class	Weingarten
$\lbrace 4\rbrace$	$-\displaystyle\frac{5}{(d-3) (d-2) (d-1) d (d+1) (d+2) (d+3)}$
$\lbrace 3,1\rbrace$	$\displaystyle\frac{2 d^2-3}{(d-3) (d-2) (d-1) d^2 (d+1) (d+2) (d+3)}$
$\lbrace 2,2\rbrace$	$\displaystyle\frac{d^2+6}{(d-3) (d-2) (d-1) d^2 (d+1) (d+2) (d+3)}$
$\lbrace 2,1,1\rbrace$	$-\displaystyle\frac{1}{(d-3) (d-1) d (d+1) (d+3)}$
$\lbrace 1,1,1,1\rbrace$	$\displaystyle\frac{d^4-8 d^2+6}{(d-3) (d-2) (d-1)d^2 (d+1) (d+2)(d+3)}$

Symmetric group $S_5$

Class	Weingarten
$\lbrace 5\rbrace$	$\displaystyle\frac{14}{(d-4) (d-3) (d-2) (d-1) d (d+1) (d+2) (d+3)(d+4)}$
$\lbrace 4,1\rbrace$	$\displaystyle\frac{24-5 d^2}{(d-4) (d-3) (d-2) (d-1) d^2 (d+1) (d+2)(d+3)(d+4)}$
$\lbrace 3,2\rbrace$	$-\displaystyle\frac{2 \left(d^2+12\right)}{(d-4) (d-3) (d-2) (d-1)d^2(d+1)(d+2)(d+3)(d+4)}$
$\lbrace 3,1,1\rbrace$	$\displaystyle\frac{2}{(d-4) (d-2) (d-1) d (d+1) (d+2) (d+4)}$
$\lbrace 2,2,1\rbrace$	$\displaystyle\frac{-d^4+14 d^2-24}{(d-4) (d-3) (d-2) (d-1) d^2 (d+1) (d+2) (d+3) (d+4)}$
$\lbrace 1,1,1,1,1\rbrace$	$\displaystyle\frac{d^4-20 d^2+78}{(d-4) (d-3) (d-2) (d-1) d (d+1) (d+2) (d+3) (d+4)}$

Examples of integrals over Haar-random unitaries

Selected integrals of unitary groups ; $i,j,k$ and $\ell$ are assumed to take distinct integer values in the following.

Integral	Result
$\int dU \ U_{ij}U^\ast_{ij}$	$\displaystyle\frac{1}{d}$
$\int dU \ U_{ij}U_{kj}U^\ast_{ij}U^\ast_{kj}$	$\displaystyle\frac{1}{d(d+1)}$
$\int dU \ U_{ik}U_{k\ell}U^\ast_{ij}U^\ast_{k\ell}$	$\displaystyle\frac{1}{(d-1)(d+1)}$
$\int dU \ U_{ij}U_{k\ell}U^\ast_{i\ell}U^\ast_{kj}$	$\displaystyle\frac{-1}{(d-1)d(d+1)}$
$\int dU \ U_{ij}U_{k\ell}U_{mn}U^\ast_{ij}U^\ast_{k\ell}U^\ast_{mn}$	$\displaystyle\frac{d^2-2}{(d-2)(d-1)d(d+1)(d+2)}$
$\int dU \ U_{i\ell}U_{jm}U_{kn}U^\ast_{im}U^\ast_{jn}U^\ast_{k\ell}$	$\displaystyle\frac{2}{(d-2)(d-1)d(d+1)(d+2)}$
$\int dU \ U_{i\ell}U_{j\ell}U_{km}U^\ast_{i\ell}U^\ast_{jm}U^\ast_{k\ell}$	$\displaystyle\frac{-1}{(d-1)d(d+1)(d+2)}$
$\int dU \ U_{i\ell}U_{j\ell}U_{k\ell}U^\ast_{i\ell}U^\ast_{j\ell}U^\ast_{k\ell}$	$\displaystyle\frac{1}{d(d+1)(d+2)}$
$\int dU \ U_{ij}U_{ik}U_{i\ell}U_{im}U^\ast_{ij}U^\ast_{ik}U^\ast_{i\ell}U^\ast_{im}$	$\displaystyle\frac{1}{d(d + 1)(d + 2)(d + 3)}$

Installation

Haarpy requires Python version 3.9 or later. Installation can be done through the pip command

pip install haarpy

Compiling from source

Haarpy has the following dependencies:

• Python >= 3.9
• SymPy >= 1.12

Documentation

Haarpy documentation is available online on Read the Docs.

How to cite this work

Please cite as:

@misc{cardin2024haarpy,
  author={Cardin, Yanic and de Guise, Hubert and Quesada, Nicol{\'a}s},
  title={Haarpy, a Python library for the symbolic calculation of Weingarten functions},
  year={2024},
  publisher={GitHub},
  journal={GitHub repository},
  howpublished = {\url{https://github.com/polyquantique/haarpy}},
  version = {0.0.5}
}

Authors

• Yanic Cardin, Hubert de Guise, Nicolás Quesada.

License

Haarpy is free and open source, released under the Apache License, Version 2.0.