Cartesian harmonics

The harmonic operator $H$ of rank $n$ takes a unit vector $\mathbf a$, tensored with itself $n$ times, to its Cartesian harmonic, the Cartesian counterpart of a spherical harmonic. $H$ takes one copy of $\mathbf a$ for each of its Cartesian indices, so the weight-2 harmonic is act(H, a, a):

import numpy as np

from natto import act, get_harmonic_operator

a = np.array([0.0, 0.0, 1.0])

H = get_harmonic_operator(2)
Y = act(H, a, a)

print(f"H of weight 2: {H}\n")
print(f"H · a a, the weight-2 harmonic of a:\n{Y}")

H of weight 2: +3/4 δ_Aa δ_Bb  +3/4 δ_Ab δ_Ba  -1/2 δ_AB δ_ab

H · a a, the weight-2 harmonic of a:
[[-0.5  0.   0. ]
 [ 0.  -0.5  0. ]
 [ 0.   0.   1. ]]

As with the other operators, act evaluates $H$ into an array and contracts it with its inputs by numpy.einsum. evaluate gives the array and the rule, with one operand for each copy of $\mathbf a$:

from natto import evaluate

H_array, rule = evaluate(H)
Y = np.einsum(rule, H_array, a, a)

print(f"einsum rule: {rule}\n")
print(f"the weight-2 harmonic of a:\n{Y}")

einsum rule: abAB,...A,...B->...ab

the weight-2 harmonic of a:
[[-0.5  0.   0. ]
 [ 0.  -0.5  0. ]
 [ 0.   0.   1. ]]

By default, normalize="legendre", $H$ is scaled so that the harmonic of one unit vector contracted with the tensored copies of another gives the Legendre polynomial of the angle between them. normalize="none" leaves it unscaled.