Reduction - natto

get_reduction builds the operators of Extraction and embedding for every weight and channel in one call. For each (ell, p) it gives the embedding operator $G$ under "embedding" and the extraction operator $\widetilde G$ under "extraction", and it takes the same ell, symmetry and basis arguments.

Reducing a rank-2 tensor extracts each ICT and embeds it back into the part of the tensor it carries:

import numpy as np

from natto import act, evaluate, get_reduction

T = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 12.0]])

reduction = get_reduction(n=2)

for (ell, p), operators in reduction.items():
    G = operators["embedding"]
    G_tilde = operators["extraction"]

    X = act(G_tilde, T)
    T_part = act(G, X)

    print(f"--- weight {ell}, channel {p} ---")
    print(f"G~: {G_tilde}")
    print(f"G:  {G}")
    print(f"X, the ICT:\n{X}")
    print(f"G · X, the part of T:\n{T_part}\n")

--- weight 0, channel 1 ---
G~: +1/3 δ_AB
G:  +1 δ_AB
X, the ICT:
6.0
G · X, the part of T:
[[6. 0. 0.]
 [0. 6. 0.]
 [0. 0. 6.]]

--- weight 1, channel 1 ---
G~: +1/2 ε_ABa
G:  +1 ε_ABa
X, the ICT:
[-1.  2. -1.]
G · X, the part of T:
[[ 0. -1. -2.]
 [ 1.  0. -1.]
 [ 2.  1.  0.]]

--- weight 2, channel 1 ---
G~: +1/2 δ_Aa δ_Bb  +1/2 δ_Ab δ_Ba  -1/3 δ_AB δ_ab
G:  +1/2 δ_Aa δ_Bb  +1/2 δ_Ab δ_Ba  -1/3 δ_AB δ_ab
X, the ICT:
[[-5.  3.  5.]
 [ 3. -1.  7.]
 [ 5.  7.  6.]]
G · X, the part of T:
[[-5.  3.  5.]
 [ 3. -1.  7.]
 [ 5.  7.  6.]]

The three parts are $\operatorname{tr}\mathbf T/3$ times the identity, the antisymmetric part and the symmetric traceless part, and they sum to T.

The same reduction with arrays and einsum rules, each operator evaluated once:

for (ell, p), operators in reduction.items():
    G_array, G_rule = evaluate(operators["embedding"])
    G_tilde_array, G_tilde_rule = evaluate(operators["extraction"])

    X = np.einsum(G_tilde_rule, G_tilde_array, T)
    T_part = np.einsum(G_rule, G_array, X)

    print(f"--- weight {ell}, channel {p} ---")
    print(f"rule of G~: {G_tilde_rule}")
    print(f"rule of G:  {G_rule}")
    print(f"G · X, the part of T:\n{T_part}\n")

--- weight 0, channel 1 ---
rule of G~: AB,...AB->...
rule of G:  AB,...->...AB
G · X, the part of T:
[[6. 0. 0.]
 [0. 6. 0.]
 [0. 0. 6.]]

--- weight 1, channel 1 ---
rule of G~: aAB,...AB->...a
rule of G:  aAB,...a->...AB
G · X, the part of T:
[[ 0. -1. -2.]
 [ 1.  0. -1.]
 [ 2.  1.  0.]]

--- weight 2, channel 1 ---
rule of G~: abAB,...AB->...ab
rule of G:  abAB,...ab->...AB
G · X, the part of T:
[[-5.  3.  5.]
 [ 3. -1.  7.]
 [ 5.  7.  6.]]