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brute:
(1/8 * D_{m}{F_{n p}} * V_{q}
+ 1/8 * D_{n}{F_{m p}} * V_{q}
- 1/8 * D_{m}{F_{n q}} * V_{p}
- 1/8 * D_{m}{F_{p n}} * V_{q}
+ 1/8 * D_{m}{F_{p q}} * V_{n}
+ 1/8 * D_{m}{F_{q n}} * V_{p}
- 1/8 * D_{m}{F_{q p}} * V_{n}
- 1/8 * D_{n}{F_{m q}} * V_{p}
- 1/8 * D_{p}{F_{m n}} * V_{q}
+ 1/8 * D_{p}{F_{m q}} * V_{n}
+ 1/8 * D_{q}{F_{m n}} * V_{p}
- 1/8 * D_{q}{F_{m p}} * V_{n});
smart:
(1/4 * D_{m}{F_{n p}} * V_{q}
+ 1/8 * D_{n}{F_{m p}} * V_{q}
- 1/4 * D_{m}{F_{n q}} * V_{p}
+ 1/4 * D_{m}{F_{q n}} * V_{p} vs D_{m}{F_{p q}} V_{n}
- 1/8 * D_{n}{F_{m q}} * V_{p}
- 1/8 * D_{p}{F_{m n}} * V_{q}
+ 1/8 * D_{p}{F_{m q}} * V_{n}
+ 1/8 * D_{q}{F_{m n}} * V_{p}
- 1/8 * D_{q}{F_{m p}} * V_{n});
The boxes are isolated correctly.
The implicit antisymmetry of the original tensor is found correctly.
And even the combinatorics output is correct:
1 0 2 3
1 0 3 2
3 0 1 2
Nevertheless, the permutation with 1/4 prefactors is not correct.
Is this really correct?:
src_fd+=sym[i][j];
dst_fd+=sym.original[j];
If this is, can we use different sublengths to achieve what we want?
_ _
1 0 2 3 -> 1 0 2 3 D_{m}{F_{n p}} V_{q}
1 0 3 2 1 0 3 2 D_{m}{F_{n q}} V_{p}
3 0 1 2 2 0 3 1 D_{m}{F_{p q}} V_{n}
Yes, this works of course. Interesting duality: sublength blocks are
for permutation by location what input_asym is for permutation by
value.
n m
p
q
A:= 1/24 * V_{m n} * F_{p q}
- 1/24 * V_{m n} * F_{q p}
- 1/24 * V_{m p} * F_{n q}
+ 1/24 * V_{m p} * F_{q n}
+ 1/24 * V_{m q} * F_{n p}
- 1/24 * V_{m q} * F_{p n}
- 1/24 * V_{n m} * F_{p q}
+ 1/24 * V_{n m} * F_{q p}
+ 1/24 * V_{n p} * F_{m q}
- 1/24 * V_{n p} * F_{q m}
- 1/24 * V_{n q} * F_{m p}
+ 1/24 * V_{n q} * F_{p m}
+ 1/24 * V_{p m} * F_{n q}
- 1/24 * V_{p m} * F_{q n}
- 1/24 * V_{p n} * F_{m q}
+ 1/24 * V_{p n} * F_{q m}
+ 1/24 * V_{p q} * F_{m n}
- 1/24 * V_{p q} * F_{n m}
- 1/24 * V_{q m} * F_{n p}
+ 1/24 * V_{q m} * F_{p n}
+ 1/24 * V_{q n} * F_{m p}
- 1/24 * V_{q n} * F_{p m}
- 1/24 * V_{q p} * F_{m n}
+ 1/24 * V_{q p} * F_{n m};
B:= V_{m n} * F_{p q}
- V_{m p} * F_{n q}
+ V_{m q} * F_{n p}
+ V_{n p} * F_{m q}
- V_{n q} * F_{m p}
+ V_{p q} * F_{m n};
@(A)-@(B):
@canonicalise!(%):
@collect_terms!(%);
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