# How to initialize a tensor with permutational symmetry?

**URL:** <https://discourse.julialang.org/t/how-to-initialize-a-tensor-with-permutational-symmetry/115390>\
**Category:** Performance\
**Tags:** tensors, tensoroperations\
**Created:** [June 9, 2024, 10:58am UTC](https://discourse.julialang.org/t/how-to-initialize-a-tensor-with-permutational-symmetry/115390 "2024-06-09T10:58:42Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![Uranium238](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/uranium238/32/209706_2.png) [@Uranium238](https://discourse.julialang.org/u/Uranium238)\
**Post date:** [June 9, 2024, 10:58am UTC](https://discourse.julialang.org/t/how-to-initialize-a-tensor-with-permutational-symmetry/115390/1 "2024-06-09T10:58:42Z")

</div>

I am working on a iterative solver for non linear Coupled Cluster equations that would require a computation of lot of tensor elements via tensor contractions. For now, I am using the TensorOperations.jl for this purpose. I am trying to initialize a tensor `g` where the values of `g[I,J,K,L]` are given for only certain `[I,J,K,L]`. And the rest are supposed to be deduced from permutational symmetries. For now, I am using this make-shift code to achieve the above, where the values `g[I,J,K,L]` had been read from some data file and only the non redundant elements were provided.

```julia
for (I,J,K,L) in non_redundant_indices
        g[K,L,I,J]=g[I,J,K,L]
        g[J,I,L,K]=g[I,J,K,L]
        g[L,K,J,I]=g[I,J,K,L]
        g[J,I,K,L]=g[I,J,K,L]
        g[L,K,I,J]=g[I,J,K,L]
        g[I,J,L,K]=g[I,J,K,L]
        g[K,L,J,I]=g[I,J,K,L]
end

```

Does anyone have an about how may I do the same but maybe a bit more conveniently or elegantly perhaps? I looked a bit into TensorKit.jl and they seem to deal with Tensors as a new data type called `TensorMap` which does support some symmetry features, but I cannot figure out how to implement this permutational symmetry in those tensors and how to exploit them during tensor operations like contractions.

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<div class="post-metadata">

**Author:** ![Dan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dan/32/42581_2.png) [@Dan](https://discourse.julialang.org/u/Dan)\
**Post date:** [June 9, 2024, 1:52pm UTC](https://discourse.julialang.org/t/how-to-initialize-a-tensor-with-permutational-symmetry/115390/2 "2024-06-09T13:52:44Z")

</div>

Perhaps a good solution would be to define a new Array type which has its own `getindex` doing the necessary transformation to arrive at a non redundant index by canonizing a general index.

This would be a commonly occuring problem, so there might be a way to use an existing package for this. I’ve managed to dig up one such package [GitHub - JuliaArrays/GetindexArrays.jl: Lazy arrays with arbitrary user-defined transformations](https://github.com/JuliaArrays/GetindexArrays.jl/) .

As an example, a way to use this to represent a triangular 3x3x3 tensor (a simple version of your symmetry requirements):

```julia
using Pkg
Pkg.add(url="https://github.com/JuliaArrays/GetindexArrays.jl/")

```

```julia
using GetindexArrays, Random
B = reshape(shuffle(1:9),(3,3)) # some base matrix
M = GetindexArray((1:3,1:3,1:3)) do r,c,d
    B[minimum((r,c,d)),maximum((r,c,d))]
end

```

and now, M is a bigger tensor, using values from B (above diagonal) as non redundant elements:

```julia
julia> B
3×3 Matrix{Int64}:
 4 2 1
 9 3 6
 5 8 7

julia> M
3×3×3 GetindexArray{Any, 3, var"#13#14", Tuple{UnitRange{Int64}, UnitRange{Int64}, UnitRange{Int64}}} with indices 1:3×1:3×1:3:
[:, :, 1] =
 4 2 1
 2 2 1
 1 1 1

[:, :, 2] =
 2 2 1
 2 3 6
 1 6 6

[:, :, 3] =
 1 1 1
 1 6 6
 1 6 7

```