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docs/make.jl

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@@ -12,8 +12,8 @@ makedocs(; modules=[TensorKit, TensorKitSectors],
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sitename="TensorKit.jl",
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authors="Jutho Haegeman",
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warnonly=[:missing_docs, :cross_references],
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format=Documenter.HTML(; prettyurls=get(ENV, "CI", nothing) == "true",
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mathengine=MathJax(), assets=["assets/custom.css"]),
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format=Documenter.HTML(; prettyurls=true, mathengine=MathJax(),
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assets=["assets/custom.css"]),
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pages=pages,
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pagesonly=true)
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docs/src/man/categories.md

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@@ -47,7 +47,7 @@ circuit convention). Throughout this manual, we stick to this latter convention
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not very common in manuscripts on category theory):
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```@raw html
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<img src="img/diagram_morphism.svg" alt="composition" class="color-invertible"/>
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<img src="../img/diagram_morphism.svg" alt="composition" class="color-invertible"/>
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```
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The direction of the arrows, which become important once we introduce duals, are also
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and target:
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```@raw html
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<img src="img/diagram-tensorproduct.svg" alt="tensorproduct" class="color-invertible"/>
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<img src="../img/diagram-tensorproduct.svg" alt="tensorproduct" class="color-invertible"/>
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```
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Another relevant example is the category ``\mathbf{SVect}_𝕜``, which has as objects *super
@@ -263,7 +263,7 @@ coevaluation of ``V`` and ``W``, such that ``{}^{∨}W ⊗ {}^{∨}V`` is at lea
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Graphically, we represent the exact pairing and snake rules as
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```@raw html
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<img src="img/diagram-leftdual.svg" alt="left dual" class="color-invertible"/>
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<img src="../img/diagram-leftdual.svg" alt="left dual" class="color-invertible"/>
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```
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Note that we denote the dual objects ``{}^{∨}V`` as a line ``V`` with arrows pointing in the
@@ -276,7 +276,7 @@ associated pairings, the right evaluation ``\tilde{ϵ}_V: V ⊗ V^{∨} → I``
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``\tilde{η}_V: I → V^{∨} ⊗ V``, satisfying
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```@raw html
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<img src="img/diagram-rightdual.svg" alt="right dual" class="color-invertible"/>
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<img src="../img/diagram-rightdual.svg" alt="right dual" class="color-invertible"/>
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```
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In particular, one could choose ``\tilde{ϵ}_{{}^{∨}V} = ϵ_V`` and thus define ``V`` as the
@@ -287,7 +287,7 @@ If objects ``V`` and ``W`` have left (respectively right) duals, than for a morp
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*transpose* ``{}^{∨}f ∈ \mathrm{Hom}({}^{∨}V, {}^{∨}W)`` (respectively ``f^{∨} ∈ \mathrm{Hom}(V^{∨}, W^{∨})``) as
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```@raw html
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<img src="img/diagram-transpose.svg" alt="transpose" class="color-invertible"/>
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<img src="../img/diagram-transpose.svg" alt="transpose" class="color-invertible"/>
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```
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where on the right we also illustrate the mapping from
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They are graphically represented as
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```@raw html
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<img src="img/diagram-trace.svg" alt="trace" class="color-invertible"/>
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<img src="../img/diagram-trace.svg" alt="trace" class="color-invertible"/>
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```
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and they do not need to coincide. Note that
@@ -404,13 +404,13 @@ The braiding isomorphism ``τ_{V,W}`` and its inverse are graphically represente
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lines ``V`` and ``W`` crossing over and under each other:
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```@raw html
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<img src="img/diagram-braiding.svg" alt="braiding" class="color-invertible"/>
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<img src="../img/diagram-braiding.svg" alt="braiding" class="color-invertible"/>
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```
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such that we have
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```@raw html
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<img src="img/diagram-braiding2.svg" alt="braiding relations" class="color-invertible"/>
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<img src="../img/diagram-braiding2.svg" alt="braiding relations" class="color-invertible"/>
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```
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where the expression on the right hand side, ``τ_{W,V}∘τ_{V,W}`` can generically not be
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``τ_{V^*,W} = λ_{W ⊗ V^*} ∘ (ϵ_V ⊗ \mathrm{id}_{W ⊗ V^*}) ∘ (\mathrm{id}_{V^*} ⊗ τ_{V,W}^{-1} ⊗ \mathrm{id}_{V^*}) ∘ (\mathrm{id}_{V^*⊗ W} ⊗ η_V) ∘ ρ_{V^* ⊗ W}^{-1}``, i.e.
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```@raw html
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<img src="img/diagram-braidingdual.svg" alt="braiding dual" class="color-invertible"/>
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<img src="../img/diagram-braidingdual.svg" alt="braiding dual" class="color-invertible"/>
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```
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**Balanced categories** ``C`` are braided categories that come with a **twist** ``θ``, a
@@ -460,7 +460,7 @@ where we omitted the necessary left and right unitors and associators. Graphical
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twists and their inverse (for which we refer to [^turaev]) are then represented as
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```@raw html
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<img src="img/diagram-twists.svg" alt="twists" class="color-invertible"/>
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<img src="../img/diagram-twists.svg" alt="twists" class="color-invertible"/>
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```
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The graphical representation also makes it straightforward to verify that
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or graphically
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```@raw html
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<img src="img/diagram-pivotalfromtwist.svg" alt="pivotal from twist" class="color-invertible"/>
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<img src="../img/diagram-pivotalfromtwist.svg" alt="pivotal from twist" class="color-invertible"/>
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```
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where we have drawn ``θ`` as ``θ^{\mathrm{l}}`` on the left and as ``θ^{\mathrm{r}}`` on
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their original orientation before the mirror operation):
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```@raw html
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<img src="img/diagram-dagger.svg" alt="dagger" class="color-invertible"/>
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<img src="../img/diagram-dagger.svg" alt="dagger" class="color-invertible"/>
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```
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where for completeness we have also depicted the graphical representation of the transpose,
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Graphically, we represent these relations as
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```@raw html
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<img src="img/diagram-fusion.svg" alt="fusion" class="color-invertible"/>
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<img src="../img/diagram-fusion.svg" alt="fusion" class="color-invertible"/>
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```
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and also refer to the inclusion and projection maps as splitting and fusion tensor,
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i.e. graphically:
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```@raw html
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<img src="img/diagram-Fmove.svg" alt="Fmove" class="color-invertible"/>
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<img src="../img/diagram-Fmove.svg" alt="Fmove" class="color-invertible"/>
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```
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The matrix ``F^{abc}_d`` is thus a unitary matrix. The pentagon coherence equation can also
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``F^{ab1}_c``, which are graphically represented as
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```@raw html
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<img src="img/diagram-Fmove1.svg" alt="Fmove1" class="color-invertible"/>
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<img src="../img/diagram-Fmove1.svg" alt="Fmove1" class="color-invertible"/>
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```
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In the case of group representations, i.e. the category ``\mathbf{Rep}_{\mathsf{G}}``, the
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*Frobenius-Schur indicator*. Graphically, we represent this isomorphism and its relations as
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```@raw html
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<img src="img/diagram-Zisomorphism.svg" alt="Zisomorphism" class="color-invertible"/>
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<img src="../img/diagram-Zisomorphism.svg" alt="Zisomorphism" class="color-invertible"/>
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```
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We can now discuss the relation between the exact pairing and the fusion and splitting
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graphical representation is more enlightning:
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```@raw html
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<img src="img/diagram-ZtoF.svg" alt="ZtoF" class="color-invertible"/>
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<img src="../img/diagram-ZtoF.svg" alt="ZtoF" class="color-invertible"/>
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```
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With these definitions, we can now also evaluate the action of the evaluation map on the
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splitting tensors, namely
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```@raw html
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<img src="img/diagram-splittingfusionrelation.svg" alt="splittingfusionrelation" class="color-invertible"/>
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<img src="../img/diagram-splittingfusionrelation.svg" alt="splittingfusionrelation" class="color-invertible"/>
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```
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where again bar denotes complex conjugation in the second line, and we introduced two new
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``f = d_a^{-1} \mathrm{tr}(f) \mathrm{id}_a``, i.e.
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```@raw html
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<img src="img/diagram-Brelation.svg" alt="Brelation" class="color-invertible"/>
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<img src="../img/diagram-Brelation.svg" alt="Brelation" class="color-invertible"/>
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```
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allows to conclude that
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or graphically
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```@raw html
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<img src="img/diagram-braidingR.svg" alt="braidingR" class="color-invertible"/>
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<img src="../img/diagram-braidingR.svg" alt="braidingR" class="color-invertible"/>
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```
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The hexagon coherence axiom for the braiding and the associator can then be reexpressed in
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```@raw html
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<img src="img/diagram-simpletwist.svg" alt="simpletwist" class="color-invertible"/>
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<img src="../img/diagram-simpletwist.svg" alt="simpletwist" class="color-invertible"/>
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```
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Henceforth, we reserve ``θ_a`` for the scalar value itself. Note that ``θ_a = θ_{\bar{a}}``
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Schur indicator via ``θ_a χ_a R^{aa}_1 =1``, because of
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```@raw html
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<img src="img/diagram-twistfrobeniusschur.svg" alt="twistfrobeniusschur" class="color-invertible"/>
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<img src="../img/diagram-twistfrobeniusschur.svg" alt="twistfrobeniusschur" class="color-invertible"/>
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```
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For the recurring example of ``\mathbf{Rep}_{\mathsf{G}}``, the braiding acts simply as the

docs/src/man/sectors.md

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manipulations, which we summarize here:
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```@raw html
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<img src="img/tree-summary.svg" alt="summary" class="color-invertible"/>
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<img src="../img/tree-summary.svg" alt="summary" class="color-invertible"/>
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```
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As always, we refer to the subsection on
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⊗ V_3) ⊗ … )⊗ V_{N_1})``, the corresponding fusion and splitting trees take the form
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```@raw html
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<img src="img/tree-simple.svg" alt="double fusion tree" class="color-invertible"/>
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<img src="../img/tree-simple.svg" alt="double fusion tree" class="color-invertible"/>
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for the specific case ``N_1=4`` and ``N_2=3``. We can separate this tree into the fusing
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of splitting and fusion tree would look as
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```@raw html
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<img src="img/tree-extended.svg" alt="extended double fusion tree" class="color-invertible"/>
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<img src="../img/tree-extended.svg" alt="extended double fusion tree" class="color-invertible"/>
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The presence of these isomorphisms will be important when we start to bend lines, to move
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linear combination of new fusion trees in canonical order:
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```@raw html
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<img src="img/tree-artinbraid.svg" alt="artin braid" class="color-invertible"/>
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<img src="../img/tree-artinbraid.svg" alt="artin braid" class="color-invertible"/>
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While the upper path is the most intuitive, it requires two recouplings or F-moves (one
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`levels=(1,2,3,4,5)` and `permutation=(5,3,1,4,2)`, the corresponding braid is given by
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```@raw html
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<img src="img/tree-braidinterface.svg" alt="braid interface" class="color-invertible"/>
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<img src="../img/tree-braidinterface.svg" alt="braid interface" class="color-invertible"/>
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that is, the first sector or space goes to position 3, and crosses over all other lines,
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that we need is summarized in
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```@raw html
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<img src="img/tree-linebending.svg" alt="line bending" class="color-invertible"/>
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<img src="../img/tree-linebending.svg" alt="line bending" class="color-invertible"/>
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We will only need the B-symbol and not the A-symbol. Applying the left evaluation on the
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need to take the presence of these isomorphisms into account. Indeed, we obtain the relation
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```@raw html
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<img src="img/tree-linebending2.svg" alt="dual line bending" class="color-invertible"/>
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<img src="../img/tree-linebending2.svg" alt="dual line bending" class="color-invertible"/>
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Hence, bending an `isdual` sector from the splitting tree to the fusion tree yields an
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the fusion and splitting tree:
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```@raw html
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<img src="img/tree-repartition.svg" alt="repartition" class="color-invertible"/>
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<img src="../img/tree-repartition.svg" alt="repartition" class="color-invertible"/>
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```
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The result is returned as a dictionary with keys `(f1′, f2′)` and the corresponding `coeff`

docs/src/man/tensors.md

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@@ -149,7 +149,7 @@ fusiontrees((a1, …, aN₁), c)` as ``X^{a_1, …, a_{N₁}}_{c,α}`` where
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[corresponding section](@ref ss_fusiontrees). The tensor is then represented as
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```@raw html
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<img src="img/tensor-storage.svg" alt="tensor storage" class="color-invertible"/>
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<img src="../img/tensor-storage.svg" alt="tensor storage" class="color-invertible"/>
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In this diagram, we have indicated how the tensor map can be rewritten in terms of a block
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(codomain) side. In more detail, it is given by
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```@raw html
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<img src="img/tensor-unitary.svg" alt="tensor unitary" class="color-invertible"/>
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<img src="../img/tensor-unitary.svg" alt="tensor unitary" class="color-invertible"/>
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```
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Indeed, remembering that ``V_i = ⨁_{a_i} R_{a_i} ⊗ ℂ^{n_{a_i}}`` with ``R_{a_i}`` the
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```@raw html
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<img src="img/tensor-transpose.svg" alt="transpose" class="color-invertible"/>
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<img src="../img/tensor-transpose.svg" alt="transpose" class="color-invertible"/>
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```
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In categorical language, the reason for this extra twist is that we use the left
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We sketch such a rearrangement in the following picture
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```@raw html
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<img src="img/tensor-bosoniccontraction.svg" alt="tensor unitary" class="color-invertible"/>
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<img src="../img/tensor-bosoniccontraction.svg" alt="tensor unitary" class="color-invertible"/>
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Hence, we can now specify such a tensor diagram, henceforth called a tensor contraction or
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network. For the particular case of matrix matrix multiplication, which also captures more general settings by appropriotely combining spaces into a single line, we indeed find
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```@raw html
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<img src="img/tensor-contractionreorder.svg" alt="tensor contraction reorder" class="color-invertible"/>
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<img src="../img/tensor-contractionreorder.svg" alt="tensor contraction reorder" class="color-invertible"/>
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```
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or thus, the following to lines of code yield the same result

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