Skeletal fusion categories and symbol conventions

Let $\mathcal C$ be a split fusion category over a field $k$, and choose representatives $S_1,\ldots,S_r$ of its simple objects. Decompositions into these simples give a skeletal linear model: objects become multiplicity vectors, morphisms become blocks of matrices over $k$, and the tensor structure is described by fusion rules and associator matrices in chosen bases. TensorCategories.jl implements this model as SixJCategory. We use the definitions of semisimple and fusion categories in Etingof et al. (2015), Chapters 1, 2, and 4. The implementation of this model and its use in the package are described by Mäurer and Thiel (2024), §2.

The name SixJCategory refers to this general skeletal data model. It allows arbitrary fusion multiplicities, and its associator entries need not be literal Wigner $6j$-symbols.

Objects and morphisms

An object

\[\label{eq:skeletal-object-decomposition} X=\bigoplus_{i=1}^r S_i^{\oplus m_i}\]

is represented by the vector $(m_1,\ldots,m_r)$ of nonnegative integers. In the implementation this vector is X.components, and C[i] denotes the chosen representative $S_i$. The zero object has all multiplicities zero, and direct sums add multiplicity vectors.

To keep formulas and code legible, the manual also uses a simple label such as $a$ for its position in simples(C) when it occurs inside an array access. Thus C.ass[a,b,c,d] means the block indexed by the positions of the four simples $a,b,c,d$; Julia code must of course supply the corresponding integers.

A SixJCategory is mutable, and its objects retain that particular category as their parent. Two independently constructed categories can carry identical arrays without being the same parent; transport objects explicitly between them.

If

\[\label{eq:skeletal-target-decomposition} Y=\bigoplus_{i=1}^r S_i^{\oplus n_i},\]

then

\[\label{eq:skeletal-homspace} \operatorname{Hom}_{\mathcal C}(X,Y) \cong\bigoplus_{i=1}^r\operatorname{Mat}_{m_i\times n_i}(k).\]

A morphism $f:X\to Y$ is therefore stored as one $m_i\times n_i$ matrix for each simple $S_i$. These matrices act on row coordinates, so the block representing $g\circ f$ is the block for $f$ multiplied by the block for $g$.

This description uses $\operatorname{End}_{\mathcal C}(S_i)=k$. In a non-split semisimple category, the blocks have coefficients in the division algebras $\operatorname{End}_{\mathcal C}(S_i)$ instead. SixJCategory implements the split case; see scalar extension and splitting for the non-split setting.

The matrix blocks describe maps between multiplicity spaces. Their existence does not require a monoidal fiber functor $\mathcal C\to\operatorname{Vec}_k$; see Fiber functors and semisimple coordinates.

Fusion rules

The fusion multiplicities are the structure constants of the Grothendieck ring in its simple-object basis:

\[\label{eq:skeletal-fusion-rule} S_i\otimes S_j\cong \bigoplus_l S_l^{\oplus N_{ij}^{\,l}}, \qquad N_{ij}^{\,l} =\dim_k\operatorname{Hom}_{\mathcal C}(S_i\otimes S_j,S_l).\]

The integer array C.tensor_product[i,j,l] stores $N_{ij}^{\,l}$. By bilinearity,

\[\label{eq:skeletal-tensor-multiplicity} [X\otimes Y:S_l]=\sum_{i,j}m_i n_jN_{ij}^{\,l}.\]

The simple unit has a distinguished index $u$ and satisfies

\[\label{eq:skeletal-unit-fusion} N_{ui}^{\,j}=N_{iu}^{\,j}=\delta_{ij}.\]

The program stores this index separately through set_one!.

For morphisms, tensor products use Kronecker products of matrix blocks, with one copy for each binary fusion channel. Coordinates in these copies require a choice of basis in every space $\operatorname{Hom}_{\mathcal C}(S_i\otimes S_j,S_l)$.

Associativity of the fusion rules requires

\[\label{eq:skeletal-associativity-dimensions} \sum_eN_{ab}^{\,e}N_{ec}^{\,d} = \sum_fN_{bc}^{\,f}N_{af}^{\,d}.\]

The equality of these dimensions is not yet an associator. To write its matrix, one must first choose and order bases in the binary fusion spaces.

Fusion bases and structural matrices

Scalar structural data arise only after bases have been chosen in the binary fusion spaces. We now fix those bases, use them to define the matrices of the associator, braiding, and pivotal structure, and only then name their scalar entries.

Binary fusion bases

The implementation uses the fixed order $(S_1,\ldots,S_r)$ returned by simples(C). For simple objects $a,b,e$, put

\[\label{eq:binary-fusion-space} V_{ab}^{e}=\operatorname{Hom}_{\mathcal C}(a\otimes b,e), \qquad N_{ab}^{e}=\dim_k V_{ab}^{e},\]

and choose an ordered projection basis

\[\label{eq:binary-projection-basis} p^{ab}_{e,\mu}:a\otimes b\longrightarrow e, \qquad 1\leq \mu\leq N_{ab}^{e}.\]

Let

\[\label{eq:binary-splitting-basis} s_{ab}^{e,\mu}:e\longrightarrow a\otimes b\]

be the composition-dual splitting basis. Thus

\[\label{eq:composition-dual-bases} p^{ab}_{e,\mu}\circ s_{ab}^{e,\nu} =\delta_{\mu,\nu}\operatorname{id}_e, \qquad \sum_{e,\mu}s_{ab}^{e,\mu}\circ p^{ab}_{e,\mu} =\operatorname{id}_{a\otimes b}.\]

These binary bases determine bases for the two projection spaces of a triple tensor product. For a fixed output simple $d$, define

\[\label{eq:left-triple-projection} L_{e,\mu,\nu} =p^{ec}_{d,\nu}\circ \bigl(p^{ab}_{e,\mu}\otimes\operatorname{id}_c\bigr) :(a\otimes b)\otimes c\longrightarrow d\]

and

\[\label{eq:right-triple-projection} R_{f,\rho,\sigma} =p^{af}_{d,\sigma}\circ \bigl(\operatorname{id}_a\otimes p^{bc}_{f,\rho}\bigr) :a\otimes(b\otimes c)\longrightarrow d.\]

The left paths are ordered lexicographically as $(e,\mu,\nu)$, with $\nu$ varying fastest. The right paths are ordered as $(f,\rho,\sigma)$, with $\sigma$ varying fastest. Intermediate simples follow the order of simples(C).

The associator matrix

Let

\[\label{eq:associator-map} \alpha_{a,b,c}:(a\otimes b)\otimes c \longrightarrow a\otimes(b\otimes c)\]

be the associator. Its block with output $d$ is the matrix $A=A^{abc}_d$ defined by

\[\label{eq:associator-projection-bases} R_{f,\rho,\sigma}\circ\alpha_{a,b,c} =\sum_{e,\mu,\nu} A^{abc}_d[(e,\mu,\nu),(f,\rho,\sigma)] L_{e,\mu,\nu}.\]

Thus rows are left paths and columns are right paths. In the documentation's index notation, the stored block is C.ass[a,b,c,d]. Its size is

\[\label{eq:associator-block-size} \left(\sum_e N_{ab}^{e}N_{ec}^{d}\right) \times \left(\sum_f N_{bc}^{f}N_{af}^{d}\right).\]

Associativity of the fusion rules makes the two numbers equal.

Let $L^{e,\mu,\nu}$ and $R^{f,\rho,\sigma}$ be the triple-product splittings induced by the composition-dual binary bases. Equation $\eqref{eq:associator-projection-bases}$ is equivalent to

\[\label{eq:associator-splitting-bases} \alpha_{a,b,c}\circ L^{e,\mu,\nu} =\sum_{f,\rho,\sigma} A^{abc}_d[(e,\mu,\nu),(f,\rho,\sigma)] R^{f,\rho,\sigma}.\]

The entries of $A^{abc}_d$ are the $F$-symbols in this manual. Explicitly,

\[\label{eq:F-symbol-definition} \left[F^{abc}_d\right]_{(e,\mu,\nu),(f,\rho,\sigma)} :=A^{abc}_d[(e,\mu,\nu),(f,\rho,\sigma)].\]

Equation $\eqref{eq:associator-splitting-bases}$ replaces a left-associated splitting tree by a linear combination of right-associated splitting trees; this change of basis is the $F$-move. The first multi-index in $\eqref{eq:F-symbol-definition}$ labels the input tree and the second labels the output tree. At the coordinate-free level, this is the associativity isomorphism on the splitting spaces $H^e_{ab}=\operatorname{Hom}(e,a\otimes b)$ in Etingof et al. (2015), §4.9, Eqs. (4.12)–(4.13). The same direction and coefficient convention are used in Bonderson et al. (2008), Eq. (2.14) and Barkeshli et al. (2019), Eq. (10). Both references work with unitary anyon models and orthonormal splitting bases. The same coefficient convention makes sense for the composition-dual bases used here over an arbitrary splitting field.

The function F_symbols(C; convention=...) returns all admissible coefficients, including zeros, as a Julia dictionary. The two accepted dictionary conventions are:

With convention=:bonderson, the value is precisely the entry in $\eqref{eq:F-symbol-definition}$ named by the two fusion paths in the key. The default is convention=:column_major_packing, the layout used by TensorCategories data files. In that layout the suffix records the order used to pack the entries of $A$ in Julia column-major order: for fixed $a,b,c,d$, the keys are traversed in the nested order $(e,f,\nu,\mu,\rho,\sigma)$ and matched with successive entries of vec(A). Programmatically, this pairs the column-major list of matrix entries with a nested enumeration of the admissible basis morphisms. The keys record that enumeration rather than a pair of direct fusion-path indices. Since the admissible path lists depend on $(a,b,c,d)$ and need not coincide, conversion to direct path indices is block-dependent and cannot be given by a fixed permutation of the key positions. The data exchange page gives the equivalent entry-by-entry reconstruction rule. Changing convention changes the dictionary keys and their interpretation, not the matrix $A$, the chosen bases, or the gauge. numeric_F_symbols uses the same keyword.

The package stores matrices for row coordinates: whenever the represented composition is defined,

matrix(g ∘ f) == matrix(f) * matrix(g)

A reader who instead represents splitting-space vectors by columns therefore uses $A^{\mathsf T}$ as the matrix of the associator. This transpose is only a coordinate convention. It is neither an inverse nor a complex conjugate, and composition-dual bases need not be Hermitian-adjoint bases.

The braiding matrix

For simple objects $a,b,d$, let $B=B^{ab}_d$ be defined by

\[\label{eq:braiding-projection-bases} p^{ba}_{d,\nu}\circ c_{a,b} =\sum_{\mu} B^{ab}_d[\mu,\nu]p^{ab}_{d,\mu}.\]

In composition-dual splitting bases this is

\[\label{eq:braiding-splitting-bases} c_{a,b}\circ s_{ab}^{d,\mu} =\sum_{\nu}B^{ab}_d[\mu,\nu]s_{ba}^{d,\nu}.\]

The entries of $B^{ab}_d$ are the $R$-symbols:

\[\label{eq:R-symbol-definition} \left[R^{ab}_d\right]_{\mu,\nu}:=B^{ab}_d[\mu,\nu].\]

Equation $\eqref{eq:braiding-splitting-bases}$ is the $R$-move induced by the braiding. The row index $\mu$ labels the input splitting vector $s_{ab}^{d,\mu}$ and its composition-dual projection $p^{ab}_{d,\mu}$; the column index $\nu$ similarly labels the output splitting vector and dual projection for $b\otimes a$. This is precisely the convention of Bonderson et al. (2008), Eqs. (2.30)–(2.32), abbreviated BSS below: their operator $R_{ab}$ sends the splitting basis for $a\otimes b$ to that for $b\otimes a$, and

\[\label{eq:bonderson-R-identification} B^{ab}_d[\mu,\nu] =\left[R^{ab}_d\right]^{\mathrm{BSS}}_{\mu,\nu}\]

after identifying the chosen splitting bases. With the same index notation, the structural matrix is C.braiding[a,b,d].

The order in the superscript is not uniform in the physics literature. Barkeshli et al. (2019), Eqs. (27)–(28), abbreviated BBCW below, define $R^{ab}_d:V^{ba}_d\to V^{ab}_d$. With matching bases, their notation is therefore related to the package's by

\[\label{eq:barkeshli-R-identification} B^{ab}_d[\mu,\nu] =\left[R^{ba}_d\right]^{\mathrm{BBCW}}_{\mu,\nu}.\]

The function R_symbols(C; convention=...) returns all admissible coefficients, including zeros. The two accepted dictionary conventions are:

Without multiplicities, both conventions use the key [a,b,d] and return the single entry of $B^{ab}_d$. Thus the two multiplicity indices are transposed in the default dictionary layout. This is a packing rule, not an inverse braiding or a change of gauge. numeric_R_symbols uses the same keyword.

Both functions return an ordinary Dict, which does not retain the selected convention. Code that stores or passes the dictionary separately from the category must therefore retain the convention as accompanying metadata.

The order of $a$ and $b$ matters. The inverse of $B^{ab}_d$ represents the inverse map from $b\otimes a$ to $a\otimes b$; it is not generally $B^{ba}_d$.

Pivotal coefficients

A pivotal structure is a monoidal natural isomorphism

\[\label{eq:pivotal-structure-map} j:\operatorname{id}_{\mathcal C}\Longrightarrow(-)^{**}\]

(Etingof et al., 2015; Definition 4.7.7). In the skeletal model the chosen duality has $S_i^{**}=S_i$. Since $S_i$ is split simple, the component of $j$ is a scalar:

\[\label{eq:P-symbol-definition} j_{S_i}=P_i\operatorname{id}_{S_i},\qquad P_i\in k^\times.\]

TensorCategories.jl calls the scalars $P_i$ the $P$-symbols. They are stored as C.pivotal[i], and P_symbols(C) returns the dictionary Dict([i] => P_i). This function name refers specifically to the components in $\eqref{eq:P-symbol-definition}$; the term “pivotal symbols” is also used elsewhere for different data attached to trivalent fusion spaces.

The coefficients $P_i$ are additional structure: they are not determined by the $F$- or $R$-symbols. They must make $j$ monoidal. If

\[\label{eq:double-dual-tensorator} \phi_{X,Y}:(X\otimes Y)^{**}\longrightarrow X^{**}\otimes Y^{**}\]

is the package's monoidal structure of the double-dual functor, the required identity is

\[\label{eq:pivotal-monoidality} j_X\otimes j_Y=\phi_{X,Y}\circ j_{X\otimes Y}.\]

The initializer six_j_category sets $P_i=1$ for every simple object, and set_pivotal! replaces these components. Validation is explicit: use is_pivotal(C; check=true) to check $\eqref{eq:pivotal-monoidality}$. Sphericality is the additional equality of the left and right pivotal traces and can be checked with is_spherical(C; check=true). Since a $P$-symbol dictionary has one scalar per simple object rather than matrix entries indexed by fusion paths, P_symbols has no convention keyword.

When skeletonize(C) constructs a skeletal model from a concrete category, the ordered simple representatives and the bases of $\operatorname{Hom}(S_i\otimes S_j,S_k)$ determine the skeletal associator, braiding, and duality. The same choices must therefore be used when the pivotal structure is transported. Let $S_i$ be the source representative, let $\widehat S_i$ be the corresponding skeletal simple, and let $j^{(0)}_{\widehat S_i}=\operatorname{id}_{\widehat S_i}$ denote the reference double-dual identification before its pivotal coefficient is changed. The transported coefficient is

\[\label{eq:skeletal-pivotal-transport} P_i= \frac{\operatorname{Tr}_{L}(j_{S_i})} {\operatorname{Tr}_{L}(j^{(0)}_{\widehat S_i})}.\]

This trace ratio is the coordinate of the transported map in the one-dimensional split-simple double-dual Hom space. Its denominator is nonzero: in a semisimple tensor category the trace of an isomorphism from a simple object to its double dual is nonzero (Etingof et al., 2015; Proposition 4.8.4). This remains true in positive characteristic under the split semisimple hypotheses required by skeletonize. With ball-valued coefficients, an enclosure containing zero means that the chosen working precision does not resolve this nonzero trace; skeletonization then asks for higher precision instead of retaining default coefficients.

The values $P_i$ are coordinates in the resulting skeletal gauge. Changing the multiplicity-space bases can change them, even though the transported pivotal category remains equivalent. In particular, an all-one pivotal vector is a statement about a chosen presentation, not an invariant of the category.

Unit normalization

SixJCategory uses strict unit constraints in its skeletal coordinates. The normalized convention requires every associator block with a unit input to be an identity matrix, and the public associator function treats such inputs as strict. The low-level setters accept check=false for prevalidated input. With check=true, set_one! and set_associator! check unit normalization. The full pentagon is checked separately by pentagon_axiom(C).

Pentagon and hexagon equations

In a multiplicity-free category, write $\mathcal F^{abc}_d[e,f]$ for the coefficient in $\eqref{eq:associator-splitting-bases}$, and write $\mathcal R^{ab}_d$ for the coefficient in $\eqref{eq:braiding-splitting-bases}$. With inadmissible fusion paths interpreted as zero, the pentagon equation is

\[\label{eq:pentagon-multiplicity-free} \mathcal F^{fcd}_e[g,l]\,\mathcal F^{abl}_e[f,k] =\sum_h \mathcal F^{abc}_g[f,h]\, \mathcal F^{ahd}_e[g,k]\, \mathcal F^{bcd}_k[h,l].\]

This is the multiplicity-free specialization of the standard indexed pentagon equation in Barkeshli et al. (2019), Eq. (12). The two hexagon equations, in the same convention, are

\[\label{eq:hexagon-positive-multiplicity-free} \mathcal R^{ca}_e\, \mathcal F^{acb}_d[e,g]\, \mathcal R^{cb}_g =\sum_f \mathcal F^{cab}_d[e,f]\, \mathcal R^{cf}_d\, \mathcal F^{abc}_d[f,g]\]

and

\[\label{eq:hexagon-negative-multiplicity-free} (\mathcal R^{ac}_e)^{-1}\, \mathcal F^{acb}_d[e,g]\, (\mathcal R^{bc}_g)^{-1} =\sum_f \mathcal F^{cab}_d[e,f]\, (\mathcal R^{fc}_d)^{-1}\, \mathcal F^{abc}_d[f,g].\]

These agree with Bonderson (2007), Eqs. (2.57)–(2.58), with the label order and $R$-symbol superscripts used here.

Equations $\eqref{eq:hexagon-positive-multiplicity-free}$ and $\eqref{eq:hexagon-negative-multiplicity-free}$ result by suppressing the multiplicity indices in those full equations. The compact formulas printed as Bonderson (2007), Eqs. (2.78)–(2.79) reverse the ordered superscripts of the $R$-symbols relative to the defining $R$-move in Eq. (2.54) and the full hexagon equations. The convention here follows Eq. (2.54), Eqs. (2.57)–(2.58), and the categorical hexagon.

Equations $\eqref{eq:pentagon-multiplicity-free}$–$\eqref{eq:hexagon-negative-multiplicity-free}$ display the multiplicity-free scalar form. With fusion multiplicities, the coherence conditions are the same pentagon and hexagon identities between structural morphisms, using the full matrices and all four binary-basis indices. The methods pentagon_axiom(C) and hexagon_axiom(C) evaluate those morphism equations without a multiplicity-free assumption.

Changes of fusion bases

A gauge transformation changes the basis of every binary fusion space $V_{ab}^{e}$. It consequently changes both triple-product bases. If

\[\label{eq:gauge-basis-change} L'_u=\sum_s L_sU_{s,u}, \qquad R'_v=\sum_t R_tV_{t,v},\]

then the associator block in the new bases is

\[\label{eq:associator-gauge-change} A'=U^{-1}AV.\]

The matrices $U$ and $V$ are assembled from the binary basis changes, block by block over the intermediate channels. The same binary basis choices determine the transformation of $B$. Consequently, raw $F$- and $R$-symbol entries are not gauge invariants. The corresponding entrywise formulas for changes of binary splitting bases are Bonderson (2007), Eqs. (2.75)–(2.76).

Relation to other published conventions

The definitions in $\eqref{eq:F-symbol-definition}$ and $\eqref{eq:R-symbol-definition}$ are the published anyon conventions cited above. Other sources may instead use projection trees, reverse the associator, or place the output coordinate first. Those choices change the displayed matrix without changing the underlying structural morphism.

The fusion spaces in Osborne et al. (2019), §2, p. 2, and §4, pp. 3–4 are $\operatorname{Hom}(d,a\otimes b)$. Its associator map is written in column coordinates with the output channel as the first matrix index. With composition-dual bases and matching labels, that matrix is $A^{\mathsf T}$. This explains the transpose between the package's structural matrices and the convention used for the $H_3$ formulas in that reference.

A different published convention expresses the $F$-move on projection rather than splitting trees. The diagrammatic equations in Ardonne and Slingerland (2010), Eq. (2) and Barter et al. (2022), Eq. (3) suppress associators. Restoring the parenthesized sources with $\alpha:(a\otimes b)\otimes c\to a\otimes(b\otimes c)$ gives

\[\label{eq:inverse-associator-projection-matrix} L_u\circ\alpha^{-1}=\sum_vM^{F}_{u,v}R_v.\]

Here $u=(e,\mu,\nu)$ and $v=(f,\rho,\sigma)$ have the path order fixed above. Solving $\eqref{eq:associator-projection-bases}$ for $L_u\circ\alpha^{-1}$ shows that this matrix is related to the package's structural matrix by

\[\label{eq:projection-inverse-conversion} M^{F}=(A^{-1})^{\mathsf T}.\]

Thus, in the package's row-coordinate realization, the associator block corresponding to the projection-inverse matrix $M^F$ is $A=(M^F)^{-\mathsf T}$. The dictionary conventions described above change the association of keys with entries of $A$; they do not apply the inverse-transpose operation in $\eqref{eq:projection-inverse-conversion}$.

There is an analogous distinction for the braiding. The package follows the direct splitting-space $R$-move in $\eqref{eq:braiding-splitting-bases}$. If instead one represents the map on projection spaces induced by the inverse braiding and defines $M^R$ by

\[\label{eq:inverse-braiding-projection-matrix} p^{ab}_{d,\mu}\circ c_{a,b}^{-1} =\sum_\nu M^R_{\mu,\nu}p^{ba}_{d,\nu},\]

then solving $\eqref{eq:braiding-projection-bases}$ gives

\[\label{eq:projection-inverse-braiding-conversion} M^R=(B^{-1})^{\mathsf T}.\]

The package stores $B$, not $M^R$. This inverse transpose is independent of the transpose used by convention=:column_major_packing, which only assigns the entries of $B$ to dictionary keys. The same published source explicitly distinguishes the braiding from its inverse and defines its $R$-symbols using the direct move (Bonderson et al., 2008; Eqs. (2.30)–(2.32)).

These comparisons concern mathematical matrix conventions. The dictionary packing used by the database is a separate software format, described on the data exchange page.

Constructing and checking skeletal data

Conversely, arrays of plausible dimensions do not yet define a fusion category. The fusion multiplicities must give an associative unital fusion ring with duals. Every associator block must be an invertible matrix of the prescribed size, the unit blocks must have the normalization fixed above, and all pentagon equations must hold. Braiding requires invertible blocks of the prescribed sizes satisfying both hexagon equations. Pivotal and spherical structures require their own coherence conditions.

The call six_j_category(K,N,names) creates a mutable container with fusion array $N$, identity associator blocks of the required sizes, and all-one pivotal components. The names argument may be omitted, in which case the labels are X1, X2, and so on. The shorter call six_j_category(K,names) sets only the coefficient ring, rank, labels, and all-one pivotal components; a subsequent set_tensor_product! call installs the fusion array and initializes the identity associator blocks. Neither form sets the tensor unit or certifies any coherence axiom. After entering data, use pentagon_axiom(C), hexagon_axiom(C), and the relevant checked structural predicates.

These initializer methods accept a Julia Ring, but the split fusion-category interpretation on this page and algorithms that use dimensions of Hom spaces require $K$ to be a field. The initializer does not check that $N$ is a nonnegative, associative, unital fusion table or that the number of supplied names matches its rank.

The worked examples now apply these definitions to explicit categories. The data exchange section specifies the exact key packing and serialization metadata.