F-Matrix Factory for FusionRings

class sage.combinat.root_system.f_matrix.FMatrix(fusion_ring, fusion_label='f', var_prefix='fx', inject_variables=False)

Bases: object

Return an F-Matrix factory for a FusionRing.

INPUT:

• FR – a FusionRing

• fusion_label – (optional) a string used to label basis elements of the FusionRing associated to self

  See FusionRing.fusion_labels()

• var_prefix – (optional) a string indicating the desired prefix for variables denoting F-symbols to be solved

• inject_variables – (default: False) a boolean indicating whether to inject variables (FusionRing basis element labels and F-symbols) into the global namespace

The FusionRing or Verlinde algebra is the Grothendieck ring of a modular tensor category [BaKi2001]. Such categories arise in conformal field theory or in the representation theories of affine Lie algebras, or quantum groups at roots of unity. They have applications to low dimensional topology and knot theory, to conformal field theory and to topological quantum computing. The FusionRing captures much information about a fusion category, but to complete the picture, the F-matrices or 6j-symbols are needed. For example these are required in order to construct braid group representations.

We only undertake to compute the F-matrix if the FusionRing is multiplicity free meaning that the Fusion coefficients \(N^{ij}_k\) are bounded by 1. For Cartan Types \(X_r\) and level \(k\), the multiplicity-free cases are given by the following table.

Cartan Type	\(k\)
\(A_1\)	any
\(A_r, r\geq 2\)	\(\leq 2\)
\(B_r, r\geq 2\)	\(\leq 2\)
\(C_2\)	\(\leq 2\)
\(C_r, r\geq 3\)	\(\leq 1\)
\(D_r, r\geq 4\)	\(\leq 2\)
\(G_2,F_4,E_r\)	\(\leq 2\)

Beyond this limitation, computation of the F-matrix can involve very large systems of equations. A rule of thumb is that this code can compute the F-matrix for systems with \(\leq 14\) simple objects (primary fields) on a machine with 16 GB of memory. (Larger examples can be quite time consuming.)

The FusionRing and its methods capture much of the structure of the underlying tensor category. But an important aspect that is not encoded in the fusion ring is the associator, which is a homomorphism \((A\otimes B)\otimes C\to A\otimes(B\otimes C)\) that requires an additional tool, the F-matrix or 6j-symbol. To specify this, we fix a simple object \(D\) and represent the transformation

\[\text{Hom}(D,(A\otimes B)\otimes C) \to \text{Hom}(D,A\otimes(B\otimes C))\]

by a matrix \(F^{ABC}_D\). This depends on a pair of additional simple objects \(X\) and \(Y\). Indeed, we can get a basis for \(\text{Hom}(D,(A\otimes B)\otimes C)\) indexed by simple objects \(X\) in which the corresponding homomorphism factors through \(X\otimes C\), and similarly \(\text{Hom}(D,A\otimes(B\otimes C))\) has a basis indexed by \(Y\), in which the basis vector factors through \(A\otimes Y\).

See [TTWL2009] for an introduction to this topic, [EGNO2015] Section 4.9 for a precise mathematical definition, and [Bond2007] Section 2.5 for a discussion of how to compute the F-matrix. In addition to [Bond2007], worked out F-matrices may be found in [RoStWa2009] and [CHW2015].

The F-matrix is only determined up to a gauge. This is a family of embeddings \(C \to A\otimes B\) for simple objects \(A,B,C\) such that \(\text{Hom}(C, A\otimes B)\) is nonzero. Changing the gauge changes the F-matrix though not in a very essential way. By varying the gauge it is possible to make the F-matrices unitary, or it is possible to make them cyclotomic.

Due to the large number of equations we may fail to find a Groebner basis if there are too many variables.

EXAMPLES:

sage: I = FusionRing("E8",2,conjugate=True)
sage: I.fusion_labels(["i0","p","s"],inject_variables=True)
sage: f = FMatrix(I,inject_variables=True); f
creating variables fx1..fx14
Defining fx0, fx1, fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11, fx12, fx13
F-Matrix factory for The Fusion Ring of Type E8 and level 2 with Integer Ring coefficients

We have injected two sets of variables to the global namespace. We created three variables i0, p, s to represent the primary fields (simple elements) of the FusionRing. Creating the FMatrix factory also created variables fx1, fx2, ..., fx14 in order to solve the hexagon and pentagon equations describing the F-matrix. Since we called FMatrix with the parameter inject_variables=True, these have been injected into the global namespace. This is not necessary for the code to work but if you want to run the code experimentally you may want access to these variables.

EXAMPLES:

sage: f.fmatrix(s,s,s,s)
[fx10 fx11]
[fx12 fx13]

The F-matrix has not been computed at this stage, so the F-matrix \(F^{sss}_s\) is filled with variables fx10, fx11, fx12, fx13. The task is to solve for these.

As explained above The F-matrix \((F^{ABC}_D)_{X,Y}\) two other variables \(X\) and \(Y\). We have methods to tell us (depending on \(A,B,C,D\)) what the possibilities for these are. In this example with \(A=B=C=D=s\) both \(X\) and \(Y\) are allowed to be \(i_0\) or \(s\).

sage: f.f_from(s,s,s,s), f.f_to(s,s,s,s)
([i0, p], [i0, p])

The last two statments show that the possible values of \(X\) and \(Y\) when \(A=B=C=D=s\) are \(i_0\) and \(p\).

The F-matrix is computed by solving the so-called pentagon and hexagon equations. The pentagon equations reflect the Mac Lane pentagon axiom in the definition of a monoidal category. The hexagon relations reflect the axioms of a braided monoidal category, which are constraints on both the F-matrix and on the R-matrix. Optionally, orthogonality constraints may be imposed to obtain an orthogonal F-matrix.

sage: f.get_defining_equations("pentagons")[1:3]
[fx9*fx12 - fx2*fx13, fx3*fx8 - fx4*fx9]
sage: f.get_defining_equations("hexagons")[1:3]
[fx11*fx12 + (-zeta128^32)*fx13^2 + (-zeta128^48)*fx13,
 fx10*fx11 + (-zeta128^32)*fx11*fx13 + (-zeta128^16)*fx11]
sage: f.get_orthogonality_constraints()[1:3]
[fx1^2 - 1, fx2^2 - 1]

There are two methods available to compute an F-matrix. The first, find_cyclotomic_solution() uses only the pentagon and hexagon relations. The second, find_orthogonal_solution() uses additionally the orthogonality relations. There are some differences that should be kept in mind.

find_cyclotomic_solution() currently works only with smaller examples. For example the FusionRing for \(G_2\) at level 2 is too large. When it is available, this method produces an F-matrix whose entries are in the same cyclotomic field as the underlying FusionRing.

sage: f.find_cyclotomic_solution()
Setting up hexagons and pentagons...
Finding a Groebner basis...
Solving...
Fixing the gauge...
adding equation... fx1 - 1
adding equation... fx11 - 1
Done!

We now have access to the values of the F-matrix using the methods fmatrix() and fmat():

sage: f.fmatrix(s,s,s,s)
[(-1/2*zeta128^48 + 1/2*zeta128^16)                                  1]
[                               1/2  (1/2*zeta128^48 - 1/2*zeta128^16)]
sage: f.fmat(s,s,s,s,p,p)
(1/2*zeta128^48 - 1/2*zeta128^16)

find_orthogonal_solution() is much more powerful and is capable of handling large cases, sometimes quickly but sometimes (in larger cases) after hours of computation. Its F-matrices are not always in the cyclotomic field that is the base ring of the underlying FusionRing, but sometimes in an extension field adjoining some square roots. When this happens, the FusionRing is modified, adding an attribute _basecoer that is a coercion from the cyclotomic field to the field containing the F-matrix. The field containing the F-matrix is available through field().

sage: f = FMatrix(FusionRing("B3",2))
sage: f.find_orthogonal_solution(verbose=False,checkpoint=True)     # not tested (~100 s)
sage: all(v in CyclotomicField(56) for v in f.get_fvars().values()) # not tested
True

sage: f = FMatrix(FusionRing("G2",2))
sage: f.find_orthogonal_solution(verbose=False) # long time (~11 s)
sage: f.field()                                 # long time
Algebraic Field

FR()

Return the FusionRing associated to self.

EXAMPLES:

sage: f = FMatrix(FusionRing("D3",1))
sage: f.FR()
The Fusion Ring of Type D3 and level 1 with Integer Ring coefficients

attempt_number_field_computation()

Based on the CartanType of self and data known on March 17, 2021, determine whether to attempt to find a NumberField containing all the F-symbols.

This method is used by find_orthogonal_solution() to determine a field containing all F-symbols. See field() and get_non_cyclotomic_roots().

For certain FusionRing, the number field computation does not terminate in reasonable time. In these cases, we report F-symbols as elements of the AlgebraicField QQbar.

EXAMPLES:

sage: f = FMatrix(FusionRing("F4",2))
sage: f.attempt_number_field_computation()
False
sage: f = FMatrix(FusionRing("G2",1))
sage: f.attempt_number_field_computation()
True

Note

In certain cases, F-symbols are found in the associated FusionRing’s cyclotomic field and a NumberField computation is not needed. In these cases this method returns True but the find_orthogonal_solution() solver does not undertake a NumberField computation.

certify_pentagons(use_mp=True, verbose=False)

Obtain a certificate of satisfaction for the pentagon equations, up to floating-point error.

This method converts the computed F-symbols (available through get_fvars()) to native Python floats and then checks whether the pentagon equations are satisfied using floating point arithmetic.

When self.FR().basis() has many elements, verifying satisfaction of the pentagon relations exactly using get_defining_equations() with option="pentagons" may take a long time. This method is faster, but it cannot provide mathematical guarantees.

EXAMPLES:

sage: f = FMatrix(FusionRing("C3",1))  # long time
sage: f.find_orthogonal_solution()     # long time
Computing F-symbols for The Fusion Ring of Type C3 and level 1 with Integer Ring coefficients with 71 variables...
Set up 134 hex and orthogonality constraints...
Partitioned 134 equations into 17 components of size:
[12, 12, 6, 6, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1]
Elimination epoch completed... 10 eqns remain in ideal basis
Elimination epoch completed... 0 eqns remain in ideal basis
Hex elim step solved for 51 / 71 variables
Set up 121 reduced pentagons...
Elimination epoch completed... 18 eqns remain in ideal basis
Elimination epoch completed... 5 eqns remain in ideal basis
Pent elim step solved for 64 / 71 variables
Partitioned 5 equations into 1 components of size:
[4]
Elimination epoch completed... 0 eqns remain in ideal basis
Partitioned 6 equations into 6 components of size:
[1, 1, 1, 1, 1, 1]
Computing appropriate NumberField...
sage: f.certify_pentagons()            # long time (~1.5s)
Success!!! Found valid F-symbols for The Fusion Ring of Type C3 and level 1 with Integer Ring coefficients

clear_equations()

Clear the list of equations to be solved.

EXAMPLES:

sage: f = FMatrix(FusionRing("E6",1))
sage: f.get_defining_equations('hexagons', output=False)
sage: len(f.ideal_basis)
6
sage: f.clear_equations()
sage: len(f.ideal_basis) == 0
True

clear_vars()

Reset the F-symbols.

EXAMPLES:

sage: f = FMatrix(FusionRing("C4", 1))
sage: fvars = f.get_fvars()
sage: some_key = sorted(fvars)[0]
sage: fvars[some_key]
fx0
sage: fvars[some_key] = 1
sage: f.get_fvars()[some_key]
1
sage: f.clear_vars()
sage: f.get_fvars()[some_key]
fx0

equations_graph(eqns=None)

Construct a graph corresponding to the given equations.

Every node corresponds to a variable and nodes are connected when the corresponding variables appear together in an equation.

INPUT:

• eqns – a list of polynomials

Each polynomial is either an object in the ring returned by get_poly_ring() or it is a tuple of pairs representing a polynomial using the internal representation.

If no list of equations is passed, the graph is built from the polynomials in self.ideal_basis. In this case the method assumes the internal representation of a polynomial as a tuple of pairs is used.

This method is crucial to find_orthogonal_solution(). The hexagon equations, obtained using get_defining_equations(), define a disconnected graph that breaks up into many small components. The find_orthogonal_solution() solver exploits this when undertaking a Groebner basis computation.

OUTPUT:

A Graph object. If a list of polynomial objects was given, the set of nodes in the output graph is the subset polynomial ring generators appearing in the equations.

If the internal representation was used, the set of nodes is the subset of indices corresponding to polynomial ring generators. This option is meant for internal use by the F-matrix solver.

EXAMPLES:

sage: f = FMatrix(FusionRing("A3",1))
sage: f.get_poly_ring().ngens()
27
sage: he = f.get_defining_equations('hexagons')
sage: graph = f.equations_graph(he)
sage: graph.connected_components_sizes()
[6, 3, 3, 3, 3, 3, 3, 1, 1, 1]

f_from(a, b, c, d)

Return the possible \(x\) such that there are morphisms \(d \to x \otimes c \to (a\otimes b)\otimes c\).

INPUT:

• a,b,c,d – basis elements of the associated FusionRing

EXAMPLES:

sage: fr = FusionRing("A1", 3, fusion_labels="a", inject_variables=True)
sage: f = FMatrix(fr)
sage: f.fmatrix(a1,a1,a2,a2)
[fx6 fx7]
[fx8 fx9]
sage: f.f_from(a1,a1,a2,a2)
[a0, a2]
sage: f.f_to(a1,a1,a2,a2)
[a1, a3]

f_to(a, b, c, d)

Return the possible \(y\) such that there are morphisms \(d\to a\otimes y\to a\otimes(b\otimes c)\).

INPUT:

• a,b,c,d – basis elements of the associated FusionRing

EXAMPLES:

sage: b22 = FusionRing("B2",2)
sage: b22.fusion_labels("b",inject_variables=True)
sage: B=FMatrix(b22)
sage: B.fmatrix(b2,b4,b2,b4)
[fx266 fx267 fx268]
[fx269 fx270 fx271]
[fx272 fx273 fx274]
sage: B.f_from(b2,b4,b2,b4)
[b1, b3, b5]
sage: B.f_to(b2,b4,b2,b4)
[b1, b3, b5]

field()

Return the base field containing the F-symbols.

When self is initialized, the field is set to be the cyclotomic field of the FusionRing associated to self.

The field may change after running find_orthogonal_solution(). At that point, this method could return the associated FusionRing’s cyclotomic field, an appropriate NumberField that was computed on the fly by the F-matrix solver, or the AlgebraicField QQbar.

Depending on the CartanType of self, the solver may need to compute an extension field containing certain square roots that do not belong to the associated FusionRing’s cyclotomic field.

In certain cases we revert to QQbar because the extension field computation does not seem to terminate. See attempt_number_field_computation() for more details.

The method get_non_cyclotomic_roots() returns a list of roots defining the extension of the FusionRing’s cyclotomic field needed to contain all F-symbols.

EXAMPLES:

sage: f = FMatrix(FusionRing("G2",1))
sage: f.field()
Cyclotomic Field of order 60 and degree 16
sage: f.find_orthogonal_solution(verbose=False)
sage: f.field()
Number Field in a with defining polynomial y^32 - ... - 22*y^2 + 1
sage: phi = f.get_qqbar_embedding()
sage: [phi(r).n() for r in f.get_non_cyclotomic_roots()]
[-0.786151377757423 - 8.92806368517581e-31*I]

Note

Consider using self.field().optimized_representation() to obtain an equivalent NumberField with a defining polynomial with smaller coefficients, for a more efficient element representation.

find_cyclotomic_solution(equations=None, algorithm='', verbose=True, output=False)

Solve the the hexagon and pentagon relations to evaluate the F-matrix.

This method (omitting the orthogonality constraints) produces output in the cyclotomic field, but it is very limited in the size of examples it can handle: for example, \(G_2\) at level 2 is too large for this method. You may use find_orthogonal_solution() to solve much larger examples.

INPUT:

• equations – (optional) a set of equations to be solved; defaults to the hexagon and pentagon equations

• algorithm – (optional) algorithm to compute Groebner Basis

• output – (default: False) output a dictionary of F-matrix values; this may be useful to see but may be omitted since this information will be available afterwards via the fmatrix() and fmat() methods.

EXAMPLES:

sage: f = FMatrix(FusionRing("A2",1,fusion_labels="a",inject_variables=True),inject_variables=True)
creating variables fx1..fx8
Defining fx0, fx1, fx2, fx3, fx4, fx5, fx6, fx7
sage: f.find_cyclotomic_solution(output=True)
Setting up hexagons and pentagons...
Finding a Groebner basis...
Solving...
Fixing the gauge...
adding equation... fx4 - 1
Done!
{(a2, a2, a2, a0, a1, a1): 1,
 (a2, a2, a1, a2, a1, a0): 1,
 (a2, a1, a2, a2, a0, a0): 1,
 (a2, a1, a1, a1, a0, a2): 1,
 (a1, a2, a2, a2, a0, a1): 1,
 (a1, a2, a1, a1, a0, a0): 1,
 (a1, a1, a2, a1, a2, a0): 1,
 (a1, a1, a1, a0, a2, a2): 1}

After you successfully run find_cyclotomic_solution() you may check the correctness of the F-matrix by running get_defining_equations() with option='hexagons' and option='pentagons'. These should return empty lists of equations.

EXAMPLES:

sage: f.get_defining_equations("hexagons")
[]
sage: f.get_defining_equations("pentagons")
[]

find_orthogonal_solution(checkpoint=False, save_results='', warm_start='', use_mp=True, verbose=True)

Solve the the hexagon and pentagon relations, along with orthogonality constraints, to evaluate an orthogonal F-matrix.

INPUT:

• checkpoint – (default: False) a boolean indicating whether the computation should be checkpointed. Depending on the associated CartanType, the computation may take hours to complete. For large examples, checkpoints are recommended. This method supports “warm” starting, so the calculation may be resumed from a checkpoint, using the warm_start option.

  Checkpoints store necessary state in the pickle file "fmatrix_solver_checkpoint_" + key + ".pickle", where key is the result of get_fr_str().

  Checkpoint pickles are automatically deleted when the solver exits a successful run.

• save_results – (optional) a string indicating the name of a pickle file in which to store calculated F-symbols for later use.

  If save_results is not provided (default), F-matrix results are not stored to file.

  The F-symbols may be saved to file after running the solver using save_fvars().

• warm_start – (optional) a string indicating the name of a pickle file containing checkpointed solver state. This file must have been produced by a previous call to the solver using the checkpoint option.

  If no file name is provided, the calculation begins from scratch.

• use_mp – (default: True) a boolean indicating whether to use multiprocessing to speed up calculation. The default value True is highly recommended, since parallel processing yields results much more quickly.

• verbose – (default: True) a boolean indicating whether the solver should print out intermediate progress reports.

OUTPUT:

This method returns None. If the solver runs successfully, the results may be accessed through various methods, such as get_fvars(), fmatrix(), fmat(), etc.

EXAMPLES:

sage: f = FMatrix(FusionRing("B5",1), fusion_label="b", inject_variables=True)
creating variables fx1..fx14
Defining fx0, fx1, fx2, fx3, fx4, fx5, fx6, fx7, fx8, fx9, fx10, fx11, fx12, fx13
sage: f.find_orthogonal_solution()
Computing F-symbols for The Fusion Ring of Type B5 and level 1 with Integer Ring coefficients with 14 variables...
Set up 25 hex and orthogonality constraints...
Partitioned 25 equations into 5 components of size:
[4, 3, 3, 3, 1]
Elimination epoch completed... 0 eqns remain in ideal basis
Hex elim step solved for 10 / 14 variables
Set up 7 reduced pentagons...
Elimination epoch completed... 0 eqns remain in ideal basis
Pent elim step solved for 12 / 14 variables
Partitioned 0 equations into 0 components of size:
[]
Partitioned 2 equations into 2 components of size:
[1, 1]
sage: f.fmatrix(b2, b2, b2, b2)
[ 1/2*zeta80^30 - 1/2*zeta80^10 -1/2*zeta80^30 + 1/2*zeta80^10]
[ 1/2*zeta80^30 - 1/2*zeta80^10  1/2*zeta80^30 - 1/2*zeta80^10]
sage: f.fmat(b2, b2, b2, b2, b0, b1)
-1/2*zeta80^30 + 1/2*zeta80^10

Every F-matrix \(F^{a,b,c}_d\) is orthogonal and in many cases real. We may use fmats_are_orthogonal() and fvars_are_real() to obtain correctness certificates.

EXAMPLES:

sage: f.fmats_are_orthogonal()
True
sage: f.fvars_are_real()
True

In any case, the F-symbols are obtained as elements of the associated FusionRing’s CyclotomicField, a computed NumberField, or QQbar. Currently, the field containing the F-symbols is determined based on the CartanType associated to self. See attempt_number_field_computation() for details.

findcases(output=False)

Return unknown F-matrix entries.

If run with output=True, this returns two dictionaries; otherwise it just returns the number of unknown values.

EXAMPLES:

sage: f = FMatrix(FusionRing("G2",1,fusion_labels=("i0","t")))
sage: f.findcases()
5
sage: f.findcases(output=True)
({fx4: (t, t, t, t, t, t),
 fx3: (t, t, t, t, t, i0),
 fx2: (t, t, t, t, i0, t),
 fx1: (t, t, t, t, i0, i0),
 fx0: (t, t, t, i0, t, t)},
 {(t, t, t, i0, t, t): fx0,
  (t, t, t, t, i0, i0): fx1,
  (t, t, t, t, i0, t): fx2,
  (t, t, t, t, t, i0): fx3,
  (t, t, t, t, t, t): fx4})

fmat(a, b, c, d, x, y, data=True)

Return the F-Matrix coefficient \((F^{a,b,c}_d)_{x,y}\).

EXAMPLES:

sage: f=FMatrix(FusionRing("G2",1,fusion_labels=("i0","t"),inject_variables=True))
sage: [f.fmat(t,t,t,t,x,y) for x in f._FR.basis() for y in f._FR.basis()]
[fx1, fx2, fx3, fx4]
sage: f.find_cyclotomic_solution(output=True)
Setting up hexagons and pentagons...
Finding a Groebner basis...
Solving...
Fixing the gauge...
adding equation... fx2 - 1
Done!
{(t, t, t, i0, t, t): 1,
 (t, t, t, t, i0, i0): (-zeta60^14 + zeta60^6 + zeta60^4 - 1),
 (t, t, t, t, i0, t): 1,
 (t, t, t, t, t, i0): (-zeta60^14 + zeta60^6 + zeta60^4 - 1),
 (t, t, t, t, t, t): (zeta60^14 - zeta60^6 - zeta60^4 + 1)}
sage: [f.fmat(t,t,t,t,x,y) for x in f._FR.basis() for y in f._FR.basis()]
[(-zeta60^14 + zeta60^6 + zeta60^4 - 1),
 1,
 (-zeta60^14 + zeta60^6 + zeta60^4 - 1),
 (zeta60^14 - zeta60^6 - zeta60^4 + 1)]

fmatrix(a, b, c, d)

Return the F-Matrix \(F^{a,b,c}_d\).

INPUT:

• a,b,c,d – basis elements of the associated FusionRing

EXAMPLES:

sage: f = FMatrix(FusionRing("A1",2,fusion_labels="c",inject_variables=True))
sage: f.fmatrix(c1,c1,c1,c1)
[fx0 fx1]
[fx2 fx3]
sage: f.find_cyclotomic_solution(verbose=False);
adding equation... fx4 - 1
adding equation... fx10 - 1
sage: f.f_from(c1,c1,c1,c1)
[c0, c2]
sage: f.f_to(c1,c1,c1,c1)
[c0, c2]
sage: f.fmatrix(c1,c1,c1,c1)
[ (1/2*zeta32^12 - 1/2*zeta32^4) (-1/2*zeta32^12 + 1/2*zeta32^4)]
[ (1/2*zeta32^12 - 1/2*zeta32^4)  (1/2*zeta32^12 - 1/2*zeta32^4)]

fmats_are_orthogonal()

Verify that all F-matrices are orthogonal.

This method should always return True when called after running find_orthogonal_solution().

EXAMPLES:

sage: f = FMatrix(FusionRing("D4",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: f.fmats_are_orthogonal()
True

fvars_are_real()

Test whether all F-symbols are real.

EXAMPLES:

sage: f = FMatrix(FusionRing("A1",3))           # long time
sage: f.find_orthogonal_solution(verbose=False) # long time
sage: f.fvars_are_real()                        # long time
True

get_coerce_map_from_fr_cyclotomic_field()

Return a coercion map from the associated FusionRing’s cyclotomic field into the base field containing all F-symbols (this could be the FusionRing’s CyclotomicField, a NumberField, or QQbar).

EXAMPLES:

sage: f = FMatrix(FusionRing("G2",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: f.FR().field()
Cyclotomic Field of order 60 and degree 16
sage: f.field()
Number Field in a with defining polynomial y^32 - 6*y^30 - 7*y^28 + 62*y^26 - 52*y^24 - 308*y^22 + 831*y^20 + 7496*y^18 + 18003*y^16 - 2252*y^14 + 42259*y^12 - 65036*y^10 + 29368*y^8 - 3894*y^6 + 377*y^4 - 22*y^2 + 1
sage: phi = f.get_coerce_map_from_fr_cyclotomic_field()
sage: phi.domain() == f.FR().field()
True
sage: phi.codomain() == f.field()
True

When F-symbols are computed as elements of the associated FusionRing’s base CyclotomicField, we have self.field() == self.FR().field() and this method returns the identity map on self.field().

sage: f = FMatrix(FusionRing("A2",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: phi = f.get_coerce_map_from_fr_cyclotomic_field()
sage: f.field()
Cyclotomic Field of order 48 and degree 16
sage: f.field() == f.FR().field()
True
sage: phi.domain() == f.field()
True
sage: phi.is_identity()
True

get_defining_equations(option, worker_pool=None, output=True)

Get the equations defining the ideal generated by the hexagon or pentagon relations.

INPUT:

• option – a string determining equations to be set up:

  • 'hexagons' - get equations imposed on the F-matrix by the hexagon relations in the definition of a braided category

  • 'pentagons' - get equations imposed on the F-matrix by the pentagon relations in the definition of a monoidal category

• worker_pool – (default: None) a Pool object of the Python multiprocessing module

• output – (default: True) a boolean indicating whether results should be returned, where the equations will be polynomials.

  Otherwise, the constraints are appended to self.ideal_basis. They are stored in the internal tuple representation. The output=False option is meant mostly for internal use by the F-matrix solver.

EXAMPLES:

sage: f = FMatrix(FusionRing("B2",1))
sage: f.get_defining_equations('hexagons')
[fx0 - 1,
 fx10^2 + (-zeta32^8)*fx11*fx12 + (-zeta32^12)*fx10,
 fx11*fx12 + (-zeta32^8)*fx13^2 + (zeta32^12)*fx13,
 fx2 + 1,
 fx7 + 1,
 fx3*fx8 - fx6,
 fx1*fx5 + fx2,
 fx6 - 1,
 fx4*fx9 + fx7,
 fx10*fx11 + (-zeta32^8)*fx11*fx13 + (zeta32^4)*fx11,
 fx10*fx12 + (-zeta32^8)*fx12*fx13 + (zeta32^4)*fx12]
sage: pe = f.get_defining_equations('pentagons')
sage: len(pe)
33

get_fr_str()

Auto-generate an identifying key for saving results.

EXAMPLES:

sage: f = FMatrix(FusionRing("B3",1))
sage: f.get_fr_str()
'B31'

get_fvars()

Return a dictionary of F-symbols.

The keys are sextuples \((a,b,c,d,x,y)\) of basis elements of self.FR() and the values are the corresponding F-symbols \((F^{a,b,c}_d)_{xy}\).

These values reflect the current state of a solver’s computation.

EXAMPLES:

sage: f = FMatrix(FusionRing("A2",1), inject_variables=True)
creating variables fx1..fx8
Defining fx0, fx1, fx2, fx3, fx4, fx5, fx6, fx7
sage: f.get_fvars()[(f1, f1, f1, f0, f2, f2)]
fx0
sage: f.find_orthogonal_solution(verbose=False)
sage: f.get_fvars()[(f1, f1, f1, f0, f2, f2)]
1

get_fvars_by_size(n, indices=False)

Return the set of F-symbols that are entries of an \(n \times n\) matrix \(F^{a,b,c}_d\).

INPUT:

• \(n\) – a positive integer

• indices – (default: False) a boolean.

If indices is False (default), this method returns a set of sextuples \((a,b,c,d,x,y)\) identifying the corresponding F-symbol. Each sextuple is a key in the dictionary returned by get_fvars().

Otherwise the method returns a list of integer indices that internally identify the F-symbols. The indices=True option is meant for internal use.

EXAMPLES:

sage: f = FMatrix(FusionRing("A2",2), inject_variables=True)
creating variables fx1..fx287
Defining fx0, ..., fx286
sage: f.largest_fmat_size()
2
sage: f.get_fvars_by_size(2)
{(f2, f2, f2, f4, f1, f1),
 (f2, f2, f2, f4, f1, f5),
 ...
 (f4, f4, f4, f4, f4, f0),
 (f4, f4, f4, f4, f4, f4)}

get_fvars_in_alg_field()

Return F-symbols as elements of the AlgebraicField QQbar. This method uses the embedding defined by get_qqbar_embedding() to coerce F-symbols into QQbar.

EXAMPLES:

sage: f = FMatrix(FusionRing("G2",1), fusion_label="g", inject_variables=True)
creating variables fx1..fx5
Defining fx0, fx1, fx2, fx3, fx4
sage: f.find_orthogonal_solution(verbose=False)
sage: f.field()
Number Field in a with defining polynomial y^32 - ... - 22*y^2 + 1
sage: f.get_fvars_in_alg_field()
{(g1, g1, g1, g0, g1, g1): 1,
 (g1, g1, g1, g1, g0, g0): 0.61803399? + 0.?e-8*I,
 (g1, g1, g1, g1, g0, g1): -0.7861514? + 0.?e-8*I,
 (g1, g1, g1, g1, g1, g0): -0.7861514? + 0.?e-8*I,
 (g1, g1, g1, g1, g1, g1): -0.61803399? + 0.?e-8*I}

get_non_cyclotomic_roots()

Return a list of roots that define the extension of the associated FusionRing’s base CyclotomicField containing all the F-symbols.

OUTPUT:

The list of non-cyclotomic roots is given as a list of elements of the field returned by field().

If self.field() == self.FR().field() then this method returns an empty list.

EXAMPLES:

sage: f = FMatrix(FusionRing("E6",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: f.field() == f.FR().field()
True
sage: f.get_non_cyclotomic_roots()
[]
sage: f = FMatrix(FusionRing("F4",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: f.field() == f.FR().field()
False
sage: phi = f.get_qqbar_embedding()
sage: [phi(r).n() for r in f.get_non_cyclotomic_roots()]
[-0.786151377757423 + 1.73579267033929e-59*I]

When self.field() is a NumberField, one may use get_qqbar_embedding() to embed the resulting values into QQbar.

get_orthogonality_constraints(output=True)

Get equations imposed on the F-matrix by orthogonality.

INPUT:

• output – a boolean

OUTPUT:

If output=True, orthogonality constraints are returned as polynomial objects.

Otherwise, the constraints are appended to self.ideal_basis. They are stored in the internal tuple representation. The output=False option is meant mostly for internal use by the F-matrix solver.

EXAMPLES:

sage: f = FMatrix(FusionRing("B4", 1))
sage: f.get_orthogonality_constraints()
[fx0^2 - 1,
 fx1^2 - 1,
 fx2^2 - 1,
 fx3^2 - 1,
 fx4^2 - 1,
 fx5^2 - 1,
 fx6^2 - 1,
 fx7^2 - 1,
 fx8^2 - 1,
 fx9^2 - 1,
 fx10^2 + fx12^2 - 1,
 fx10*fx11 + fx12*fx13,
 fx10*fx11 + fx12*fx13,
 fx11^2 + fx13^2 - 1]

get_poly_ring()

Return the polynomial ring whose generators denote F-symbols we seek.

EXAMPLES:

sage: f = FMatrix(FusionRing("B6",1))
sage: f.get_poly_ring()
Multivariate Polynomial Ring in fx0, ..., fx13 over
 Cyclotomic Field of order 96 and degree 32

get_qqbar_embedding()

Return an embedding from the base field containing F-symbols (the associated FusionRing’s CyclotomicField, a NumberField, or QQbar) into QQbar.

This embedding is useful for getting a better sense for the F-symbols, particularly when they are computed as elements of a NumberField. See also get_non_cyclotomic_roots().

EXAMPLES:

sage: f = FMatrix(FusionRing("F4",1), fusion_label="f", inject_variables=True)
creating variables fx1..fx5
Defining fx0, fx1, fx2, fx3, fx4
sage: f.find_orthogonal_solution()
Computing F-symbols for The Fusion Ring of Type F4 and level 1 with Integer Ring coefficients with 5 variables...
Set up 10 hex and orthogonality constraints...
Partitioned 10 equations into 2 components of size:
[4, 1]
Elimination epoch completed... 0 eqns remain in ideal basis
Hex elim step solved for 4 / 5 variables
Set up 0 reduced pentagons...
Pent elim step solved for 4 / 5 variables
Partitioned 0 equations into 0 components of size:
[]
Partitioned 1 equations into 1 components of size:
[1]
Computing appropriate NumberField...
sage: phi = f.get_qqbar_embedding()
sage: phi(f.fmat(f1,f1,f1,f1,f1,f1)).n()
-0.618033988749895 + 3.63089268571980e-21*I

get_radical_expression()

Return radical expression of F-symbols for easy visualization

EXAMPLES:

sage: f = FMatrix(FusionRing("G2",1))
sage: f.FR().fusion_labels("g", inject_variables=True)
sage: f.find_orthogonal_solution(verbose=False)
sage: radical_fvars = f.get_radical_expression()       # long time (~1.5s)
sage: radical_fvars[g1, g1, g1, g1, g1, g0]            # long time
-sqrt(1/2*sqrt(5) - 1/2)

largest_fmat_size()

Get the size of the largest F-matrix \(F^{abc}_d\).

EXAMPLES:

sage: f = FMatrix(FusionRing("B3",2))
sage: f.largest_fmat_size()
4

load_fvars(filename)

Load previously computed F-symbols from a pickle file.

See save_fvars() for more information.

EXAMPLES:

sage: f = FMatrix(FusionRing("A2",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: fvars = f.get_fvars()
sage: K = f.field()
sage: filename = f.get_fr_str() + "_solver_results.pickle"
sage: f.save_fvars(filename)
sage: del f
sage: f2 = FMatrix(FusionRing("A2",1))
sage: f2.load_fvars(filename)
sage: fvars == f2.get_fvars()
True
sage: K == f2.field()
True
sage: os.remove(filename)

Note

save_fvars(). This method does not work with intermediate checkpoint pickles; it only works with pickles containing all F-symbols, i.e. those created by save_fvars() and by specifying an optional save_results parameter for find_orthogonal_solution().

save_fvars(filename)

Save computed F-symbols for later use.

INPUT:

• filename – a string specifying the name of the pickle file to be used

The current directory is used unless an absolute path to a file in a different directory is provided.

Note

This method should only be used after successfully running one of the solvers, e.g. find_cyclotomic_solution() or find_orthogonal_solution().

When used in conjunction with load_fvars(), this method may be used to restore state of an FMatrix object at the end of a successful F-matrix solver run.

EXAMPLES:

sage: f = FMatrix(FusionRing("A2",1))
sage: f.find_orthogonal_solution(verbose=False)
sage: fvars = f.get_fvars()
sage: K = f.field()
sage: filename = f.get_fr_str() + "_solver_results.pickle"
sage: f.save_fvars(filename)
sage: del f
sage: f2 = FMatrix(FusionRing("A2",1))
sage: f2.load_fvars(filename)
sage: fvars == f2.get_fvars()
True
sage: K == f2.field()
True
sage: os.remove(filename)

singletons()

Find \(x_i\) that are automatically nonzero, because their F-matrix is \(1 \times 1\).

EXAMPLES:

sage: f = FMatrix(FusionRing("E7",1))
sage: singles = f.singletons()
sage: all(f.fmatrix(*f._var_to_sextuple[v][:4]).nrows() == 1 for v in singles)
True