The Fusion Ring of the Drinfeld Double of a Finite Group#

class sage.algebras.fusion_rings.fusion_double.FusionDouble(G, prefix='s')#

Bases: CombinatorialFreeModule

The fusion ring corresponding to the Drinfeld double of a finite group.

This is the fusion ring of the modular tensor category of modules over the Drinfeld double of a finite group. Usage is similar to FusionRing; we refer the reader to that class for more information.

INPUT:

G – a finite group
prefix – (default: 's') a prefix for the names of simple objects
inject_varables – (optional) set to True to create variables for the simple objects

REFERENCES:

EXAMPLES:

sage: G = DihedralGroup(5)
sage: H = FusionDouble(G, inject_variables=True)
sage: H.basis()
Finite family {0: s0, 1: s1, 2: s2, 3: s3, 4: s4, 5: s5, 6: s6, 7: s7, 8: s8,
               9: s9, 10: s10, 11: s11, 12: s12, 13: s13, 14: s14, 15: s15}
sage: for x in H.basis():
....:     print ("%s : %s"%(x,x^2))
....:
s0 : s0
s1 : s0
s2 : s0 + s1 + s3
s3 : s0 + s1 + s2
s4 : s0 + s2 + s3 + s6 + s7 + s8 + s9 + s10 + s11 + s12 + s13 + s14 + s15
s5 : s0 + s2 + s3 + s6 + s7 + s8 + s9 + s10 + s11 + s12 + s13 + s14 + s15
s6 : s0 + s1 + s11
s7 : s0 + s1 + s13
s8 : s0 + s1 + s15
s9 : s0 + s1 + s12
s10 : s0 + s1 + s14
s11 : s0 + s1 + s6
s12 : s0 + s1 + s9
s13 : s0 + s1 + s7
s14 : s0 + s1 + s10
s15 : s0 + s1 + s8
sage: s4*s5
s1 + s2 + s3 + s6 + s7 + s8 + s9 + s10 + s11 + s12 + s13 + s14 + s15
sage: s4.ribbon()
1
sage: s5.ribbon()
-1
sage: s8.ribbon()
zeta5^3

If the fusion double is multiplicity-free, meaning that the fusion coefficients \(N_k^{ij}\) are bounded by \(1\), then the F-matrix may be computed, by solving the pentagon and hexagon relations as described in [Bond2007] and [Ab2022], just as for FusionRing. There is a caveat here, since even if the fusion rules are multiplicity-free, if there are too many F-matrix values to compute, even if many of them are zero, in the current implementation singular cannot create enough variables. At least, this code can compute the F-matrix for the Fusion Double of the symmetric group \(S_3\), duplicating the result of [CHW2015].

sage: G1 = SymmetricGroup(3)
sage: H1 = FusionDouble(G1, prefix="u", inject_variables=True)
sage: F = H1.get_fmatrix()

The above commands create the F-matrix. You can compute all of the F-matrices with the command:

sage: H1.find_orthogonal_solution()  # not tested (10-15 minutes)

Individual F-matrices may be computed thus:

sage: F.fmatrix(u3, u3, u3, u4)  # not tested

See FMatrix for more information.

Unfortunately beyond \(S_3\) the number of simple objects is seemingly impractical. Although the FusionDouble class and its methods work well for groups of moderate size, the FMatrix may not be computable. For the dihedral group of order 8, there are already 22 simple objects, and the F-matrix seems out of reach. The actual limitation is that singular will not create a polynomial ring in more than \(2^{15}-1 = 32767\) symbols, and there are more than this many F-matrix values to be computed for the dihedral group of order 8, so in the current implementation, this FusionRing is out of reach.

It is an open problem to classify the finite groups whose fusion doubles are multiplicity-free. Abelian groups, dihedral groups, dicyclic groups, and all groups of order 16 are multiplicity-free. On the other hand, for groups of order 32, some are multiplicity-free and others are not. These can all be constructed using SmallPermutationGroup.

EXAMPLES:

sage: G = SmallPermutationGroup(16,9)
sage: F = FusionDouble(G, prefix="b",inject_variables=True)
sage: b13^2 # long time (4s)
b0 + b2 + b4 + b15 + b16 + b17 + b18 + b24 + b26 + b27

D_minus(base_coercion=True)#

Return the positive square root of self.global_q_dimension() as an element of self.field().

For the Drinfeld double of a finite group \(G\), this equals the cardinality of \(G\). This is also equal to \(\sum d_i^2 \theta_i^{\pm 1}\), where \(i\) runs through the simple objects, \(d_i\) is the quantum dimension, and \(\theta_i\) is the twist. This sum with \(\theta_i\) is denoted \(p_-\) in [BaKi2001] Chapter 3.

EXAMPLES:

sage: FusionDouble(DihedralGroup(7)).total_q_order()
14

D_plus(base_coercion=True)#

Return the positive square root of self.global_q_dimension() as an element of self.field().

EXAMPLES:

sage: FusionDouble(DihedralGroup(7)).total_q_order()
14

class Element#

Bases: IndexedFreeModuleElement

char()#

Return the character \(\chi\) corresponding to self.

The data determining a simple object consists of a conjugacy class representative \(g\) and an irreducible character \(\chi\) of the centralizer of \(g\).

See also

g()

EXAMPLES:

sage: G = DihedralGroup(5)
sage: H = FusionDouble(G, prefix="f", inject_variables=True)
sage: f10.g()
(1,2,3,4,5)
sage: f10.char()
Character of Subgroup generated by [(1,2,3,4,5)] of
    (Dihedral group of order 10 as a permutation group)

dual()#

Return the dual of self.

This method is only available for simple objects.

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: H = FusionDouble(G, prefix="j")
sage: [x for x in H.basis() if x == x.dual()]
[j0, j1, j8, j9]

g()#

The data determining a simple object consists of a conjugacy class representative \(g\) and an irreducible character \(\chi\) of the centralizer of \(g\).

Returns the conjugacy class representative of the underlying group corresponding to a simple object. See also char().

EXAMPLES:

sage: G = QuaternionGroup()
sage: H = FusionDouble(G, prefix="e", inject_variables=True)
sage: e10.g()
(1,3)(2,4)(5,7)(6,8)
sage: e10.char()
Character of Subgroup generated by [(1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6)]
    of (Quaternion group of order 8 as a permutation group)

is_simple_object()#

Determine whether self is a simple object (basis element) of the fusion ring.

EXAMPLES:

sage: H = FusionDouble(CyclicPermutationGroup(2), prefix="g", inject_variables=True)
sage: [x.is_simple_object() for x in [g0, g1, g0+g1]]
[True, True, False]

q_dimension(base_coercion=True)#

Return the q-dimension of self.

This method is only available for simple objects.

EXAMPLES:

sage: G = AlternatingGroup(4)
sage: H = FusionDouble(G)
sage: [x.q_dimension() for x in H.basis()]
[1, 1, 1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
sage: sum(x.q_dimension()^2 for x in H.basis()) == G.order()^2
True

ribbon(base_coercion=True)#

The twist or ribbon of the simple object.

EXAMPLES:

sage: H = FusionDouble(CyclicPermutationGroup(3))
sage: [i.ribbon() for i in H.basis()]
[1, 1, 1, 1, zeta3, -zeta3 - 1, 1, -zeta3 - 1, zeta3]

twist(reduced=True)#

Return a rational number \(h\) such that \(\theta = e^{i \pi h}\) is the twist of self.

The quantity \(e^{i \pi h}\) is also available using ribbon().

This method is only available for simple objects.

EXAMPLES:

sage: Q=FusionDouble(CyclicPermutationGroup(3))
sage: [x.twist() for x in Q.basis()]
[0, 0, 0, 0, 2/3, 4/3, 0, 4/3, 2/3]
sage: [x.ribbon() for x in Q.basis()]
[1, 1, 1, 1, zeta3, -zeta3 - 1, 1, -zeta3 - 1, zeta3]

N_ijk(i, j, k)#

The symmetric invariant of three simple objects.

This is the dimension of

\[Hom(i \otimes j \otimes k, s_0),\]

where \(s_0\) is the unit element (assuming prefix='s'). Method of computation is through the Verlinde formula, deducing the values from the known values of the S-matrix.

EXAMPLES:

sage: A = FusionDouble(AlternatingGroup(4),prefix="a",inject_variables=True)
sage: [A.N_ijk(a10,a11,x) for x in A.basis()]
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]

Nk_ij(i, j, k, use_characters=False)#

Return the fusion coefficient \(N^k_{ij}\).

INPUT:

i, j, k – basis elements
use_characters – (default: False) see the algorithm description below

ALGORITHM:

If use_characters=False, then this is computed using the Verlinde formula:

\[N^k_{ij} = \sum_l \frac{s(i, \ell)\, s(j, \ell)\, \overline{s(k, \ell)}}{s(I, \ell)}.\]

Otherwise we use a character theoretic method to compute the fusion coefficient \(N_{ij}^k\) as follows. Each simple object, for example \(i\) corresponds to a conjugacy class \(\mathcal{C}_i\) of the underlying group \(G\), and an irreducible character \(\chi_i\) of the centralizer \(C(g_i)\) of a fixed representative \(g_i\) of \(\mathcal{C}_i\). In addition to the fixed representative \(g_k\) of the class \(\mathcal{C}_i\) and \(\mathcal{C}_j\), the formula will make use of variable elements \(h_i\) and \(h_j\) that are subject to the condition \(h_i h_j = g_k\). See [GoMa2010] equation (7).

\[\begin{split}\frac{|\mathcal{C}_k|}{|G|} \sum_{\substack{h_i\in\mathcal{C}_i \\ h_j\in\mathcal{C}_j \\ h_ih_j=g_k}} \lvert C(h_i)\cap C(h_j) \rvert \, \langle \chi_i^{(h_i)} \chi_j^{(h_j)}, \chi_k \rangle_{C(h_i)\cap C(h_j)},\end{split}\]

where \(\chi_i^{(h_i)}\) is the character \(\chi_i\) of \(C(g_i)\) conjugated to a character of \(C(h_i)\), when \(h_i\) is a conjugate of the fixed representative \(g_i\). More exactly, there exists \(r_i\) such that \(r_i g_i r_i^{-1} = h_i\), and then \(\chi_i^{(h_i)}(x) = \chi_i(r_i^{-1}xr_i)\), and this definition does not depend on the choice of \(r_i\).

Note

This should be functionally equivalent, and testing shows that it is, but it is slower.

EXAMPLES:

sage: A = FusionDouble(AlternatingGroup(4),prefix="aa",inject_variables=True)
sage: [A.Nk_ij(aa8,aa10,x) for x in A.basis()]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1]

sage: B = FusionDouble(CyclicPermutationGroup(2))
sage: all(B.Nk_ij(x,y,z,use_characters=True) == B.Nk_ij(x,y,z)
....:     for x in B.basis() for y in B.basis() for z in B.basis())
True

dual(i)#

Return the dual object i^\ast to i.

The dual is also available as an element method of i.

EXAMPLES:

sage: K = FusionDouble(CyclicPermutationGroup(3),prefix="k")
sage: [(x,K.dual(x)) for x in K.basis()]
[(k0, k0),
(k1, k2),
(k2, k1),
(k3, k6),
(k4, k8),
(k5, k7),
(k6, k3),
(k7, k5),
(k8, k4)]
sage: all(K.dual(x)==x.dual() for x in K.basis())
True

field()#

Returns a cyclotomic field large enough to contain the values of R-matrices and twists that can arise for this fusion ring.

EXAMPLES:

sage: FusionDouble(SymmetricGroup(3)).field()
Cyclotomic Field of order 24 and degree 8

fvars_field()#

Return a field containing the CyclotomicField computed by field() as well as all the F-symbols of the associated FMatrix factory object.

This method is only available if self is multiplicity-free.

EXAMPLES:

sage: FusionDouble(SymmetricGroup(3)).fvars_field()
Cyclotomic Field of order 24 and degree 8

get_fmatrix(*args, **kwargs)#

Construct an FMatrix factory to solve the pentagon and hexagon relations and organize the resulting F-symbols.

EXAMPLES:

sage: f = FusionDouble(SymmetricGroup(3)).get_fmatrix(); f
F-Matrix factory for The Fusion Ring of the Drinfeld Double of
    Symmetric group of order 3! as a permutation group

global_q_dimension(base_coercion=True)#

Return the global quantum dimension, which is the sum of the squares of the quantum dimensions of the simple objects. For the Drinfeld double, it is the square of the order of the underlying quantum group.

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: H = FusionDouble(G)
sage: H.global_q_dimension()
576
sage: sum(x.q_dimension()^2 for x in H.basis())
576

group()#

Return the underlying group.

EXAMPLES:

sage: FusionDouble(DiCyclicGroup(4)).group()
Dicyclic group of order 16 as a permutation group

inject_variables()#

Create variables for the simple objects in the global name space.

EXAMPLES:

sage: F = FusionDouble(DiCyclicGroup(3), prefix="d")
sage: F.inject_variables()
sage: d0 + d1 + d5
d0 + d1 + d5

is_multiplicity_free(verbose=False)#

Return True if all fusion coefficients are at most 1.

EXAMPLES:

sage: FusionDouble(SymmetricGroup(3)).is_multiplicity_free()
True
sage: FusionDouble(SymmetricGroup(4)).is_multiplicity_free()
False

sage: FusionDouble(SymmetricGroup(3)).is_multiplicity_free(True)
Checking multiplicity freeness
True
sage: FusionDouble(SymmetricGroup(4)).is_multiplicity_free(True)
Checking multiplicity freeness
N(s2,s13,s13) = 2
False

one_basis()#

The unit element of the ring, which is the first basis element.

EXAMPLES:

sage: FusionDouble(CyclicPermutationGroup(2), prefix="h").one()
h1

product_on_basis(a, b)#

Return the product of two basis elements corresponding to keys \(a\) and \(b\).

INPUT:

a`, ``b – keys for the dictionary self._names representing simple objects

EXAMPLES:

sage: Q=FusionDouble(SymmetricGroup(3),prefix="q",inject_variables=True)
sage: q3*q4
q1 + q2 + q5 + q6 + q7
sage: Q._names
{0: 'q0', 1: 'q1', 2: 'q2', 3: 'q3', 4: 'q4', 5: 'q5', 6: 'q6', 7: 'q7'}
sage: Q.product_on_basis(3,4)
q1 + q2 + q5 + q6 + q7

r_matrix(i, j, k, base_coercion=True)#

Return the R-matrix entry corresponding to the subobject k in the tensor product of i with j. This method is only correct if the fusion coefficient N_{ij}^k\leq 1. See the FusionRing method for more information, including the reason for this caveat, and the algorithm.

EXAMPLES:

sage: C = FusionDouble(SymmetricGroup(3),prefix="c",inject_variables=True)
sage: c4*c5
c3 + c4
sage: [C.r_matrix(c4,c5,k) for k in [c3,c4]]
[-zeta24^6, 1]
sage: c6^2
c0 + c1 + c6
sage: [C.r_matrix(c6,c6,k) for k in [c0,c1,c6]]
[zeta3, -zeta3, -zeta3 - 1]

root_of_unity(r, base_coercion=True)#

Return \(e^{i\pi r}\) as an element of self.field() if possible.

INPUT:

r – a rational number

EXAMPLES:

sage: H = FusionDouble(DihedralGroup(6))
sage: H.field()
Cyclotomic Field of order 24 and degree 8
sage: for n in [1..7]:
....:     try:
....:         print (n,H.root_of_unity(2/n))
....:     except ValueError as err:
....:         print (n,err)
....:
1 1
2 -1
3 zeta24^4 - 1
4 zeta24^6
5 not a root of unity in the field
6 zeta24^4
7 not a root of unity in the field

s_ij(i, j, unitary=False, base_coercion=True)#

Return the element of the S-matrix of this fusion ring corresponding to the given elements.

Without the unitary option set true, this is the unnormalized S-matrix entry, denoted \(\tilde{s}_{ij}\), in [BaKi2001] Chapter 3. The normalized S-matrix entries are denoted \(s_{ij}\).

INPUT:

i, j, – a pair of basis elements
unitary – (default: False) set to True to obtain the unitary S-matrix

EXAMPLES:

sage: D = FusionDouble(SymmetricGroup(3), prefix="t", inject_variables=True)
sage: [D.s_ij(t2, x) for x in D.basis()]
[2, 2, 4, 0, 0, -2, -2, -2]
sage: [D.s_ij(t2, x, unitary=True) for x in D.basis()]
[1/3, 1/3, 2/3, 0, 0, -1/3, -1/3, -1/3]

s_ijconj(i, j, unitary=False, base_coercion=True)#

Return the conjugate of the element of the S-matrix given by self.s_ij(elt_i, elt_j, base_coercion=base_coercion).