SOME PROPERTIES OF THE REPRESENTATION CATEGORY OF TWISTED DRINFELD DOUBLES OF FINITE GROUPS

A criterion for a simple object of the representation category Rep(Dω(G)) of the twisted Drinfeld double Dω(G) to be a generator is given, where G is a finite group and ω is a 3-cocycle on G. A description of the adjoint category of Rep(Dω(G)) is also given. Mathematics Subject Classification (2020): 18M20


Introduction
Modular tensor categories arise in several diverse areas such as quantum group theory, vertex operator algebras, and rational conformal field theory. Let G be a finite group, let D(G) denote the Drinfeld double of G, a quasi-triangular semisimple Hopf algebra, and let Rep(D(G)) denote the category of finite-dimensional complex representations of D(G). The category Rep(D(G)) is a modular tensor category [1], and it is perhaps the most accessible constructions of a modular tensor category. As such, it is desirable to have a thorough understanding of this category.
In this paper, we make a contribution towards this goal. The category Rep(D(G)) is equivalent to the G-equivariantization of Vec G , and it is also equivalent to the center Z(Vec G ) of the tensor category Vec G of finite-dimensional G-graded complex vector spaces.
In the papers [3,4], R. Dijkgraaf, V. Pasquier, and P. Roche introduce a quasitriangular semisimple quasi-Hopf algebra D ω (G), often called the twisted Drinfeld double of G, where ω is a 3-cocycle on G. When ω = 1 this quasi-Hopf algebra coincides with the Drinfeld double D(G) considered above. The category Rep(D ω (G)) of finite-dimensional complex representations of D ω (G) is a modular tensor category. Analogous to the ω = 1 case, the category Rep(D ω (G)) is equivalent to the G-equivariantization of Vec ω G , and it is also equivalent to the center Z(Vec ω G ) of the tensor category Vec ω G of finite-dimensional G-graded complex vector spaces with associativity constraint defined using ω. Every braided group-theoretical fusion 224 DEEPAK NAIDU category is equivalent to a full fusion subcategory of some Rep(D ω (G)), and all such subcategories were parametrized in the paper [12]. This paper contains two main results, stated below. The first gives a criterion for a simple object of Rep(D ω (G)) to be a generator, and the second gives a description of the adjoint category of Rep(D ω (G)).
Theorem. Let G be a finite group, let ω be a normalized 3-cocycle on G, and let (a, χ) be a simple object of Rep(D ω (G)). Then (a, χ) is a generator of Rep(D ω (G)) if and only if the following two conditions hold.
(a) The normal closure of a in G is equal to G. and

Organization:
In Section 2, we recall basic facts about the modular tensor category Rep(D ω (G)).
In Section 3, we prove the first theorem above, and in Section 4, we prove the second theorem.

Convention and notation:
Throughout this paper we work over the field C of complex numbers. The multiplicative group of nonzero complex numbers is denoted C × . Let G be a finite group. The identity element of G is denoted e, and the center of G is denoted Z(G).
For any character χ of G, the degree of χ is denoted deg χ, the complex conjugate of χ is denoted χ, and the kernel of χ is denoted Ker χ. Let µ be a 2-cocycle on G with coefficients in C × . The set of irreducible µ-characters of G is denoted Irr µ (G).
When µ = 1, we write Irr(G) instead of Irr 1 (G). Finally, the coboundary operator on the space of cochains of G with coefficients in C × is denoted d.

Drinfeld doubles of finite groups
Let G be a finite group. As stated earlier, the category Rep(D(G)) of finitedimensional representations of the Drinfeld double D(G) is a modular tensor category [1]. The simple objects of Rep(D(G)) are in bijection with the set of pairs (a, χ), where a is a representative of a conjugacy class of G, and χ is an irreducible character of the centralizer C G (a) of a in G. The S-matrix and the T -matrix of Rep(D(G)) are square matrices indexed by the simple objects of Rep(D(G)), and are given by the following formulas [1,2]. Then for all a, b, c, d ∈ G, and ω(a, b, c) = 1 if a, b, or c is the identity element. Replacing ω by a cohomologous 3-cocycle, if necessary, we may assume that the values of ω are roots of unity.
The 3-cocycle condition on ω ensures that the relation holds for all a, x, y, z ∈ G. Therefore, for any a ∈ G, the restriction of β a to the centralizer C G (a) of a in G is a normalized 2-cocycle, that is, for all x, y, z ∈ C G (a), and β a (x, y) = 1 if x or y is the identity element.
Direct calculations using the 3-cocycle condition of ω show that for all a, b, x, y ∈ G. For all a ∈ G, the functions β a and γ a are equal when restricted to C G (a). Therefore, we have for all a, b ∈ Z(G), and x, y ∈ G. .
where g = x −1 ax, g = y −1 by, and C G (a) denotes the conjugacy class of a in G.

Tensor generators
In this section, we give a criterion for a simple object (a, χ) of Rep(D ω (G)) to be a tensor generator, that is, the full fusion subcategory given by the intersection of all fusion subcategories of Rep(D ω (G)) that contain (a, χ) is Rep(D ω (G)).
Let C be a modular tensor category with braiding c. Two objects X, Y ∈ C centralize each other if Let D be a full (not necessarily tensor) subcategory of C. In the paper [10], M.
Müger defined the centralizer of D in C as the full subcategory of C, denoted D , consisting of all objects in C that centralize every object in D. That is, It was shown in [10] that D is a fusion subcategory, and that if D is a fusion subcategory, then D = D; we refer to this result as the double centralizer theorem.
We recall the following result from [11]. (a) The conjugacy classes of a and b commute elementwise.
Below, we record a special case of the result above.
If b = e or χ = 1, then the condition above is equivalent to the condition Proof. If b = e or χ = 1, then the equality in condition (i) is equivalent to the equality χ (a) = deg χ , which is equivalent to condition (i ).
Suppose that (a, χ) and (b, χ ) centralize each other. Putting g = e in condition Conversely, suppose that condition (i) holds. Since b lies in the center of G, we (a) The normal closure of a in G is equal to G. The converse clearly holds, by Theorem 3.3 Next, we address the twisted case. Of course, the untwisted case above is a special case of the twisted case below, but we find it instructive to treat the untwisted case separately, as in [11] and [12]. We recall the following result from [11].

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We will need the following result from [12].
Lemma 3.6. Let G be a finite group, let ω be a normalized 3-cocycle on G, and let a, b, x ∈ G. If ab = ba, then −1 a, x) .
Lemma 3.7. Let G be a finite group, and let µ be a normalized 2-cocycle on G.

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Applying the 2-cocycle condition of µ to the triple (x, y −1 , yx −1 ax) gives Making this substitution in the expression above yields establishing (a).
Applying the 2-cocycle condition of µ to the triple (x, Making this substitution in the expression Applying the 2-cocycle condition of µ to the triple ( µ(a,x) , and so the expression above is equal to  Conversely, suppose that condition (i) holds. Since b lies in the center of G, we know that C G (b) = G, β b is a 2-cocycle on G, and χ is a β b -character of G.

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(b) of Proposition 3.5 holds. Let x, y ∈ G. Since b lies in the center of G, the left-hand side of condition (b) of Proposition 3.5 reduces to Let ρ : G → GL(V ) be a projective β b -representation of G whose character is χ , and let z ∈ G. Then and so Taking the trace of both sides, we get Putting z = xy −1 in the equation above, we get Substituting the expression above in (3), we get Using Lemma 3.6, we see that the expression above is equal to Applying Lemma 3.7 with µ = β b , and noting that   Example 3.11. Let G be a finite group, and let ω be a normalized 3-cocycle on G.
If G has trivial center, and a is an element of G whose normal closure is G, then, by Corollary 3.10, for every irreducible β a -character of C G (a), the simple object (a, χ) is a generator of Rep(D ω (G)). We give three related examples below.
(a) Take G = S n , the symmetric group on n letters, with n ≥ 3. Then S n has trivial center, and the normal closure of the transposition σ = (12) is S n .
(b) Let n ≥ 3 be an odd integer, and take G = Dih n , the dihedral group of order 2n generated by the elements a and b subject to the relations a n = e, b 2 = e, and ba = a −1 b. Then Dih n has trivial center, and the normal closure of the element b is Dih n . Therefore, for every irreducible β b -character of C Dihn (b) = {e, b}, the simple object (b, χ) is a generator of Rep(D ω (Dih n )).
Note that, since the Schur multiplier of a cyclic group is trivial, the 2-cocycle β b is cohomologically trivial, and so χ may be identified with an ordinary character.
(c) Suppose that G is nonabelian and simple. Then G has trivial center, and the normal closure of every nontrivial element a is G. Therefore, for every nontrivial element a, and for every irreducible β a -character of C G (a), the simple object (a, χ) is a generator of Rep(D ω (G)). Schur [13] and independently H. Jordan [9] obtained the characters of the special linear groups over arbitrary finite fields [8]. We use the exposition given in [5]. The group SL(2, p) has exactly p + 4 distinct irreducible characters. For the purpose of this example, we will only need a portion of the character table of SL(2, p). Set = (−1) (p−1)/2 . The table below gives the values of the irreducible characters evaluated at the identity matrix I and at the matrix Y , omitting identical columns.
It is easily verified that X = Y −1 (XY −1 X −1 )Y −1 , and since the matrices X and Y generate SL(2, p), it follows that the normal closure of Y is SL(2, p).
The conjugacy class of Y contains (p 2 − 1)/2 elements, and so the centralizer of Y in SL(2, p) has order 2p. The matrix Y has order p, so the matrix −Y has order 2p, and it follows that the centralizer of Y in SL(2, p) is a cyclic group with generator −Y .

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Let ζ be a primitive 2p-th root of unity. For each 1 ≤ i ≤ 2p, let χ i : −Y → C × denote the group homomorphism that sends −Y to ζ i . Then the χ i constitute all of the irreducible characters of the centralizer of Y in SL(2, p). We have Suppose that i is odd. We will show that the simple object (Y, χ i ) is a generator of Rep(D(SL(2, p))). We have shown above that condition (

Adjoint category
In this section, we describe the adjoint category of Rep(D ω (G)). The case where ω = 1 was addressed in the paper [12]. For a fusion category C, adjoint category of C, denoted C ad , is the full fusion subcategory of C generated by all subobjects of X ⊗ X * , where X runs through simple objects of C. For example, for a finite group G, we have Rep(G) ad ∼ = Rep(G/Z(G)).
Lemma 4.1. Let G be a finite group, and let ω be a normalized 3-cocycle on G.
The set Proof. Since β e = 1, the identity element e lies in Z ω (G). Let a, b ∈ Z ω (G). Define a function τ a,b : G → C × by τ a,b (x) = β x (a, b). It follows from (2) that showing that β ab and β a β b are cohomologous, where d denotes the coboundary operator. Since β a and β b are cohomologically trivial, the same is true for β ab , and so ab ∈ Z ω (G).
The following definition is taken from [12].
Definition 4.2. Let G be a finite group, let ω be a normalized 3-cocycle on G, let K and H be normal subgroups of G that commute elementwise, and let B : for all x, y ∈ K and u, v ∈ H. We say that B is G-invariant if for all g ∈ G, x ∈ K, and u ∈ H.
We refer the reader to [12] for an explanation of the G-invariance property and the apparent lack of symmetry. It was shown in [12] that the fusion subcategories of Rep(D ω (G)) are parametrized by triples (K, H, B), where K and H are normal subgroups of G that commute elementwise, and B is an ω-bicharacter on K × H.
where (B op ) −1 : Lemma 4.3. Let G be a finite group, and let ω be a normalized 3-cocycle on G.
Proof. Part (a) was proved in [12]. To see (b), let g, x, y ∈ G, and let a ∈ Z ω (G).
Since we also have β ab = dσ ab , we deduce that the functions σ ab and σ a · σ b · τ a,b are equal when restricted to H, that is, for all x ∈ H, σ ab (x) = σ a (x)σ b (x)τ a,b (x), equivalently, which is the second condition in the definition of ω-character.
Since β a is cohomologically trivial, it is symmetric, and so the expression above reduces to σ a (gxg −1 ). By Lemma 4.3, (β a ) g = β a , and so d(σ a ) g = (β a ) g = β a = dσ a , where the superscript denotes the conjugation action. Therefore, the functions (σ a ) g and σ a are equal when restricted to H, and so σ a (gxg −1 ) = σ a (x) = B(a, b), proving that B is G-invariant.
A fusion category C is said to be pseudounitary if its categorical dimension coincides with its Frobenius-Perron dimension [6]. In this case, C admits a canonical spherical structure with respect to which categorical dimensions of objects coincide with their Frobenius-Perron dimensions [6]. The category Rep(D ω (G)) is pseudounitary. In the paper [7], S. Gelaki and D. Nikshych showed that, for a pseudounitary modular category C, the adjoint subcategory C ad and the full maximal pointed subcategory C pt are centralizers of each other, that is,