MODULAR GROUP IMAGES ARISING FROM DRINFELD DOUBLES OF DIHEDRAL GROUPS

We show that the image of the representation of the modular group SL(2,Z) arising from the representation category Rep(D(G)) of the Drinfeld double D(G) is isomorphic to the group PSL(2,Z/nZ) × S3, when G is either the dihedral group of order 2n or the dihedral group of order 4n for some odd integer n ≥ 3. Mathematics Subject Classification (2020): 18M20


Introduction
The modular group SL(2, Z) is the group of all 2 × 2 matrices of determinant 1 whose entries belong to the ring Z of integers. The modular group is known to play a significant role in conformal field theory [3]. Every two-dimensional rational conformal field theory gives rise to a finite-dimensional representation of the modular group, and the kernel of this representation has been of much interest. In particular, the question whether the kernel is a congruence subgroup of SL(2, Z) has been investigated by several authors. For example, A. Coste and T. Gannon in their paper [4] showed that under certain assumptions the kernel is indeed a congruence subgroup. In the present paper, we consider the kernel of the representation of the modular group arising from Drinfeld doubles of dihedral groups.
The group SL(2, Z) is generated by the matrices In fact, the relations above are defining relations for the modular group, that is, the modular group has the presentation X, Y | X 4 = 1, (XY ) 3 = X 2 .
Let G be a finite group, let D(G) denote the Drinfeld double of G, a quasitriangular 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 comes equipped with a pair of invertible matrices In their paper [9], Y. Sommerhäuser and Y. Zhu showed that the kernels of the representations of the modular group arising from factorizable semisimple Hopf algebras and from Drinfeld doubles of semisimple Hopf algebras are congruence subgroups of SL(2, Z). Later, S-H Ng and P. Schauenburg generalized the results of Y. Sommerhäuser and Y. Zhu to spherical fusion categories [8]. It follows from results in [9] that the kernel of ρ is a congruence subgroup. As a consequence, the image of ρ is finite. Our work gives a direct proof of this fact for dihedral groups of certain orders. Specifically, we show that if G is either the dihedral group of order 2n or the dihedral group of order 4n for some odd integer n ≥ 3, then the image of ρ is isomorphic to the group PSL(2, Z/nZ) × S 3 , where PSL(2, Z/nZ) denotes the projective special linear group, S 3 denotes the symmetric group on three letters, and Z/nZ denotes the ring of integers modulo n.

Organization:
In Section 2, we recall basic facts about the modular tensor category Rep(D(G)), and a description of the S-matrix and T -matrix for the dihedral groups.
In Section 3, we recall a presentation of the projective special linear group PSL(2, Z/nZ), and a description of its normal subgroups.
Section 4 contains our main result in which we establish that when G is either the dihedral group of order 2n or the dihedral group of order 4n for some odd integer n ≥ 3, the image of ρ is isomorphic to the group PSL(2, Z/nZ) × S 3 .

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 × . We will use the Kronecker symbol δ x,y , which is equal to 1 if x = y and zero otherwise. For any character α of a group, deg α denotes the degree of α, and α denotes the complex conjugate of α. We denote by [x] n the image of the integer x in the ring Z/nZ of integers modulo n; on occasions we will suppress the brackets as well as the subscript n. Rep(D(G)), and are given by the following formulas [1,5].
where G(x, y) denotes the set {g ∈ G | xgyg −1 = gyg −1 x}. The function G(x, y) → G(y, x) that sends each element g to g −1 is a bijection, and as a consequence the matrix S is symmetric.
We have where exp(G) denotes the exponent of G. In fact, the order of T is precisely exp(G) [5].
There is an involution * on the set of simple objects of Rep(D(G)) given by (x, α) * = (g −1 x −1 g, χ g ), where g is an element of G such that g −1 x −1 g is the element chosen to represent the conjugacy class of x −1 . The so-called charge conjugation matrix is the square matrix C indexed by the simple objects of Rep(D(G)) defined by C (x,α),(y,β) = δ (x,α) * ,(y,β) . We have S 2 = C [1].

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The S-matrix and T -matrix of Rep(D(G)) when G = G 1 × G 2 for finite groups G 1 and G 2 are given by the Kronecker products S 1 ⊗ S 2 and T 1 ⊗ T 2 , where S i and T i denote the S-matrix and T -matrix of Rep(D(G i )), i = 1, 2.
Example 2.1. Let n be an integer with n ≥ 3, and let Dih n denote 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.
(a) Suppose that n is even. Then there are (n/2) + 3 conjugacy classes in Dih n , and they are We choose the elements e, a k (1 ≤ k ≤ n/2), b, and ab as representatives of the conjugacy classes. The centralizers of these elements are C(e) = Dih n C(a n/2 ) = Dih n C(a k ) = a (1 ≤ k < n/2) C(b) = {e, b, a n/2 , a n/2 b} C(ab) = {e, ab, a n/2 , a 1+n/2 b}.
The center of Dih n is the subgroup {e, a n/2 }, and the character table of Dih n is e a k b ab n , a primitive nth root of unity. For each 1 ≤ i ≤ n, let α i : a → C × denote the group homomorphism that sends a to ζ i . For i, j ∈ {0, 1}, let β i,j : {e, b, a n/2 , a n/2 b} → C × denote the group homomorphism that sends b to (−1) i and sends a n/2 to (−1) j , and let γ i,j : {e, ab, a n/2 , a 1+n/2 b} → C × denote the group homomorphism that sends ab to (−1) i and sends a n/2 to The simple objects of Rep(D(Dih n )) are in bijection with the set consisting of the following pairs.
Then the S-matrix is given by the first three tables below, and the T -matrix is given by the fourth table below.
S (e, χj) (e, ψj) (a n/2 , χj) (a n/2 , ψj) (a , αj) Suppose that n is odd. Then there are (n + 3)/2 conjugacy classes in Dih n , and they are We choose the elements e, a k (1 ≤ k ≤ (n − 1)/2), and b as representatives of the conjugacy classes. The centralizers of these elements are The center of Dih n is trivial in this case, and the character table of Dih n is e a k b χ 0 1 1 1 n , a primitive nth root of unity. For each 1 ≤ i ≤ n, let denote the group homomorphism that sends a to ζ i . For i ∈ {0, 1}, let The simple objects of Rep(D(Dih n )) are in bijection with the set consisting of the pairs and the S-matrix and the T -matrix are given by the following tables.

Projective special linear groups
For any commutative ring R, the special linear group SL(2, R) is the group consisting of all 2 × 2 matrices a b c d with a, b, c, d ∈ R such that ad − bc = 1. Let n be a positive integer. Of particular interest is the special linear group SL(2, Z/nZ), where Z/nZ is the ring of integers modulo n. The order of SL(2, Z/nZ) for n ≥ 2 is given by the following formula [6].
where p runs over all primes that divide n.
If n is odd, then is a presentation of SL(2, Z/nZ) [2], and if n is a power of 2, then is a presentation of SL(2, Z/nZ) [9,4], where k runs over all odd integers between 1 and n, W (k) = XY X −1 Y k XY , and is an integer such that k ≡ 1 (mod n).
The matrices where the entries of the matrices are identified with their images in the ring Z/nZ, satisfy the relations in both the cases above.

DEEPAK NAIDU
The projective special linear group PSL(2, Z/nZ) is defined by where I is the identity matrix. If n is odd, then is a presentation of PSL(2, Z/nZ) [2]. The cosets satisfy the relations given above.
The group homomorphism SL(2, Z) → SL(2, Z/nZ) induced by the ring homomorphism Z → Z/nZ is surjective [6]. It follows that for each positive divisor r of n, the group homomorphism r is minimal with this property. We record the following result for later use.  Let n = n 1 n 2 · · · n t denote the decomposition of n into a product of powers of distinct primes. Choose integers u 1 , u 2 , . . . , u t such that n is a ring isomorphism, and it induces a group isomorphism Using the natural group isomorphism We will use this isomorphism in the next section.
The following result was proved by D. L. McQuillan in the paper [7].   We note that the center of PSL(2, Z/nZ) consists of all cosets of the form ± ( a 0 0 a ) with a 2 ≡ 1 (mod n) [7], and the unique Sylow 2-subgroup of SL(2, Z/3Z) is the

Main result
Let n be an odd integer with n ≥ 3, and let Dih n denote 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. In this section, we determine the group structure of the image of the representation of the modular group SL(2, Z) arising from the modular tensor category Rep(D(Dih n )). For a description of the simple objects of Rep(D(Dih n )) and the corresponding S-matrix and T -matrix we refer the reader to Example 2.1.
The charge conjugation matrix associated to Rep(D(Dih n )) is the identity, and so S 2 = I Therefore, the relations in (1) reduce to the following.
Since the group Dih n has exponent 2n, the relation in (2) gives T 2n = I. We note that, in fact, the order of T is precisely 2n.
We will need the matrices ST n S, ST n+1 S, and ST n−1 S, described below.
The computations involved in determining the matrices above are routine, albeit tedious. As a sample, we show the computation of one entry. The entry of the matrix ST n+1 S corresponding to the pair ((a k , α i ), (a , α j )) can be computed as follows.
where we used the formulas Proof. The matrix A is described below.
Since S and T have orders 2 and 2n, respectively, the inverse of A is T n ST n S, which is described below.
A routine calculation shows that A 2 = A −1 , and so A has order 3. The matrix B has order 2, since T has order 2n. We have BA = T n ST n ST n = A −1 B, and it follows that the group A, B is isomorphic to S 3 . Proof. The matrices P and Q are described below.

DEEPAK NAIDU
We have P −1 = T −1 ST n−1 ST −1 , whose description is easily obtained from the descriptions of ST n−1 S and T given earlier. We find that P −1 = P , and so P has order 2. The order of Q is 2n/ gcd(2n, n + 1) = 2n/ gcd(2, n + 1) = n.
A calculation shows that P QP = ST n+1 S. Using the descriptions of ST n+1 S and T given earlier, we immediately see that the matrices ST n+1 S and T n commute. Then and it follows that (P Q) 3 = I.
The matrix P Q n+1 2 P Q and its square are described below.
A routine calculation shows that the product of the two matrices above is the identity. and Q = T n+1 . The group P, Q generated by P and Q is isomorphic to the group PSL(2, Z/nZ).
Proof. As stated earlier, is a presentation of PSL(2, Z/nZ) [2], and the cosets A central element of PSL(2, Z/nZ) is necessarily of the form ± ( u 0 0 u ) with u 2 ≡ 1 (mod n), and it is easily verified that it corresponds to (XY u ) 3 . Suppose that for some integer u with u 2 ≡ 1 (mod n), the element (XY u ) 3 is in the kernel of ϕ.
Let H be a subgroup of SL(2, Z/nZ) of level less than n. By Lemma 3.1, the subgroup H contains Ker φ n n/p for some prime p that divides n, where φ n n/p : SL(2, Z/nZ) → SL(2, Z/(n/p)Z) is the reduction homomorphism. Observe that Ker φ n n/p contains the matrix 1 n/p 0 1 , and therefore the coset ± 1 n/p 0 1 , which corresponds to Y n/p , lies in π(H). The image of Y n/p under ϕ is the matrix Q n/p .
where ζ = e 2πi/n . Therefore, we must have ζ 1+ n 3 u = ζ, equivalently, n 3 · u ≡ 0 (mod n), a contradiction. It follows that Ker ϕ can not be of the form π(K)C where C is a subgroup of the center of PSL(2, Z/nZ) and K is the image in SL(2, Z/nZ) of the unique Sylow 2-subgroup of SL(2, Z/3Z).
Having exhausted all cases, we conclude that Ker ϕ must be trivial, and hence ϕ is an isomorphism. It follows that the subgroups P, Q and A, B of S, T commute element-wise.
Therefore, the intersection of these subgroups must be contained in the center of A, B . By Lemma 4.1, the group A, B is isomorphic to S 3 , which has trivial center, and so the intersection in question must be trivial. Therefore,  As seen in the proof of Theorem 4.4, the subgroups P, Q and A, B commute element-wise, and so the subgroups P ⊗ I, Q ⊗ I and A ⊗ A , B ⊗ B commute element-wise too. The group A ⊗ A , B ⊗ B is isomorphic to S 3 , which has trivial center, and so the subgroups P ⊗ I, Q ⊗ I and A ⊗ A , B ⊗ B intersect trivially.