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MODULAR GROUP IMAGES ARISING from DRINFELD DOUBLES of DIHEDRAL GROUPS Deepak Naidu 1. Introduction the Modular Group SL(2, Z) Is

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MODULAR GROUP IMAGES ARISING from DRINFELD DOUBLES of DIHEDRAL GROUPS Deepak Naidu 1. Introduction the Modular Group SL(2, Z) Is International Electronic Journal of Algebra Volume 28 (2020) 156-174 DOI: 10.24330/ieja.768210 MODULAR GROUP IMAGES ARISING FROM DRINFELD DOUBLES OF DIHEDRAL GROUPS Deepak Naidu Received: 28 October 2019; Revised: 30 May 2020; Accepted: 31 May 2020 Communicated by A. C¸i˘gdem Ozcan¨ Abstract. 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 Keywords: Drinfeld double, modular tensor category, modular group, con- gruence subgroup 1. 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 con- formal 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 particu- lar, 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 ! ! 0 −1 1 1 X = and Y = ; 1 0 0 1 and these matrices satisfy the relations X4 = I and (XY )3 = X2: MODULAR GROUP IMAGES 157 In fact, the relations above are defining relations for the modular group, that is, the modular group has the presentation hX; Y j X4 = 1; (XY )3 = X2i: 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 comes equipped with a pair of invertible matrices S and T , called the S-matrix and T -matrix of Rep(D(G)), and they satisfy the relations S4 = I and (ST )3 = S2: (1) Therefore, Rep(D(G)) gives rise to a representation ρ : SL(2; Z) ! hS; T i of the modular group such that ρ(X) = S and ρ(Y ) = T . In their paper [9], Y. Sommerh¨auserand 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. Sommerh¨auserand 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) × S3, where PSL(2; Z=nZ) denotes the projective special linear group, S3 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) × S3. 158 DEEPAK NAIDU 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 δx;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. 2. Drinfeld doubles of finite groups 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 representations of D(G). The category Rep(D(G)) is a modular tensor category [1]. The simple objects of Rep(D(G)) are in bijection with the set of pairs (x; α), where x is a representative of a conjugacy class of G, and α is an irreducible character of the centralizer CG(x) of x 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 [1,5]. 1 X −1 −1 S(x,α);(y,β) = α(gyg )β(g xg); jCG(x)jjCG(y)j g2G(x;y) α(x) T = δ δ ; (x,α);(y,β) x;y α,β deg α where G(x; y) denotes the set fg 2 G j xgyg−1 = gyg−1xg. 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 T exp(G) = I; (2) 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−1x−1g; χg), where g is an element of G such that g−1x−1g is the element chosen to represent the conjugacy class of x−1. The so-called charge conju- gation matrix is the square matrix C indexed by the simple objects of Rep(D(G)) 2 defined by C(x,α);(y,β) = δ(x,α)∗;(y,β). We have S = C [1]. MODULAR GROUP IMAGES 159 The S-matrix and T -matrix of Rep(D(G)) when G = G1 × G2 for finite groups G1 and G2 are given by the Kronecker products S1 ⊗ S2 and T1 ⊗ T2, where Si and Ti denote the S-matrix and T -matrix of Rep(D(Gi)), i = 1; 2. Example 2.1. Let n be an integer with n ≥ 3, and let Dihn denote the Dihedral group of order 2n generated by the elements a and b subject to the relations an = e, b2 = e, and ba = a−1b. (a) Suppose that n is even. Then there are (n=2)+3 conjugacy classes in Dihn, and they are feg; fan=2g; fak; a−kg (1 ≤ k < n=2); fa2kb j 0 ≤ k < n=2g; fa2k+1b j 0 ≤ k < n=2g: We choose the elements e, ak (1 ≤ k ≤ n=2), b, and ab as representatives of the conjugacy classes. The centralizers of these elements are C(e) = Dihn n=2 C(a ) = Dihn C(ak) = hai (1 ≤ k < n=2) C(b) = fe; b; an=2; an=2bg C(ab) = fe; ab; an=2; a1+n=2bg: n=2 The center of Dihn is the subgroup fe; a g, and the character table of Dihn is e ak b ab χ0 1 1 1 1 k χ1 1 (−1) 1 −1 χ2 1 1 −1 −1 k χ3 1 (−1) −1 1 2πik i 2 2 cos n 0 0 where 1 ≤ i ≤ (n=2) − 1 and 1 ≤ k ≤ n=2. 2πi Let ζ = e n , a primitive nth root of unity. For each 1 ≤ i ≤ n, let × αi : hai ! C denote the group homomorphism that sends a to ζi. For i; j 2 f0; 1g, let n=2 n=2 × βi;j : fe; b; a ; a bg ! C 160 DEEPAK NAIDU denote the group homomorphism that sends b to (−1)i and sends an=2 to (−1)j, and let n=2 1+n=2 × γi;j : fe; ab; a ; a bg ! C denote the group homomorphism that sends ab to (−1)i and sends an=2 to (−1)j. The simple objects of Rep(D(Dihn)) are in bijection with the set con- sisting of the following pairs. (e; χi) 0 ≤ i ≤ 3 (e; i) 1 ≤ i ≤ (n=2) − 1 n=2 (a ; χi) 0 ≤ i ≤ 3 n=2 (a ; i) 1 ≤ i ≤ (n=2) − 1 k (a ; αi) 1 ≤ k < n=2; 1 ≤ i ≤ n (b; βi;j) i; j 2 f0; 1g (ab; γi;j) i; j 2 f0; 1g Set 8 8 <1 if i = 0; 1 0 <1 if i = 0; 3 ∆i = and ∆i = :−1 if i = 2; 3 :−1 if i = 1; 2: Then the S-matrix is given by the first three tables below, and the T -matrix is given by the fourth table below.
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