Eclectic flavor groups

The simultaneous study of top-down and bottom-up approaches to modular flavor symmetry leads necessarily to the concept of eclectic flavor groups. These are non-trivial products of modular and traditional flavor symmetries that exhibit the phenomenon of local flavor enhancement in moduli space. We develop methods to determine the eclectic flavor groups that can be consistently associated with a given traditional flavor symmetry. Applying these methods to a large family of prominent traditional flavor symmetries, we try to identify potential candidates for realistic eclectic flavor groups and show that they are relatively rare. Model building with finite modular flavor symmetries thus appears to be much more restrictive than previously thought.


Introduction
While the parameters of the flavor sector of the Standard Model of particle physics have essentially all been determined experimentally, the origin of "flavor" remains a mystery. Models with traditional (discrete) flavor symmetries have provided various fits for masses and mixing angles of quarks and leptons [1]. More recently, it was suggested that (discrete) modular symmetries might describe the flavor structure of the lepton sector [2]. This bottom-up (BU) description is based on the consideration of finite modular groups Γ N with N = 2, 3, 4, 5. Typically, in these models, some of the lepton multiplets are described by nontrivial singlets or irreducible triplets of Γ N . This suggestion has an important impact on the field of lepton flavor physics .
To understand the origin of flavor and modular symmetries we need to consider additionally a top-down (TD) approach, based on ultraviolet complete theories. Recently, such attempts have studied modular symmetries in string derived standard-like models based on JHEP02(2020)045 heterotic orbifolds [42,43] and magnetized D-branes [44][45][46][47] in connection with the standard discrete flavor symmetries within this framework [48]. This leads to a hybrid picture where the traditional flavor group and the finite modular group combine (as a nontrivial product 1 ) to a generalized flavor group which we call "Eclectic Flavor Group". It contains the traditional flavor group (which acts universally in moduli space) as well as the corresponding modular flavor structure Γ N . This picture includes the mechanism of "Local Flavor Unification" of flavor, CP and modular symmetries with enhanced symmetries at certain locations in moduli space [42,43], see also ref. [34]. Furthermore, it potentially incorporates a different flavor structure for the quark-and lepton-sector of the Standard Model.
In the present paper we want to analyze possible relations between the bottom-up (BU) and the top-down (TD) approaches. At first sight, we are confronted with some potential obstructions. First of all, it seems to be difficult to find nontrivial singlets and irreducible triplet representations of finite modular groups within the TD approach. Such triplets, as used in the BU-case, can be identified more easily for traditional flavor groups, as e.g. in ∆(54) of ref. [48]. In addition, the TD picture does not always lead to Γ N itself (as the finite modular group), but to its double cover (T in case of Γ 3 , see refs. [42,43]). Although we have studied up to now only a limited number of TD-models, we can emphasize the following key observation: The full eclectic flavor group is a nontrivial product of the traditional flavor group, a corresponding finite modular group and a CP-like transformation. We cannot treat these symmetries separately (as mostly done in the BU-approach) and have to be aware of restrictions (for superpotential and Kähler potential [33]) from all of these components.
As a step in our search for a connection between BU-and TD-approaches, we shall develop a classification method to obtain all allowed eclectic flavor groups. This is the main goal of the present paper. We shall show that this combination of traditional flavor group and finite modular group cannot be arbitrary but has to satisfy severe consistency conditions. This can be seen already from the results of previous work [42,43], where it was observed that candidate eclectic flavor groups derive from the traditional flavor group and its outer automorphisms. 2 Based on this observation, we shall classify the possible eclectic flavor groups in a bottom-up way (for a class of prominent traditional flavor groups) and show that there is only a limited number of possibilities.
The paper is structured as follows. In section 2 we shall discuss the interplay of traditional flavor and modular symmetries and derive relevant consistency conditions. Section 3 discusses the question of "Local Flavor Unification" from the point of view of allowed modular symmetries. In section 4 we give a specific example (closely related to the T 2 /Z 3 orbifold) with traditional flavor group ∆ (54) [50], where the first number gives the order of the group. We remark several group isomorphisms: the allowed eclectic flavor groups for representative examples of several traditional flavor groups. Section 6 gives conclusions and outlook.

Extending flavor symmetries by modular symmetries
The modular group SL(2, Z) can be defined by the presentation [51] 3 and a choice of SL(2, Z) generators S and T is given by Under a modular transformation γ ∈ SL(2, Z) both, the modulus τ and matter fields ψ, transform in general nontrivially according to Here, k ∈ Q is the so-called modular weight 4 of ψ and ρ(γ 1 γ 2 ) = ρ(γ 1 )ρ(γ 2 ) is a representation of the finite modular group Γ N or of its double cover Γ N [22] for N ∈ {2, 3, 4, 5}. These finite groups are defined by the presentations All finite modular groups Γ N and Γ N with N = 2, 3, 4, 5 are listed in table 1.

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The symmetry group of a modular invariant theory is SL (2, Z). This symmetry has different realizations for the various fields of the theory: the τ modulus feels only PSL(2, Z) since ±γ ∈ SL(2, Z) yield the same transformation (2.3) for τ . In contrast, a matter field ψ transforms twofold: i) by the automorphy factor (c τ +d) −k that distinguishes between γ and −γ for general modular weight k, and ii) by a linear transformation ρ(γ) that can distinguish between γ and −γ only in the Finally, due to the transformation of the τ modulus, Yukawa couplings Y (τ ) are in general modular forms and transform similar to eq. (2.3) as where ρ Y (γ) is also a representation of the finite modular group. For a given modular weight k Y (with k Y ∈ 2N for Γ N or k Y ∈ N for Γ N ), the number of independent Yukawa couplings Y (τ ) is finite and their τ -dependence and transformations ρ Y (γ) are explicitly known, see e.g. [2,5,6,10,15,22,40]. Hence, in order to fully specify a modular invariant theory one has to choose the finite modular group Γ N or Γ N that shall host the representation matrices ρ(γ) and ρ Y (γ). Now, since the matrices ρ(γ) in eq. (2.3) and ρ Y (γ) in eq. (2.5) must build a (reducible or irreducible) representation of Γ N or Γ N , they have to satisfy the respective presentation (2.4a) or (2.4b), i.e.
(ρ(S)) N S = 1 , (ρ(T)) N = 1 , (ρ(S T)) 3 = 1 , and ρ(S 2 T) = ρ(T S 2 ) , (2.6) with N ∈ {2, 3, 4, 5} and N S = 2 for Γ N or N S = 4 for Γ N . Let us stress that the τ modulus transforms nontrivially under modular transformations eq. (2.3). In contrast to the modular group, we define the traditional flavor group G fl by those discrete transformations g ∈ G fl that leave τ invariant at all points in τ moduli space, i.e. for all τ τ where ρ(g) is a (reducible or irreducible) representation of G fl . Hence, Yukawa couplings Y (τ ) are invariant under transformations from G fl for all τ . As we show next, traditional flavor groups are naturally connected to finite modular groups. To see this, we apply the modular S transformation eq. (2.3) twice and obtain Since the τ modulus is invariant under S 2 , S 2 is by definition part of the traditional flavor group. Moreover, the matter fields ψ transform in general nontrivially, (−1) −k (ρ(S)) 2 = 1,

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see also appendix A. Thus, finite modular groups naturally yield traditional flavor groups, and one might wonder how one can in general combine a traditional flavor group consistently with a finite modular group.
To answer this question, we derive a constraint on the extension of a traditional flavor group by a finite modular group, inspired by the discussions in refs. [17,[55][56][57], where symmetries are extended by CP. In detail, we start with a given traditional flavor group G fl and try to extend this group consistently by two generators ρ(S) and ρ(T) of some finite modular group. To do so, let us consider two chains of transformations of the form "modular, flavor, inverse modular", for g ∈ G fl . The τ modulus is invariant under both chains of transformations. Since we do not want to enhance the traditional flavor group G fl to a larger traditional flavor group G fl by including new generators ρ(S)ρ(g)ρ(S) −1 and ρ(T)ρ(g)ρ(T) −1 , we see from eq. (2.9) that ρ(S) ρ(g) ρ(S) −1 ∈ G fl and ρ(T) ρ(g) ρ(T) −1 ∈ G fl (2.10) must belong to the traditional flavor group G fl for all g ∈ G fl . In other words, due to eq. (2.10) the traditional flavor group G fl must be a normal subgroup of the combined group generated by ρ(S), ρ(T) and ρ(g), which we call the eclectic flavor group. Moreover, we find Thus, we can sharpen the constraint (2.10) as ρ(S) ρ(g) ρ(S) −1 = ρ(u S (g)) and ρ(T) ρ(g) ρ(T) −1 = ρ(u T (g)) , (2.12) where due to eq. (2.11) the maps u S and u T are automorphisms of the traditional flavor group G fl . Since ρ(S) and ρ(T) are assumed to generate a finite modular group, it follows from eq. (2.12) that the automorphisms u S and u T have to satisfy the defining relations (2.6), i.e. 13) with N S = 2 for Γ N and N S = 4 for Γ N . Note that the identity 1 in eq. (2.13) has to be understood as the trivial automorphism, 1: g → g for all g ∈ G fl , and not as an inner automorphism. As a consequence, the finite modular group defined by eq. (2.13) must be a subgroup of the full automorphism group of the traditional flavor group G fl . Now, we can consider two cases. First, if the traditional flavor group commutes with the finite modular group, eq. (2.12) yields ρ(g) = ρ(u S (g)) and ρ(g) = ρ(u T (g)) , (2.14) for all g ∈ G fl . Thus, both modular transformations S and T in eq. (2.12) correspond to the trivial automorphism of G fl , u S = u T = 1, and the eclectic flavor group is just given by the direct product extension G fl × Γ N or G fl × Γ N . In this case, the finite modular group can be chosen freely.

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Motivated by the TD approach however, we are interested only in the case where the traditional flavor group does not commute with the finite modular group. In order to satisfy condition (2.12) in this case, the traditional flavor group must have nontrivial automorphisms as candidates for u S and u T . In general, an automorphism can be inner or outer, where for an outer automorphism u there exists no h u ∈ G fl such that Thus, we have to decide whether u S and u T are inner or outer automorphisms of the traditional flavor group G fl . Let us first assume that u S and u T are inner automorphisms of the traditional flavor group G fl . In this case, the automorphisms u S and u T would be defined as for some fixed h S , h T ∈ G fl . Now, assume that u S and u T satisfy eq. (2.13) and, hence, generate some finite modular group. Then, the action of the modular generators S and T can always be compensated by an element of the flavor group. To be specific, consider the transformations and similarly for T, where we used ρ(S) = ρ(h S ) that follows from eqs. (2.12) and (2.16). Hence, the generators of the finite modular group can be redefined such that the representation of the finite modular group on matter fields ψ is trivial, ρ(γ) = 1. Since this group would be a trivial extension, we demand in the following that u S and u T are outer automorphisms.
Once this requirement is met, u S and u T are subject to eqs. (2.12) and (2.13), which impose strong constraints on the possible extensions of the traditional flavor group by a finite modular group.

The eclectic extension
From our previous discussion, one can classify for a given traditional flavor group 5 G fl all nontrivial extensions by finite modular groups as follows: i) First, one determines the automorphisms of G fl and chooses two particular outer automorphisms u S and u T , whose specific properties shall be motivated and explained in detail in the next section in the context of CP.
ii) Then, one checks whether u S and u T satisfy the presentation of a finite modular group as given in eq. (2.13).
iii) Finally, for a given representation ρ(g) of the traditional flavor group, one constructs ρ(S) and ρ(T) explicitly using eq. (2.12) such that ρ(S) and ρ(T) satisfy the presentation eq. (2.6) of the same finite modular group as u S and u T .

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The multiplicative closure of the traditional flavor group G fl and its compatible finite modular group Γ N (or Γ N ) is called eclectic flavor group, where a potential extension by a CP-like transformation will be discussed in the next section. Let us stress that the eclectic flavor group is not a direct product of G fl and Γ N (or Γ N ) -in other words, G fl does not commute with Γ N (or Γ N ).

Combining with CP
One can combine the modular group SL(2, Z) with a CP-like transformation by introducing a new generator K * , which, on the level of the 2 × 2 matrices given in eq. (2.2), can be realized as such that SL(2, Z) is enhanced to GL(2, Z) [17]. Under K * , the τ modulus and the matter fields ψ(x) transform as see ref. [42] and also [17,43,59]. Demanding that the CP-like transformation be of order and therefore In general, the additional generator ρ(K * ) of CP does not commute with Γ N (or Γ N ). To see this, let us first consider the chain of transformations for some modular transformation γ ∈ SL(2, Z) that we determine next. Under the chain of transformations eq. (2.22) the τ modulus transforms as τ → − 1 /τ. Thus, γ = S or S −1 from eq. (2.2). Eq. (2.22c) implies that the solution is γ = S −1 . In summary, we have found that K * S K * = S −1 on the level of GL(2, Z). Consequently, the finite modular group has to be extended by ρ(K * ) satisfying

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By repeating these steps for T, we find K * T K * = T −1 and, as the final result, that the conditions (2.6) get extended by This enhances the finite modular group Γ N to Γ * N (and enhances Γ N to Γ * N ), defined as All finite modular groups with a CP-like extension and N = 2, 3, 4, 5 are listed in table 1. Next, we discuss how Γ * N can be made compatible with the traditional flavor group G fl , cf. [55][56][57]. With the additional element K * , our previous discussion proceeds directly. First, eq. (2.9) includes now also the chain of transformations As shown in ref. [57], eq. (2.27) is satisfied by a class-inverting outer automorphism of the traditional flavor group G fl . However, one can also have a situation in which not all irreducible representations of G fl appear in the theory and there exists an automorphism u K * satisfying eq. (2.27) only for the representation(s) ρ present in the spectrum. Such an automorphism could then be seen as a ρ-restricted class-inverting automorphism. 7 Now, since this type of outer automorphisms necessarily doubles the dimensions of (some of) the representations and because in the TD approach they give rise to CP-like transformations [42,43], we reserve these ρ-restricted class-inverting automorphisms of G fl exclusively for CP. Consequently, the automorphisms u K * , u S and u T of the traditional flavor group G fl have to satisfy (in addition to eq. (2.13)) the conditions In more detail, by applying the definitions of the automorphisms u S and u K * in eqs. (2.12) and (2.27), the second relation is obtained as follows: (2.29) 7 We thank Andreas Trautner for useful comments on the properties of class-inverting automorphisms.

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As before, for each traditional flavor group G fl , it is possible to classify all finite modular groups Γ * N (and Γ * N ), endowed with a CP-like transformation, that are compatible with G fl . One must first create all automorphisms of G fl and choose outer automorphisms u S , u T and u K * satisfying the presentation of an enhanced finite modular group given by eqs. (2.13) and (2.28). Finally, one must explicitly find the representations ρ(S), ρ(T) and ρ(K * ) that fulfill eqs. (2.12), (2.24) and (2.27).
Let us make a remark on the non-Abelian structure of the CP-like extension of finite modular groups. Despite the fact that eq. (2.25a) indicates that, for N > 2, the generator K * of CP does not commute with S and T, the CP-like extension of Γ N becomes Γ * N = Γ N × Z 2 for several finite modular groups. In detail, the Z 2 factors in the Γ * N finite modular groups S 3 × Z 2 , S 4 × Z 2 and A 5 × Z 2 are generated by K * , (T 2 S) 2 K * and S(T 2 ST) 2 TK * , respectively, see table 1.

Local flavor unification
We are considering a setting, where modular and traditional flavor symmetries do not commute. This gives rise to the picture of "Local Flavor Unification" [42,43]: at so-called self-dual points or lines in moduli space τ the finite modular symmetry is broken spontaneously to those subgroups that leave τ invariant. In contrast, the traditional flavor symmetry, by definition, leaves the modulus τ invariant and, hence, remains unbroken everywhere in moduli space. As modular and traditional flavor symmetries do not commute, the unbroken modular transformations yield nontrivial enhancements of the traditional flavor symmetry to the so-called unified flavor symmetries at the self-dual points in moduli space.
Let us begin the discussion with the finite modular group Γ N or Γ N , i.e. without taking CP-like transformations into account. Then, if the modulus is stabilized at τ = i, the following modular transformation remains unbroken see also the related discussion in ref. [34]. At this point in moduli space, matter fields transform as Thus, at τ = i the traditional flavor symmetry G fl is enhanced by the generator S. Since ρ(S) is either of order N S = 2 or N S = 4, see eq. (2.6), the unified flavor symmetry at τ = i is a nontrivial extension of G fl by Z N S [42,43], or even of higher order depending on the modular weight k ∈ Q.
If the modulus is stabilized at τ = exp ( 2πi /3), we find the following unbroken modular transformation

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Thus, at τ = exp ( 2πi /3) the traditional flavor symmetry is enhanced by S T and matter fields transform as ψ g −→ ρ(g) ψ for g ∈ G fl and ψ g −→ exp ( 2πik /3) ρ(g) ψ for g = S T (3.4) under the unified flavor symmetry at τ = exp ( 2πi /3). Note that ρ(S T) = ρ(S)ρ(T) is of order 3, see eq. (2.6). Thus, the unified flavor symmetry at τ = exp ( 2πi /3) is a nontrivial product of G fl and a Z 3 generated by S T (if k ∈ Z). If one includes the CP-like transformation, additional modular transformations can remain unbroken. For example, at vertical lines in moduli space given by τ = n B/2 +i √ 3r /2 with n B ∈ Z and r ∈ R we find that T n B K * remains unbroken [42,43], i.e.
In particular, for n B = 0, i.e. on the vertical line τ = i √ 3r /2 we obtain an unbroken modular transformation K * , while at τ = i eq. (3.1) yields an unbroken S transformation. Moreover, if one moves away from τ = i but stays on the circle τ = exp(iα), only the combined transformation K * S remains unbroken At these lines in moduli space, a CP-like transformation is unbroken. However, it is easy to break CP spontaneously by moving τ away from these symmetry enhanced lines in moduli space.  A three-dimensional representation of ∆(54) is given by where ω := exp 2πi /3. Since the group of outer automorphisms  As a remark, all of these groups have three-dimensional irreducible representations. 9 Note the change of convention compared to ref. [43].

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CP-like modular extension. Following the discussion of section 2.2, it is possible to extend the identified modular symmetry T with a CP-like transformation K * based on the ∆(54) outer automorphism that satisfies the conditions (2.28) with u S and u T given by eqs. (4.4). In this case, one can see from eq. (2.27) that the representation ρ(K * ) of eq. (2.19) relating the field ψ(x) in the 3 representation of ∆(54) with its conjugate ψ(x P ) in the 3 representation is given by Since K * relates ψ(x) and ψ(x P ), it is more convenient to rewrite S, T and K * in the 3 ⊕ 3 representation as where ρ(S) and ρ(T) are given by eq. (4.5) and ρ(K * ) by eq. (4.9). One can easily verify that ρ 6 (S), ρ 6 (T) and ρ 6 (K * ) generate the finite modular group Γ * In extra-dimensional models, the traditional flavor symmetry ∆(54) can originate from strings on a T 2 /Z 3 orbifold [48,68]. As shown in refs. [42,43], ∆(54) is accompanied nontrivially in this setting by a T or a GL(2, 3) modular symmetry, depending on whether the CP-like transformation K * is taken into account, in full agreement with our discussion here, see also ref. [69].

Results
We have selected a representative set of traditional flavor symmetries that have been used in model building. These include This list is not exhaustive, but covers the most promising traditional flavor symmetries considered so far. More examples could be added and analyzed upon request. We then classify for each group all nontrivial extensions by finite modular symmetries that fulfill the restrictions discussed in section 2. This then allows the identification of allowed eclectic flavor groups that could be obtained in a bottom-up procedure. Our results are listed in table 2. Surprisingly it turns out that many prominent traditional flavor symmetries, such as S 3 , D 12 ∼ = S 3 × Z 2 , T 7 , S 4 , Z 9 Z 3 , SL(2, 4) and A 5 , do not allow for any nontrivial modular extension and thus are not included in table 2. Among the groups we have studied, only with CP --   admit a CP-like transformation in the eclectic flavor group. This is a quite restrictive situation. A nontrivial extension of the traditional flavor group (as required from the top-down argumentation) is limited to just a few specific cases. Flavor model building should thus be based on very few examples with an eclectic flavor group that includes a nontrivial finite modular group. Since our list of traditional flavor groups is not exhaustive, it remains to be seen in future work whether more nontrivial cases can be found. Traditional flavor groups with a sizable group of outer automorphism are particularly suited for an eclectic extension.

Conclusions
In the present paper we made an effort to match the bottom-up (BU) and top-down (TD) approaches of flavor models based on finite discrete modular symmetries. Up to now, the BU-approach considered finite modular groups Γ N , where some of the quarks and leptons JHEP02(2020)045 transform as (irreducible) triplet or nontrivial singlet representations of Γ N . This led to an excellent description of the flavor structure of the lepton sector. Efforts towards an ultraviolet completion (TD approach) were based on string theory. Up to now only few explicit TD models have been constructed. The analysis within string theory, however, leads to a general qualitative picture with the clear message that finite modular symmetries do not appear in isolated form, but are accompanied by a traditional (non-modular) flavor group. This then leads us to the concept of "Eclectic Flavor Groups" as a nontrivial product of traditional flavor symmetry and finite modular symmetry.
Given this observation, we might now reconsider the BU-approach and classify candidates for eclectic flavor groups from bottom-up. Surprisingly, the number of these candidates turns out to be very small. Only a few examples are known (see table 2). This is the main result of the present paper. The fact that we cannot disentangle traditional flavor symmetries and modular symmetries is consistent with the picture of "Local Flavor Unification", where we find an enhanced symmetry at specific regions in moduli space. This would naturally allow different flavor structures for quarks and leptons, where quarks (leptons) are predominantly described by traditional (modular) flavor groups.
It should be stressed that the concept of eclectic flavor groups is more predictive than the consideration of modular symmetries alone. Terms allowed by the modular group might be forbidden by the selection rules of the traditional flavor group. In the ∆(54)example discussed in section 4, four independent trilinear superpotential couplings (of 2 ⊕1 representations of T ) allowed by the T modular symmetry are reduced to a single one due to the presence of ∆(54) [70]. The enhanced restrictions from eclectic flavor groups are especially relevant for the form of the Kähler potential. In ref. [33] it was pointed out that general terms in the Kähler potential reduce the predictivity of models based on finite modular symmetries. This problem could be solved within the eclectic flavor picture with more restrictions on the Kähler potential due to the nontrivial combination of finite modular groups and traditional flavor groups.

A Remarks
In this appendix, we comment on some inaccuracies in the literature on modular symmetries in model building. First, it is important to note that the modular S transformation is, in general, not of order 2, even though S 2 acts trivially on τ . In detail, from eq. (2.3) we get Then, S 2 acts as as expected. Let us compare eq. (A.2b) and eq. (A.3b) in some detail. In eq. (A.2b), the transformation by ρ(S) 2 is trivial for Γ N but can be nontrivial for Γ N . Moreover, the factor (−1) −k is nontrivial for a general modular weight k ∈ Q that is not even. Consequently, we see that in general S and S −1 are different transformations for matter fields, even though they act identically on the τ modulus.

B Explicit examples
B.1 Traditional flavor symmetry Q 8 Let us consider the traditional flavor symmetry Q 8 of order 8 (GAP ID [8,4]). We choose the irreducible two-dimensional representation of Q 8 given by [58] The full automorphism group of Q 8 is S 4 with 24 elements. Out of these, we can identify two outer automorphisms that generate the finite modular group S 3 , i.e. Then, there are two choices of matrices ρ(S) and ρ(T) that realize these automorphisms via eq. (2.12) and generate S 3 , being for α = ±1. For both choices of α, ρ(S) and ρ(T) build a doublet of S 3 . Moreover, the eclectic flavor group, generated by ρ(A), ρ(B), ρ(S) and ρ(T), turns out to be GL (2,3). It is interesting to note that the Q 8 traditional flavor symmetry does not allow for an eclectic extension with CP.

B.2 Traditional flavor symmetry
We choose a (reducible) three-dimensional representation of Z 3 × Z 3 given by The full automorphism group of Z 3 × Z 3 is GL(2, 3) with 48 elements. Since the group of inner automorphisms of Z 3 ×Z 3 is trivial, all elements of GL(2, 3) are outer automorphisms. It turns out that there are two classes of outer automorphisms that generate finite modular groups, either without or with CP: which satisfies eq. (2.24). The action on the matter fields (ψ, ψ) T is realized by rewriting all modular generators in the six-dimensional representation, as in eq. (4.10). In this six-dimensional representation, one can easily confirm using GAP that the eclectic flavor group including CP is [108,17].

B.3 Traditional flavor symmetry A 4
The generators of the traditional flavor symmetry A 4 (GAP ID [12,3]) can be given in the triplet representation by the matrices The full automorphism group of A 4 is S 4 , which contains only two finite modular groups generated by the outer automorphisms: i) Two outer automorphisms that generate the finite modular group Γ 2 ∼ = S 3 are In the representation (B.9), these automorphisms can be expressed as where ω := exp 2πi /3. The full automorphism group of T is S 4 . One can choose the following two outer automorphisms of this group to generate the finite modular group Γ 2 ∼ = S 3 : From eq. (2.6), one can easily see that ρ(S) and ρ(T) generate the finite modular group Γ 2 ∼ = S 3 . Notice however that ρ(S) = ρ(C) and ρ(T) = ρ(A) 2 ρ(C). As in eq. (2.17), we observe here that the action of the S 3 modular generators is compensated by the ∆(54) elements represented by ρ(C) −1 and ρ(A 2 C) −1 . Therefore, the action of the finite modular group S 3 based on inner automorphisms can be given by the trivial representation ρ(γ) = 1, which amounts to a trivial extension of the traditional flavor symmetry. As a side remark, a careful inspection reveals that there are no outer automorphisms of ∆(54) that generate a Γ 2 finite modular group and simultaneously satisfy eqs. (2.6) and (2.12). Thus, the ∆(54) traditional flavor group cannot be enhanced nontrivially by a Γ 2 finite modular group.
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