Explicit and spontaneous breaking of SU(3) into its finite subgroups
CCP3-Origins-2011-033DIAS-2011-25
Explicit and spontaneous breaking of SU (3) into itsfinite subgroups Alexander Merle a & Roman Zwicky b a KTH Royal Institute of Technology, School of Engineering Sciences,Department of Theoretical Physics, AlbaNova University Center,Roslagstullsbacken 21, 106 91 Stockholm, Sweden b School of Physics & Astronomy, University of Southampton,Highfield, Southampton SO17 1BJ, UK
Abstract:
We investigate the breaking of SU (3) into its subgroups from the viewpoints of explicitand spontaneous breaking. A one-to-one link between these two approaches is given bythe complex spherical harmonics, which form a complete set of SU (3)-representation func-tions. An invariant of degrees p and q in complex conjugate variables corresponds to asinglet, or vacuum expectation value, in a ( p, q )-representation of SU (3). We review theformalism of the Molien function, which contains information on primary and secondaryinvariants. Generalizations of the Molien function to the tensor generating functions arediscussed. The latter allows all branching rules to be deduced. We have computed allprimary and secondary invariants for all proper finite subgroups of order smaller than512, for the entire series of groups ∆(3 n ), ∆(6 n ), and for all crystallographic groups.Examples of sufficient conditions for breaking into a subgroup are worked out for theentire T n [ a ] -, ∆(3 n )-, ∆(6 n )-series and for all crystallographic groups Σ( X ). The cor-responding invariants provide an alternative definition of these groups. A Mathematicapackage, SUtree , is provided which allows the extraction of the invariants, Molien andgenerating functions, syzygies, VEVs, branching rules, character tables, matrix ( p, q ) SU (3) -representations, Kronecker products, etc. for the groups discussed above. [email protected] [email protected] a r X i v : . [ h e p - ph ] N ov ontents SU (3) database 12 SUtree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Invariants of ∆(3 n ) and ∆(6 n ) . . . . . . . . . . . . . . . . . . . . . . . 15 SU (3) → Σ( X ) , ∆(6 n ) , ∆(3 n ) , T n [ a ] SU (3) . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Breaking to crystallographic groups Σ( X ) . . . . . . . . . . . . . . . . . . 194.3 Breaking to C - and D -groups (∆(6 n ), ∆(3 n ), and T n [ a ] ) . . . . . . . . . . 214.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 F (cid:44) → SU (3) X ), T n [ a ] , ∆(3 n ), and ∆(6 n ) . . . . . . . . 31 B.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36B.2 The subgroup tree within the group database . . . . . . . . . . . . . . . . 38
C Tensor generating function 45
C.1 S as an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47C.2 Branching rules for SO (3) → F and SU (3) → F . . . . . . . . . . . . . 48C.2.1 Examples of branching rules for SO (3) → S . . . . . . . . . . . . . 491 From the Molien function to invariants in practice 50
D.1 A manageable ambiguity of the Molien function . . . . . . . . . . . . . . . 50D.2 Degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50D.2.1 Degeneracies of invariants of lower degrees. . . . . . . . . . . . . . 51D.2.2 Degeneracies of invariants of the same degree . . . . . . . . . . . . 51
E Multiple representations 51F Conjectures concerning the T n [ a ] -groups 52G SU (3) 53 G.1 The complex spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . 53G.2 Construction of explicit ( p, q ) representations . . . . . . . . . . . . . . . . 55G.2.1 Polyomial basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55G.2.2 Gell-Mann basis on polynomial space . . . . . . . . . . . . . . . . . 562
Introduction
The aim of this work is to study the breaking of a group G , referred to as the mastergroup ,into one of its subgroups in the frameworks of explicit and spontaneous symmetry break-ing. As the master group we have in mind G = SU (3) and as subgroups proper finitesubgroups thereof, denoted by F , are considered. The methods that we use are generalbut some of the results, such as the necessary and sufficient conditions for SU (3) → F ,are specific to SU (3). The focus on the latter is motivated by the hope that patterns inthe flavour sector of the Standard Model (SM) are linked with such a symmetry. Thishope was fueled ever since the tribimaximal mixing matrix for the lepton sector has beenproposed [1], resulting in many model studies based on discrete subgroups of SU (3) [2].Discrete Abelian [3] and non-Abelian [4] symmetries further arise in string theory.Let us begin with a group theoretic introduction. A group can be defined either alge-braically, e.g. by giving relations between its generators, or through a specific (faithful)representation. For example, the well-known group O (3) can be defined from its fun-damental representation, which corresponds to a rotation in a 3-dimensional space, asfollows: O (3) = { O ∈ M ( R ) | O T O = 1 } , (1)where M ( R ) denotes the set of 3 × R . Equivalently, if we regard O ∈ M ( R ) as acting on a three-dimensional (representation) space R via the matrix-vectormultiplication (cid:126)x = ( x, y, z ) (cid:55)→ O(cid:126)x , then O (3) can be defined as the set of matrices leavingthe polynomial P = x + y + z = (cid:126)x T (cid:126)x (2)invariant. The condition (1) and the invariance of (2) are equivalent, which is easilyverified. Thus the polynomial P defines the group O (3). Note that P will be denotedby I [ SO (3)] later on. The realisation of this idea to finite groups is the main goal of thiswork. More precisely, we shall be concerned with finding a minimal number of invariantsthat enforce a faithful irreducible representation (irrep) of a group and thus can be seenas an alternative definition of the group under consideration.The so-called Molien function provides a powerful and simple tool to obtain the num-ber of algebraically independent and dependent invariants of a group. An important andsubtle question, to be discussed, is which invariants are necessary and sufficient to definea group as there are groups which have common invariants. This question will lead tothe investigation of maximal subgroups. For example, the symmetries of the permutationgroups on 4 elements, S and A ⊂ S , both leave the polynomial P = x + y + z invari-ant. Imposing P in addition to (2) leads to S and not A , since the latter is a subgroupof the former. In this work we shall provide a pragmatic solution to the problem and3hus find the necessary conditions to break from SU (3) to one of its finite subgroups via invariant polynomials , called explicit breaking . Moreover we shall show how the languageof invariant polynomials can be translated into the language of vacuum expectation values (VEVs), called spontaneous symmetry breaking (SSB) in the physics literature. This isachieved by the so-called complex spherical harmonics , which are the generalization of thespherical harmonics from SO (3) to SU (3).The paper is organized as follows: In Sec. 2 we discuss the main conceptual ideas andtools. In Sec. 3 we introduce the list of SU (3) subgroups that we are going to study andwhich are implemented in our database SUtree , as given in Tab. 5. We also present allinvariants of the ∆-groups, by which we mean the countable series ∆(3 n ) and ∆(6 n ) hereand thereafter. In Sec. 4 we discuss the necessary and sufficient conditions for breakinginto a large class of SU (3) subgroups. Some examples are given for illustration. In Sec. 5we show that our results are independent on the embedding up to trivial transformations.In Sec. 6 we conclude and give an outlook. Useful details and topics are discussed invarious appendices. In particular the tensor generating functions and branching rules arediscussed in App. C.This is a paper targeted at a physicist audience. The physics background, such asspontaneous symmetry breaking, etc. is not explained in any sufficient detail, and a lan-guage with reference to a physics background is used at times. Modulo this issue the textshould be readable for mathematicians as well. Some familiarity with basic finite grouptheory beyond the facts mentioned in App. A, such as character tables, etc. is assumed. In this section we discuss the main conceptual ideas and tools of this paper with the ex-ample of the breaking of SO (3) → S . The main topics are the explicit breaking 2.1, thebreaking via VEVs 2.2, the connection of the latter two 2.3, how to obtain all algebraicallyindependent invariants and thus VEVs 2.4, and the question of maximal subgroups 2.5,the latter being the most subtle point to handle in practice. The discussion mostly fol-lows the language of Lagrangian field theory; occasionally the mathematical perspectiveis added for clarity. The generalization of these ideas is partly obvious and the specific implementation to SU (3) will bediscussed in Sec. 4. .1 Explicit breaking The discussion in this subsection partly overlaps with the introduction. Suppose ( x, y, z ) ∈ R is the space upon which the fundamental representation of SO (3), denoted by , acts. For the breaking into S , SO (3) → S , it is is sufficient to demand invariance under thefollowing polynomial, I [ S ] = x + y + z , SO (3) → S . (3)N.B.: We have left aside how to find invariant polynomials to Sec. 2.4 and the more subtlequestion of the choice of polynomials to Sec. 2.5.How is this phrased in the language of Lagrangian field theory? We would thinkof SO (3) or SU (3) as internal symmetries of a field ( ϕ , ϕ , ϕ ) within the representa-tion space. The explicit breaking in Lagrangian language is accomplished by adding apolynomial of the invariant (3), L S = L SO (3) ( ϕ + ϕ + ϕ ) + f ( ϕ + ϕ + ϕ ) , (4)to the original SO (3) invariant Lagrangian L SO (3) , where f is a polynomial function. Thenew term can be regarded as an addition to the potential. The term explicit breakinghas to be contrasted with the term spontaneous breaking, to be discussed below, whichis more indirect. If we were to consider SO (3) in its fundamental representation and single out one direc-tion, then the symmetry breaks down to SO (2). This ought to be obvious from a spatialdrawing. How can the breaking of SO (3) to, say, S or any group different from SO (2)occur in this language? This happens when higher, not fundamental, representations areconsidered. Note that SO (3) is a subgroup of SU (3). In case we were to consider the breaking SU (3) → S wewould have to impose the polynomial P from Eq. (2) as additional invariant. When two or more invariantshave to be imposed, this can be understood as a sequential breaking, e.g., SU (3) → SO (3) → S . It is crucial here that the master groupis SO (3) and not SU (3), as otherwise SU (3) → ∆(6 · ) ⊃ ∆(6 · ) (cid:39) S with the invariant mentioned above, as we shall see in Sec. 4. As opposed to a space-time symmetries. In the case where we restrict ourselves to renormalizable terms, f ought to be linear. In Ref. [5] a few small representations were considered for SU (2) and SU (3), and it was found thatthey cannot break to any non-Abelian group except for D (cid:48) , which is the double cover of the the dihedralgroup D .
5e shall formulate this idea first in mathematical language without referring to SO (3)or SU (3). Let us choose a certain (faithful) representation of the master group G , denotedby R = R ( G ), acting on the representation space V with d G = dim( V ) >
3. Further wesingle out a certain representation vector v ∈ V and collect the following elements, H = { g ∈ G | R ( g ) v = v } . (5)It is readily verified that H constitutes a representation of a proper subgroup of G . Thegroup H is called stabilizer , isotropy group , or little group , depending on the area ofresearch.In the language of Lagrangian field theory one would add a potential to the initialkinetic term: L H = L kinetic SO (3) − U ( ϕ , .., ϕ d G ) , (6)which is R ( G )-invariant but whose extremum v , obtained by ∂∂ϕ i U ( ϕ , .., ϕ d G ) = 0 , i = 1 , .., d G ⇔ ( ϕ , ..., ϕ d G ) = v , (7)is R ( H )-invariant but not R ( G )-invariant. As the notation suggests, v in Eq. (7) corre-sponds to the representative v in Eq. (5). In the physics literature this phenomenon iscalled spontaneous symmetry breaking and v is referred to as a VEV . It is natural to ask of whether, given I [ S ] in Eq. (3), one can determine the corresponding v that leads to H = S in Eq. (5) and vice versa. In other words: Is there a link betweenexplicit breaking and spontaneous breaking? The link is readily established by noting that certain polynomial functions furnish arepresentation of the group. In the case of SO (3) this is usually given by the sphericalharmonics Y l,m . The latter correspond to a complete set of representation functions of SO (3) for l = 0 , , , ... , with representation dimensions 2 l + 1 as m ranges from m = − l to m = + l in integer steps.Expanding the invariant polynomial in spherical harmonics and using the orthogonal-ity relations, one obtains: I [ S ] = x + y + z = c (cid:32) Y , − + (cid:114) Y , + Y , (cid:33) , (8) This question was raised but deferred to later work in Ref. [10]. c is an irrelevant constant depending on the normalization of the spherical harmon-ics. This means that in an l = 4 representation, which is 9-dimensional, a direction v ∼ (1 , , , , (cid:114) , , , ,
1) (9)breaks SO (3) → S . In the language of branching rules this reads, SO (3) | S → S + ... , (10)where the dots stand for higher representations. Note that it is the VEV v (9) whichcorresponds to the trivial irrep under the restriction to S . How to obtain the other irrepsin Eq. (10), which are the ones of interest for model building, is described in App. C.2.It is the goal of this paper to generalize this to SU (3). A few explicit examples canbe found in Sec. 4.4; the complex spherical harmonics are discussed in App. G.1. Given a certain finite group H , is it possible to obtain all polynomial algebraic invariants?The answer is affirmative through the so-called Molien function [12], more generally knownas the generating function [11]. The Molien function is defined, for finite groups, as follows: M R ( H ) ( P ) ≡ |R ( H ) | (cid:88) h ∈R ( H ) − P h ) = (cid:88) m ≥ h m P m , (11)where P is a real number, R ( H ) is a representation of H , and |R ( H ) | denotes the numberof elements in that representation. Thus the Molien function is the average of the inversesof the characteristic polynomials over the group. The Molien theorem states that thepositive integer numbers h m correspond to the numbers of invariants I m of degree m thatleave the subgroup R ( H ) invariant, see e.g. [8] or [9] for a discussion within SO (3). Sinceany (polynomial) function of invariants is also an invariant, the question of minimalityimposes itself. Thus, what are the algebraically independent invariants and how do thedependencies between the others work out?It turns out that, for an n -dimensional representation, there are exactly n algebraicallyindependent invariants [7], the so-called fundamental or primary invariants [6]. Further- The concept of the Molien function finds its generalization in the generating function. For a genericreview on this powerful subject we refer the reader to Ref. [11] and references therein. In App. C thegeneralization from counting invariants to counting covariants is presented. As previously mentioned, thebranching rules can be obtained in this framework as well. secondary invariants, denoted by I as opposed to I , which are not algebraically independent. Relations of them and primary and secondary invariants areas follows [8]: I n i = f ( I m , I m , I m ) + (cid:88) j f ( j )1 ( I m , I m , I m ) · I n j , (12)where f and f ( j )1 are (polynomial) functions that depend only on the primary invariants,as indicated. Relations, as the one in Eq. (12), are called syzygies in the mathematicalliterature. Note that, once Eq. (12) is verified, we can be sure that we have found a validset of primary and secondary invariants.As a matter of fact, given a set {I m , I m , I m , I n i , .. } of primary and secondary in-variants, the Molien function can be written as [8]: {I m , I m , I m , I n i , .. } ⇒ M H ( ) ( P ) = 1 + (cid:80) i a n i P n i (1 − P m )(1 − P m )(1 − P m ) . (13)Here we have specialized to a 3-dimensional representation but the generalization shouldbe obvious. The three primary invariants are of degrees m , m , and m , respectively.Further to that there are 1 + (cid:80) i a n i secondary invariants one of which is the trivialinvariant and there are a n i invariants of degree n i . Note that the syzygies in (12) areconsistent with the fact that secondary invariants, associated with P n i , do not appear toany other power than one.The representation of the Molien function (13) is not unique and this is why the logicalarrow only goes from left to right and not the other way around. In practice invariantscan be found by following the three step procedure below:1. A form of the Molien function as in Eq. (13) is guessed. One should also check that it verifies the proposition in Eq. (14).
2. The corresponding invariants are generated (to be discussed below) and the alge-braic independence of the primary invariants is verified.
Algebraic independence of potential primary invariants can be checked with the Jacobiancriterion, Eq. (25).In the case where primary and/or secondary invariants are degenerate in degree, compli-cations may arise, c.f. App. D.2.2.
3. The syzygies from Eq. (12) are verified.If the latter step fails one has to return to the non-uniqueness of steps one and two. Issuesabout the non-uniqueness of the form of the Molien function and strategies on how to deal8ith cases when the degrees of the polynomials are degenerate are discussed in App. D.One powerful fact, c.f. proposition 2.3.6. in Ref. [8], which helps with point one is thatthe number of secondary invariants equals,number of secondary invariants ≡ (cid:88) i a n i = m · m · m | H | , (14)with an obvious generalization to an n -dimensional representation.Given the information on the degrees of the invariants from Eq. (13), how can theinvariants be constructed? This is rather straightforward, modulo ambiguities in form ofdegeneracies, by symmetrization of trial polynomials. We observe that for any (polyno-mial) function f ( x, y, z ), an invariant I ( x, y, z ) can be obtained as follows, I ( x, y, z ) = 1 |R ( H ) | (cid:88) h ∈R ( H ) f ( h ◦ x, h ◦ y, h ◦ z ) , (15)where here and thereafter ◦ denotes the action of the group on an element of the repre-sentation space. Verification of the invariance in (15) is immediate and left to the reader.This operation is known as the Reynolds operator in the mathematical literature, seee.g. [8]. Note that the form of the invariants is dependent on the embedding, e.g. on sim-ilarity transformations as discussed in Sec. 5. In practice this means that the invariantscan be obtained by taking a suitable ansatz for the function f ( x, y, z ). For our purposesthe most convenient trial functions are: f ( x, y, z ) = x n y m z − n − m , m, n ≥ . (16)N.B.: For most trial functions this invariant is going to be zero, which is the trivialinvariant. An example on how to obtain the invariants, with S , is discussed in App. D.2.1.It is now time to return to our example S and execute the three step procedureoutlined previously. The degrees of the invariants are such that no problems of the kindmentioned in point two occur.Step 1: By computing (11) and looking for poles we guess that the Molien functionas in Eq. (13) takes the following form, M S ( P ) = 1 + P (1 − P )(1 − P )(1 − P ) , (17)which satisfies proposition (14). 9tep 2: By using the Reynolds operator the primary, I [ S ] = x + y + z , I [ S ] = ( xyz ) , I [ S ] = x + y + z , (18)and secondary, I [ S ] = xyz ( x − y )( y − z )( z − x ) , (19)invariants are readily computed, and the algebraic independence of the former can beshown easily by, e.g., the Jacobian criterion [27]. Note that the first primary invariant ismerely the statement that S is a subgroup of SO (3).Step 3: The syzygy (12) is verified to be I = I I I − I I − I I I + 12 I I + 5 I I − I I I − I , (20)where we have omitted the [ S ] on the invariants. By verifying the syzygy, we havecompleted the program and shown that (17) is indeed the Molien function as in Eq. (13).This completes the exemplification of the Molien function and primary and secondaryinvariants for S . In Fig. 1 the geometric nature of these invariants is revealed in plots.In Sec. 3.2 we obtain all primary and secondary invariants including the syzygies for theentire ∆-series. In App. C the generalization from invariants to covariants, by whichwe mean tensor objects, is discussed by going from the Molien function to the tensorgenerating function. From the tensor generating function of a group the branching rulescan be deduced, as implemented in our package SUtree . In the previous discussion we have simply assumed that, by imposing the invariant (3),the group breaks from SO (3) → S . How can we be sure of that? What are the necessaryand sufficient criteria?Consider a group H and an unordered list of invariants {I [ H ] , I [ H ] , .. } associated toit, as well as the corresponding vectors { v [ H ] , v [ H ] , .. } constructed as in Eqs. (8) and (9).The certain fact is that the group H leaves I [ H ] j and v [ H ] j invariant by construction.However, there can be other groups H (cid:48) which leave them invariant as well. It is a fact [15]that in this case the group must break into the largest group, which we shall denote by H ⊂ G . Basically, there are three distinct relations between H , H (cid:48) , and H : • subgroup: H ⊂ ..H (cid:48) .. ⊂ H • supergroup: H (cid:48) ⊂ ..H.. ⊂ H This figure shows the 3-sphere (blue/dark gray) corresponding to SO (3) and (left) theequipotential surfaces (light brown/gray) corresponding to the invariant I [ S ] = x + y + z and (right) the same for I [ S ] = ( xyz ) . The two invariants, as shall be argued later in Sec. 4,are sufficient to break SO (3) → S . The polyhedric symmetries of the hexahedron (cube) andoctahedron, which are dual to each other under interchange of faces and edges, beautifullyreveal themselves in this plot. On the left the intersection of the sphere and the I -invariantcorresponds to the six faces or six edges of the hexahedron and octahedron, respectively, whereason the right the analoguous intersection corresponds to the eight edges and eight faces of thehexahedron and octahedron, respectively. This is why S is, at times, called the hexahedron or octahedron group . • no such relation: H ⊂ ..H.. ⊃ H (cid:48) As an example of the second case we mention that A and S both leave I fromEq. (18) invariant, but since A ⊂ S the master group SO (3) breaks into S with I .Note, we have assumed that there does not exist a group H (cid:48) with S ⊂ H (cid:48) ⊂ G whichleaves I invariant.Thus finding the sufficient criteria for breaking into a subgroup is a subtle issue. Thisproblem can be handled once one knows the “tree” of subgroups from the group G . Ideallywe would therefore like to know when given two groups of which one is a subgroup of theother, H ⊂ G , whether H is a maximal subgroup or whether there is another group H (cid:48) in between, H ⊂ H (cid:48) ⊂ G . In the case where G is a finite group this question can besettled by a computer algorithm using corollary 1.5.A in [13]. In the case where G is not11nite, e.g. continuous, the question cannot be answered in general. In connection withLie groups a general discussion can be found in [14], and many examples are given in [45].In the context of SO (3) the so-called Michel criterion was known for some time [15], butcounterexamples have been found [16]. In Sec. 4.1 we shall discuss strategies to cope withthe proper finite subgroups of SU (3). SU (3) database The SU (3) subgroups have been classified almost 100 years ago [18]. For a review ofthe contributions thereafter we refer the reader to the introduction of Ref. [28]. Thegenerators of the proper finite SU (3) subgroups are given in Tab. 1, taken from Ref. [17].The question to what extent these results are embedding dependent is discussed in Sec. 5.By proper we mean the groups which are not subgroups of SU (2). The invariants of the SO (3) and SU (2) subgroups have been discussed extensively in the literature in regardto applications in crystallography. There are the crystallographic types A , S , and A ,the dihedral groups D n (cid:39) Z n (cid:111) Z , and the cyclic groups Z n . For a discussion of theinvariants of these groups we refer the reader to the extensive review [19].In our database SUtree , we are going to restrict ourselves to the proper finite SU (3)subgroups of order smaller than 512, supplemented by the three crystallographic sub-groups of SO (3) mentioned above and by the two crystallographic groups Σ(216 φ ) andΣ(360 φ ) whose orders are 3 ·
216 and 3 · X ), the so-called crystallographic groups , and the countable series of C - and D -groups of which ∆(6 n ),∆(3 n ), and T n [ a ] are special cases as can be inferred from Tab. 1. This table differs fromthe one in [17] by writing the generators M, J, P, Q in terms of C - and D -type generatorsand by T n → T n [ a ] . The latter is necessary as for certain n there exist several solutions for a , e.g. for n = { , } in Tab. 5. To some degree the division into crystallographic andnon-crystallographic groups is arbitrary, as A ∼ ∆(12) and S ∼ ∆(24). We have workedout the subgroup structure or “subgroup tree” of this entire list with the generator basisand GAP, and it is given in App. B.2 in Fig. 5 and Tab. 7, where further remarks on thisprocess can be found. We do not include groups which are of the type F × Z n , where Z n denotes the cyclic group of order n . These groups can be reconstructed by using the-orem II.2 [17], which we quote in App. A for the reader’s convenience. An investigation As is nicely illustrated in [17], above the order of 512 the number of groups becomes rather large andbarely manageable. A × Z has become popular in model building recently [17]. Note that ∆(27) × Z has no faithful
12f subgroups of order smaller than 100 including all groups with direct products can befound in Ref. [20]. More subtle is the question of U (3) subgroups versus SU (3) subgroups.The classification of the former has not been completed. In particular there is more toit than F × Z n , as U (3) (cid:39) SU (3) × U (1) might suggest, e.g. S (4) (cid:39) A (cid:111) Z [17]. Inthis case the 3-dimensional irreps are obtained by multiplying a certain generator by ± i .This means that among the invariants quoted in (18) and (19) only I will remain aninvariant.For the sake of completeness let us mention the topological structures of the non-crystallographic groups: T n : Z n (cid:111) Z [23] , ∆(3 n ) : ( Z n × Z n ) (cid:111) Z [24] , ∆(6 n ) : ( Z n × Z n ) (cid:111) S [24] , (21)where (cid:111) stands for the semidirect product . The ∆-groups, sometimes called trihedralgroups , are to be seen as direct generalizations of the dihedral groups D n ⊂ SO (3), whichhave the topological structure D n ∼ Z n (cid:111) Z . The general C - and D -groups have beenshown to have the following topological structures: C ( n, a, b ) : ( Z t × Z u ) (cid:111) Z [28] ,D ( n, a, b, d, r, s ) : ( Z t × Z u ) (cid:111) S [29] . (22)An algorithm but no explicit formulae for t and u were given. For C -groups the followingis true: t ≤ u ≤ n , which is consistent with the statement in Eq. (29) in Sec. 4.3. SUtree
We have developed an accompanying software package called
SUtree with this paper.After having downloaded the software from Z and thus does not fall into the category of thetheorem II.2. If this was not the case the criteria for breaking into ∆(3(3 n ) ) groups given in Sec. 4would need further refinement. In [17] the U (3) subgroups of order smaller than 512 were considered. Moreover this referenceuncovers further series of U (3) subgroups in generalizing the ∆-series. Ref. [21] investigates a number offinite U (3) subgroups. It should be kept in mind that the semidirect product is only complete once the group-homomorphismis given. Thus the notation above does not yet determine the group. C -, D -type Σ( X )-type MBD C ( n, a, b ) E , F ( n, a, b ) CD ( n, a, b ; d, r, s ) E , F ( n, a, b ), G ( d, r, s ) D ∆(3 n ) = C ( n, , , n ≥ E , F ( n, , ∈ C ∆(6 n ) = D ( n, ,
1; 2 , , n ≥ E , F ( n, , G (2 , , ∈ DT n [ a ] = C ( n, , a ), (1 + a + a ) = n Z E , F ( n, , a ) ∈ C Σ(60) = A = I = Y E , F (2 , , H H
Σ(168) =
P SL (2 , E , M ≡ F (7 , , N J
Σ(36 φ ) E , J ≡ F (3 , , K E
Σ(72 φ ) E , J ≡ F (3 , , K , L F
Σ(216 φ ) E , J = F (3 , , P ≡ F (9 , , K G
Σ(360 φ ) E , F (2 , , Q ≡ G (6 , , H I
Table 1:
Types of finite subgroups of SU (3) which are not subgroups of SU (2) with theexceptions of A (cid:39) ∆(12), S (cid:39) ∆(24), and A (cid:39) Σ(60). Explicit generators are given inApp. B.1. For T n [ a ] : n = 3 p or n = p with p equal to the product of primes of the form3 N + 1 [22, 23]. Some further remarks on these groups and the fact that neither a nor n determine each other can be found in Sec. 4.3. This is why we have extended the notationfrom T n to T n [ a ] . The column MBD corresponds to the classification used in [18] and the lettersshould not be confused with the generators. The A and B types correspond to direct productsof Abelian factors and are not of the type considered here. ttp://theophys.kth.se/~amerle/SUtree/SUtree.html and having extracted the files, the easiest way to go is to open the example file ExampleNotebookSUtree.nb with Mathematica. In there, all features of the database are explained and exemplified.A short dialogue example can be found in Sec. 4.4.A short summary is given here: Using the group numbers, GAP numbers, or groupnames, as given in Tab. 5, one can efficiently refer to all of the 61 groups in our list. Allthe group elements are stored numerically in the database. We have used these matricesto calculate the primary invariants, secondary invariants, Molien functions, and tensorgenerating functions for all 61 groups in our database. Note that, depending on thegroup, the expressions in particular for the secondary invariants can be relatively lengthy[e.g. the one of Σ(360 φ )], and the number of secondary invariants can be quite large,too (e.g. T has 116 secondary invariants). In addition, we have also calculated allthe corresponding syzygies for the secondary invariants. Furthermore, the databasecontains a routine to calculate any ( l ) SO (3) - or ( p, q ) SU (3) -basis (cf. App. G.2), and totranslate all invariant polynomials into VEVs and vice versa. In addition, the charactertables are given for all groups, as well as the (tensor-) generating functions, from whichthe branching rules and Kronecker products can be derived with SUtree .The functions of
SUtree can be used to verify many of the following calculations. Insome cases, as will be shown below for the ∆(3 n ) and ∆(6 n ) groups, it is even possibleto use our program to guess general results that can then be proven a posteriori. ∆(3 n ) and ∆(6 n ) In this section we compute the primary and secondary invariants of the groups ∆(3 n )and ∆(6 n ), valid for any n . From Tab. 1 and Eq. (B.1) we infer that, E = , F = F ( n, ,
1) = η
00 0 η − , G = G (2 , ,
1) = − − − , (23) Note that some of our results are only true for the representation used in Tab. 5, c.f. Sec. 5. This is true except for the syzygies of the four secondary invariants of T of highest degree, aswell as for the syzygy of Σ(360 φ ). Furthermore, the syzygy of A was obtained numerically only. This is true for all groups except for T and for T , for which we did not succeed in findinggenerating functions, due to the large numbers of conjugacy classes. { E, F } and { E, F, G } generate the groups ∆(3 n ) and ∆(6 n ), respectively, with η = e πi/n . We are now going through the three step procedure of Sec. 2.4.Step 1: By computing the Molien function (11) for a few cases with lower n thefollowing Molien functions (13) suggest themselves, M ∆(3 n ) ( P )= 1 + P n (1 − P )(1 − P n )(1 − P n ) ,M ∆(6 n ) ( P )= (cid:40) P n +3 (1 − P )(1 − P n )(1 − P n ) n even , P n +3 + P n + P n +3 (1 − P )(1 − P n )(1 − P n ) n odd , (24)which verified proposition (14).Step 2: By computing a few invariants for n without problems of degeneracies, c.f.App. D.2, primary and secondary invariants have been obtained as shown in Tab. 2.The algebraic independence of the primary invariants can be verified using the Jaco-bian criterion [27]. For, e.g., the ∆(3 n ) primary invariants we get:det ∂ x I ∂ x I n ∂ x I n ∂ y I ∂ y I n ∂ y I n ∂ z I ∂ z I n ∂ z I n = 2 n ( x n − y n )( y n − z n )( z n − x n ) (cid:54) = 0 , (25)for general x, y, z .Step 3: The syzygies are given in that table as well. This completes the analysis andproves that (24) are Molien functions in the form of Eq. (13) with the interpretation ofprimary and secondary invariants.Let us end this section by pointing out that it is rather remarkable that the invariantsof ∆(3 n ) and ∆(6 n ) are expressible in such a simple manner for general n . In particularthe length of the syzygies does not depend on n and this is the reason why we were ableto compute all the data.In fact, ∆(3 n ) and ∆(6 n ) n ∈ N can be seen as the extensions of the SO (3) subgroups A and S , where the Euclidian distance (2) is generalized from a power of 2 to n : A ; S : → ∆(3 n ); ∆(6 n ) n ∈ N : x + y + z → x n + y n + z n . (26)It is like the ∆(3 n ) , ∆(6 n ) n ∈ N are the A , S of a space where distances are measured16ith x n + y n + z n . The series ∆(6 n ) n ∈ N +1 can be seen as the following extension: S : → ∆(6 n ) n ∈ N +1 : xy + yz + zx → x n y n + y n z n + z n x n , (27)since S (cid:39) ∆(6 · ). Note that S is not a proper finite SU (3) subgroup since it doesnot contain a faithful 3-dimensional irrep which, already, follows from the dimensionalitytheorem (c.f. App. A), | S | = 6 < + 3 . SU (3) → Σ( X ) , ∆(6 n ) , ∆(3 n ) , T n [ a ] In this section we provide example solutions to the problem, discussed in general terms inSec. 2.5, of selecting the invariants for a specific group H that break SU (3) → H . In Tab. 7we list the groups in our database with subgroup references from where the subgroup treecan be derived, as in Fig. 5. In the database only { S , A } and { Σ(168) , Σ(216 φ ) , Σ(360 φ ) } are maximal subgroups of SO (3) and SU (3), respectively. For all other cases there exists agroup H (cid:48) such that H (cid:40) H (cid:48) (cid:40) SU (3). The task then becomes to show that the invariantsselected for H are not invariants of any H (cid:48) as well. The group H (cid:48) could be either of finite,continuous, or of mixed type. In Sec. 4.1 potential groups H (cid:48) of continuous-type arediscussed. In Secs. 4.2 and 4.3 we give examples of sufficient invariants for T n [ a ] , ∆(3 n ),∆(6 n ), and all Σ( X ). The subtle question of why it is legitimate to work with the explicitgenerators, as given in Tab. 1, is discussed in Sec. 5.4. We have not attempted to findsufficient conditions for generic C - and D -groups. Possibly more work is needed on thestructure of these groups. SU (3) The continuous subgroups of SU (3) are SO (3) and U (2) = SU (2) × U (1), and subgroupsthereof. We observe that all groups in our list contain the generator E (cf. Tab. 1 orApp. B), which corresponds to a cyclic permutation of the three variables { x, y, z } . Letus first discuss the group SO (3). The finite subgroups of SO (3) are the well-known A , S , and A for which we have all the data, and the dihedral groups which are not invariantunder a cyclic permutation since they correspond to the symmetry of a molecule with one Note that discrete subgroups of SU (3) which are not finite, such as SU (3) elements with rationalentries, are of no interest here since they would never leave invariant the kind of polynomials we areconsidering. Some effort has been undertaken recently in Refs. [30, 31] and especially in Ref. [28]. n ) primary I = xyz , I n = x n + y n + z n I n = x n + y n + z n secondary I n = x n + y n + z n syzygy I n = 9 I n + 9 I n I n I n + I n I n − I n I n − I n I n + I n ∆(6 n ) primary I = ( xyz ) ,even n I n = x n + y n + z n I n = x n + y n + z n secondary I n +3 = xyz ( x n − y n )( y n − z n )( z n − x n )syzygy I n +3 = I (cid:104) I n − I n − I n/ I n I n + 5 I n/ I n + I n I n − I n − I n I n (cid:105) ∆(6 n ) primary I = ( xyz ) ,odd n I (cid:48) n = x n y n + y n z n + z n x n I n = x n + y n + z n secondary I n +3 = xyz ( x n + y n + z n ) I n = ( x n − y n )( y n − z n )( z n − x n ) I n +3 = xyz [( x n y n − y n x n ) + ( y n z n − z n y n ) + ( z n x n − x n z n )]syzygies I n +3 = I ( I n + 2 I (cid:48) n ) I n = I n ( I (cid:48) n ) − I (cid:48) n ) − I ( n − / I n +3 I n + 10 I ( n − / I n +3 I (cid:48) n − I n I n +3 = I I n ( I n + 2 I (cid:48) n )Table 2: Primary and secondary invariants of ∆(3 n ) and ∆(6 n ) and the corresponding syzy-gies. They have been guessed by explicit calculation of the first few cases and then proven tobe correct a posteriori by using the generators from Eq. (23). Let us stress once more that thechoice of primary invariants is in certain cases like the choice of a basis. Note that, for even n in ∆(6 n ), another choice for I n is I (cid:48) n = x n y n + y n z n + z n x n , which is symmetric as well. Itis readily verified that 2 I (cid:48) n = I n − I n . U (2) is not invariant under cyclic permutations as theembedding U (2) = SU (2) × U (1) (cid:44) → SU (3), denoted by the symbol (cid:44) → , singles outa direction and is therefore eliminated for the same reason. For groups of mixed type,only U (1) × U (1) (cid:111) Z ; S are known [18, 24], and they can be understood as the formallimits n → ∞ of ∆(3 n ); ∆(6 n ). The latter are implicitly included in our discussionthrough the ∆-groups. In summary the cyclic symmetry of our groups forbids any groupsof continuous or mixed type. Σ( X ) The partial subgroup tree in the crystallographic sector is shown in Fig. 2, from where weinfer that Σ(216 φ ), Σ(360 φ ), and Σ(168) are maximal subgroups of SU (3). This can beseen as follows: First they are maximal in the chain of crystallographic groups. Second,they cannot be subgroups of the C - and D -type groups since the latter contain irreps ofdimensions not higher than six [31], whereas Σ(216 φ ), Σ(360 φ ), and Σ(168) all containirreps of dimensions larger than six.Using our Mathematica package SUtree , the common invariants of the subgroups canbe identified, and thus breaking from SU (3) into these subgroups can be worked out. Letus list the Molien functions and the lowest invariants that break SU (3) → Σ( X ): Group Molien function Invariant of lowest degree that breaks SU (3) → Σ( X )Σ(60) P (1 − P )(1 − P )(1 − P ) ( φ x − y )( φ z − x )( φ y − z )Σ(36 φ ) P + P + P (1 − P ) (1 − P ) ( x + 2 x y − x yz + cy.) − x y z Σ(168) P (1 − P )(1 − P )(1 − P ) x z + z y + y x Σ(72 φ ) P + P (1 − P )(1 − P )(1 − P ) x + y + z − x y − y z − z x Σ(216 φ ) P + P (1 − P )(1 − P )(1 − P ) x ( y − z ) + y ( z − x ) + z ( x − y )Σ(360 φ ) P (1 − P )(1 − P )(1 − P ) x + y + z + ax y z + b + ( x y + cy.) + b − ( x z + cy.) The group SU (2) can be embedded in such a way that SU (3) → SU (2) , but then it is the same as SO (3) which we have already discussed. Note that this line of reasoning is general and much simpler than algebraic methods, which have tobe applied case by case, see Ref. [32]. The cyclicity is evident in the language of invariant polynomialsas opposed to the language of VEVs. A subtle point is that the subgroup relation Σ(36 φ ) ⊂ Σ(360 φ ) is not apparent from its generators.Thus one has to be cautious when comparing invariants. In order to verify that the invariant of degreesix proposed for the breaking SU (3) → Σ(36 φ ) is correct, one has to use the basis transformation givenin App. B.2. We have verified that, in that basis, the invariant discussed is not left invariant by Σ(360 φ ). U ! " ! ! " " ! ! " " ! Φ " ! Φ " ! Φ " ! Φ " ! " A S ! " T Figure 2:
The subgroup tree of the crystallographic groups. Note this is only a partial tree.The entire tree, within our database, is shown in Fig. 5.
In the table, we have used: φ ≡ √ , a = 3 (cid:16) − i √ (cid:17) , b ± = 38 (cid:104) ∓ √ i (cid:16) √ ± √ (cid:17)(cid:105) , (28)where “cy.” stands for cyclic permutations in the variables x , y , and z . Let us add thatthe Molien function for Σ(216 φ ) differs from the one in [18], but it is the same as in [46],where the ones for Σ(360 φ ) and Σ(168) were also presented. The reader should be ableto find invariants of higher degrees that achieve the same. A subtle point to be stressed isthat not all subgroups relations are apparent from the generators as given in Tab. 1. Thus,when comparing invariants or checking their invariance with respect to supergroups, onehas to account for this fact by similarity transformations, c.f. App. B.2, as we did for thecase Σ(36 φ ) ⊂ Σ(360 φ ) as described in an earlier footnote in this section.20 .3 Breaking to C - and D -groups ( ∆(6 n ) , ∆(3 n ) , and T n [ a ] ) Before discussing the groups ∆(3 n ) and ∆(6 n ) in more detail let us discuss some gen-eralities about subgroup structures,∆(3 n ) m ⊂ ∆(6 n ) , ∆(3 n ) ⊂ ∆(3(2 n ) ) , ∆(6 n ) ⊂ ∆(6(2 n ) ) ,C ( n, a, b ) ⊆ ∆(3 n ) , e.g. T n [ a ] m ⊂ ∆(3 n ) ,D ( m, a, b ; d, r, s ) ⊆ ∆(6 n ) , n = lcm( m, d, , (29)depicted in Fig. 3. The acronym “lcm” stands for lowest common multiple and the symbol m ⊂ for maximal subgroup. In the cases at hand this follows by virtue of Lagrange’s theorem,c.f. App. A. The first three statements are obvious from the generators. The fourth onecomes about by realizing that any C ( n, a, b ) corresponds to a ∆(3 n ) [28]. Crucially someof those representations are not faithful so that C ( n, a, b ), depending on a and b , can bea proper subgroup of ∆(3 n ). For the fifth statement we refer the reader to [31]. Thegroups ∆(3 n ), ∆(6 n ), and T n [ a ] are discussed case by case, and illustrated in Fig. 4. Breaking to ∆(6 n ) We propose that SU (3) → ∆(6 n ) by imposing I (cid:48) n [∆(6 n )] = x n y n + y n z n + z n x n , n odd , I n [∆(6 n )] = x n + y n + z n , n (cid:54) = 2 and even . (30)Note in the case where n = 2, which corresponds to S = ∆(6 · ), SU (3) → SO (3) andthus a further invariant, say I [ S ] or I [ S ] (18), has to be imposed. We have checkedthat none of the crystallographic generators in Tab. 1 leaves either I n or I (cid:48) n invariant.It remains to show that the action of F ( m, a, b ) and G ( d, r, s ) for generic parameters { m, a, b, d, r, s } together with the constraint of (30) being invariant implies that they arecontained within ∆(6 n ). • F ( m, a, b ): Let us assume that F ( m, a, b ) exists which leaves I n (30) invariant. Then21 ! n " T n a $ C ! n,a,b " ! ! " ! " " ! ! n " ! ! " ! " " D ! m,a,b;d,r,s " if n lcm ! m,d,2 " Figure 3:
Tree of subgroups in the C - and D -sector as given in Eq. (29). The dashed linesallude to the fact that there could be other subgroups in between, whereas the solid lines aremaximal subgroup relations. the following ought to be true: η an = 1 and η bn = 1 , (31)where η = e πi/m . Writing θ ∈ { a, b } , it follows from Eq. (31) that η θn = e πiθn/m ⇔ θnm ≡ k θ ∈ N . (32)This allows us to rewrite the initial generator as F ( m, a, b ) = diag (cid:0) e πia/m , e πib/m , e − πi ( a + b ) /m (cid:1) (32) = diag (cid:0) e πik a /n , e πik b /n , e − πi ( k a + k b ) /n (cid:1) = F ( n, k a , k b ) . (33)Thus we have traded the m for n by ( a, b ) → ( k a , k b ). There is no special need to22e specific about the latter two as ( k a , k b ) = (0 , n ) and second the other choices lead to smaller groups as stated in Eq. (29).Since the breaking, however, will always lead to the largest group to which one couldpossibly break, this observation completes the argument. • G ( m, a, b ): The investigation of G ( d, r, s ) calls for a distinction of odd and even n : – n even: The action of G ( d, r, s ) leaves I n invariant if and only if ( rn/d, sn/d ) ∈ Z , where we have used that for even n the phase factor, ( − n = 1, is unity.Thus we may write ( r/d, s/d ) = ( R/n, S/n ) with (
R, S ) ∈ Z , and therefore G ( d, r, s ) → G ( n, R, S ). – n odd: The very same action on I (cid:48) n lead to the conclusion that ( rn/d, sn/d ) ∈ (2 Z + 1) , which by the same argumentation leads to ( r/d, s/d ) = ( R/n, S/n )with (
R, S ) ∈ Z and therefore G ( d, r, s ) → G ( n, R, S ) as above.Making the observation that F ( n, a, b ) G (2 , ,
1) = G ( n, − a, − b ) we can infer that G ( n, R, S ) ∈ ∆(6 n ), since the latter is generated by { E, F ( n, , , G (2 , , } . Inorder to appreciate the last step it should be added that { E, F ( n, a, b ) , G (2 , , } can only be a subgroup of ∆(6 n ). Breaking to ∆(3 n ) We propose that SU (3) → ∆(3 n ) by imposing I (cid:48) n [∆(3 n )] = x n + y n + z n , n odd, I n [∆(3 n )] = x n + y n + z n , I [∆(3 n )] = xyz , n even. (34)The results follow, rather directly, from the analysis of ∆(6 n ) in the previous subsection.We will not repeat all arguments in detail. • n odd : I n is not a ∆(6 n ) invariant because of the generator G (2 , , G ( d, r, s ) does not leave I n invariant for odd n because of theminus sign in ( − δ − r − s ). Thus I n breaks SU (3) to ∆(3 n ) for odd n . • n even : Since ∆(3 n ) is a maximal subgroup of ∆(6 n ) it suffices to find oneinvariant, e.g. I [∆(3 n )], of ∆(3 n ) in order to break from ∆(6 n ) to ∆(3 n ). The case in [32] for ∆(27), n = 3, can be seen as a special case of our finding. The VEV found inthat reference ought to translate into I n . U ! " U ! " ! U ! " U ! " ! U ! " Z S " ! n " " ! n " T n a $ I a ! & I n " I I I I I n I n ! n even " I n ! n odd " $ I n ! n odd " I n ! n odd " I a I I n % I n $ ! n % " I & xyz I & ! xyz " I n & x n y n z n I n $ & x n y n y n z n z n x n I a & x a y a y a z a z a x a Figure 4:
Summary of of breaking patterns for the T n [ a ] -, ∆(3 n )-, and ∆(6 n )-groups. Thegroups U (1) × U (1) (cid:111) Z [ S ] are understood to be the formal limits of n → ∞ of ∆(3 n )[∆(6 n )]. Imposing the two invariants from Eq. (34) can be seen as a sequential breaking SU (3) I n → ∆(6 n ) I → ∆(3 n ). 24 reaking to T n [ a ] We propose that SU (3) → T n [ a ] for I a +1 [ T n [ a ] ] = x a +1 y a + y a +1 z a + z a +1 x a , a + a + 1 = 1 · n , I a +1 [ T n [ a ] ] , I n [ T n [ a ] ] = x n + y n + z n , a + a + 1 = m · n , m ∈ N + 1 . (35)Let us discuss the first case first. The generators of crystallographic type listed in Tab. 1do not leave I a +1 [ T n [ a ] ] invariant. Idem for the generator G ( d, r, s ) as it exchanges y and z but not x . Second, considering a generator F ( η, α, β ), we get three equations which addup to zero. So we effectively have two conditions:( a + 1) α + aβ = 0 mod η , ( a + 1) β + a ( − α − β ) = 0 mod η . (36)Considering α = α ( a ) and β = β ( a ), and differentiating both equations with respect to a we get a set of first order coupled homogeneous differential equations whose solutionis unique and given by ( α ( a ) , β ( a )) = (1 , a ). Reinserting this solution into (36) we get a + a + 1 = 0 mod η and, using the condition a + a + 1 = n , we get η = n if F ( η, , a )is not to be a subgroup of F ( n, , a ). This completes the argument.In the second case we have a + a + 1 = mn and we cannot conclude n = η . Imposing I a +1 [ T n [ a ] ] alone in this case will break SU (3) → C ( n · m, , a ) ⊃ T n [ a ] . This can beremedied by imposing the additional invariant I n [ T n [ a ] ] as proposed above.With respect to the classification of the T n [ a ] -series we note that that neither n deter-mines a nor does a determine n . Thus the double label seems appropriate. A few explicit examples can be found below within the basis quoted at the end of thissection. All these examples can also be found in the example notebook of
SUtree . • From (4 , SU (3) → Σ(168): (4 , (cid:39) (cid:48) I [Σ(168)] = x z + z y + y x = ( −√ v [Σ(168)] , · B (4 , ,v [Σ(168)] , = (0 , , , , , , , , , , , , − , , . (37) The fact that a (3 ,
0) = was found to break SU (3) → T [32] can be seen as a special case ofthe analysis. The VEV found in that reference ought to translate into the invariant I a +1 . In fact, for T and T , m is 3 and 7, respectively, and thus C (91 · , , (cid:39) T and C (133 · , , (cid:39) T are indeed of the T n [ a ] -series. From (4 , SU (3) → ∆(96): I [∆(96)] = x + y + z = 2 √ v [∆(96)] , · B (4 , ,v [∆(96)] , = (1 , , , , , , , , , , , , , , . (38) • From (1 , SU (3) → S : (1 , (cid:39) The irreps of S are { , (cid:48) , } and generators { E, G (2 , , } , Eq. (23). Note thatwe have used the fact that S = ∆(6 · ). From the generators we can infer that SU (3) → (cid:48) + . The branching rule is computed using the methods of App. C:(1 , SU (3) | S → ( + (cid:48) + 3 · ) S . (39)Thus there is one single invariant in that representation. The invariant is easilyguessed, I [ S ] , = xy ∗ + yx ∗ + z ∗ y + y ∗ z + x ∗ y + xz ∗ = v [ S ] , · B (1 , ,v [ S ] , = (1 , − , , , , − , − , − , (40)as it corresponds to the Weyl-symmetry of the root diagram, which is S n for SU ( n ). • Example dialogue in the Mathematica package
SUtree : In[1]:=
SetDirectory[ ” ...(your directory).../SUtree v1p0/ ” ]; In[2]:= $ RecursionLimit=260;< In[3]:= BranchingSU3[ { } , ” A ” ]; Out[3]= {{ 3, 0 } , 10, { 1, 1 } , { 3, 1 } , { 3, 1 } , { 3, 1 }} 1. The directory has to be set to the path where the package and its data direc-tory reside. 2. The package is loaded via “ < 0) and that { , } corresponds to thefirst 3-dimensional irrep in the character table.The explicit bases used above are derived from (G.3). The ordering is such that rst isinterpreted as a number with constraints (G.4), e.g. (001 , , ... , , ... , ... ). Thebases are given by: B (1 , = (cid:26) xz ∗ , − yz ∗ , xx ∗ + yy ∗ − zz ∗ √ , xy ∗ , xx ∗ − yy ∗ √ , − x ∗ y, − y ∗ z, − x ∗ z (cid:27) , B (4 , = (cid:26) x √ , − x y √ , x y , − xy √ , y √ , − x z √ , x yz √ , − xy z √ , y z √ , x z , − xyz √ , y z , − xz √ , yz √ , z √ (cid:27) . (41) F (cid:44) → SU (3) In our analysis we have chosen a particular embedding, F (cid:44) → SU (3) , (42)namely the one given in Tab. 1. It is therefore a legitimate question whether our resultsare dependent on it. We shall discuss this issue from the viewpoint of explicit breakingand not from the viewpoint of VEVs. Since the two are equivalent this is sufficient.Generically an embedding for groups, denoted by H (cid:44) → G , is an (injective) map from H to G that preserves the group structure. One distinguishes embeddings up to similaritytransformations 5.1 and those who do not fall into this class 5.2. The former case resemblesthe choice of a coordinate system and the latter corresponds to inequivalent irreps. Inthe case where the irrep is of the same dimension as the group it is embedded in, as inEq. (42), this corresponds to different irreps in the character table. In Sec. 5.3 we discussthe impact of the embedding on the Molien function and on the invariants. In Sec. 5.4 itis analyzed whether the inequivalent 3-dimensional faithful irreps of the Σ( X )-, ∆-, and T n [ a ] -groups can be distinguished with respect to each other.Before embarking on these topics, we would like to add a few more comments inconnection with larger groups and embedding into larger groups: • In this work we have restrained ourselves to 3-dimensional (irreducible) represen-tations in view of the three generations of particles in the lepton and quark sector27f the SM. If there was a fourth generation, which is possible, then we would bestudying something like: A (cid:44) → SU (4) instead of A (cid:44) → SU (3) , (43)for example. Finite SU (4) subgroups have been studied in Ref. [37]. • For model building it is interesting to consider F (cid:44) → X with | X | > 3. For examplethe chain, A + (cid:48) A (cid:44) → S (cid:44) → SU (6) , (44)could very well be part of an interesting model. The embedding theory of this kindis well developed for Lie groups, where inequivalent emebddings are characterizedby an embedding index [33] (or [34] for an alternative discussion). An exampleoften discussed in books [34, 35] is SU (2) (cid:44) → SU (3), as quoted in Sec. 4.1. Findingall embeddings is equivalent to finding all branching rules. For finite groups nocomplete theory is known to our knowledge. • A possibility, frequently used in model building, is to introduce several fields carryingdifferent irreps of SU (3) or of one of its subgroups. For this setting the embeddingup to similarity transformations does matter. This phenomenon is known under thename of vacuum alignment and is briefly outlined in App. E. Given a certain representation R ( h ) of H , which we shall denote for the sake of brevityby h only, the similarity transformation, h (cid:48) = AhA − , where A is an invertible matrix, (45)provides another representation of the group. Note that h and h (cid:48) are unitary representa-tions if and only if A is a unitary matrix, e.g. [35].Importantly the transformation (45) does not correspond to an inequivalent irrep. Inthe finite case the character and therefore the character table is left invariant. A pointwe would like to emphasize is that under (45) the invariants transform unless A ∈ R ( H ). Specht’s theorem - criteria for unitary equivalence It is an important practical question, given a set of matrices h and h (cid:48) , of whether theyare unitary equivalent, h (cid:48) = U hU † . The criteria are given by Specht’s theorem [36], which28ives sufficient conditions. For three dimensions they amount to:tr[ h ] = tr[ h (cid:48) ] , tr[ h ] = tr[ h (cid:48) ] , tr[ hh † ] = tr[ h (cid:48) h (cid:48)† ] , tr[ h ] = tr[ h (cid:48) ] , (46)tr[ h h † ] = tr[ h (cid:48) h (cid:48)† ] , tr[ h ( h ) † ] = tr[ h (cid:48) ( h (cid:48) ) † ] , tr[ h ( h ) † hh † ] = tr[ h (cid:48) ( h (cid:48) ) † h (cid:48) h (cid:48)† ] . As stated above, for the embedding type (42), inequivalent embeddings correspond todifferent 3-dimensional irreps of the group. An example is given by the two representations and of A , see e.g. [30]. We begin by observing that the Molien functions and the invariants of two complexconjugate representations and ¯ are related to each other as: M ¯ ( P ) = M ( P ) , I [¯ ] = I [ ] ∗ , (47)where the symbol ∗ denotes complex conjugation here and thereafter. This directly followsfrom (11) and (15). Note that the Molien function on the right-hand side (RHS) in theequation above is not complex conjugated for the very reason that it is real by virtue ofMolien’s theorem.Let us denote the set of matrices of a representation by { } , sometimes called the image , as opposed to for just the representation itself. In the case where two inequivalentrepresentations, say and (cid:48) , have the same image, { } = { (cid:48) } , (48)the Molien functions and the invariants are identical. They have the same Molienfunction and also the same invariants as is obvious from Eqs. (11) and (15).A particular, but not infrequent, case is when h ∈ R ( H ) ⇒ h ∗ ∈ R ( H ) (49) The fact that two inequivalent representations have the same representation matrices might be a bit ofa surprise at first thought. A simple example is Z which has three irreps, the identity and two complexconjugate pairs (cid:48) and ¯ (cid:48) which are generated by A = exp(2 πi · / 3) and A ∗ = exp(2 πi · / D (cid:39) Z (cid:111) Z in a block diagonal way, D | Z = diag( (cid:48) Z , ¯ (cid:48) Z ), then the inner automorphismlinking the two irreps is given by the Pauli matrix σ ∈ D , σ diag( (cid:48) Z , ¯ (cid:48) Z ) σ − = diag(¯ (cid:48) Z , (cid:48) Z ). vs. ¯ { } = { (cid:48) } h (cid:48) = AhA − , h ∈ H Molien function identical identical identicalInvariants complex conjugate identical change unless A ∈ H Table 3: Summary of transformation properties of Molien function and the invariants, as dis-cussed in the text, with respect to the relation as given in the first row. Most of these propertiesare easily inferred from the definitions (11) and (15). In what regards the third case it is notedthat any element can be conjugated by a separate matrix A h and the Molien function is still leftinvariant. applies, the complex conjugate representation has the same image, Eq. (48). It goeswithout saying that this is trivial and not useful if the representation is real. Note that,if an invariant is not real, then (47) and the observations above imply that the complexconjugates do not have the same image. The converse is not true. The results above aresummarized in Tab. 3.Crucially, if two irreps and (cid:48) have the same image, then the fact that they have thesame invariants, see Eq. (48), means that there is no way, in our framework, to distinguish (cid:44) → SU (3) from (cid:48) (cid:44) → SU (3) . This apparent ambiguity corresponds to the arbitrarinessof labeling of the irreps and (cid:48) . Associating SU (3) → , for example (cid:48) can be generatedfrom tensor products of the latter, since SU (3) is the fundamental irrep of SU (3) fromwhich all other irreps are generated.In connection with this observation we would like to add two remarks: First, if twoirreps have the same image this ought to imply that the Kronecker products of and (cid:48) are identical under the interchange of irreps of the same order. One can verify this for theexample of and (cid:48) for A = Σ(60) [30, 38]. Second, if we consider a higher dimensionalcase such as SU (6) → S → A + (cid:48) A , see (44), then the two irreps can be distinguished.This can be seen or described as follows: One can choose an embedding of S such thatunder A the two irreps are block diagonal, S | A = (cid:18) A (cid:48) A (cid:19) , (50)and associate the six-dimensional representation space by the variables { x , .., x } . As-suming an invariant I A ( x , x , x ) breaks S → A + 3 · A , then same invariant I A ( x , x , x ) breaks S → (cid:48) A + 3 · A . In order to determine this invariant one oughtto look at all embeddings X + X (cid:44) → S , and then go through the same reasoning as inSecs. 2.5 and 4, respectively. 30 .4 The -dimensional irreps of Σ( X ) , T n [ a ] , ∆(3 n ) , and ∆(6 n ) Equivalent embeddings It is conceivable that a similarity transformation (45) on the list of groups in Tab. 1 wouldlead to an embedding that leaves say (30) invariant and is a supergroup of ∆(6 n ). Itwould thus invalidate the condition in Eq. (30). We shall see below that, due to Schur’sLemma, c.f. App. A, this is not the case.Consider the conditions (30) and (34) for ∆(6 n ) and ∆(3 n ): These polynomialsimply that E and F ( n, , 1) are part of the groups that leave them invariant. Since E and F ( n, , 1) generate a ∆(3 n )-irrep of dimension three, by virtue of Schur’s Lemma,there does not exist a matrix, other than a multiple of the identity, that commutes with E and F ( n, , G ( d, r, s ) only andnot some AG ( d, r, s ) A − in the previous sections. The same argument holds for T n [ a ] (35)with F ( n, , 1) replaced by F ( n, , a ). Similar arguments validate the chains Σ(36 φ ) ⊂ Σ(72 φ ) ⊂ Σ(216 φ ) and Σ(60) ⊂ Σ(360 φ ), since the supergroups differ from the subgroupsby one generator only. Inequivalent embeddings In this section, we are interested in whether inequivalent embeddings of the 3-dimensionalirreps of the Σ( X )-, ∆-, and T n [ a ] -groups give rise to distinct invariants and are thus dis-tinguishable in SU (3) → Σ( X );∆ . An invaluable source for this endeavour is the diplomathesis of Patrick Ludl [30], which we shall use frequently below. The main results aresummarized in Tab. 4.We will not go through all the points but just mention a few facts. For the groupsΣ(36 φ, φ, φ ) several irreps are not in SU (3) since they are obtained from the irrep Σ(36 φ, φ, φ ) (cid:44) → SU (3) by multiplying a certain generator by − i , or − i , which violatesthe determinant condition for SU (3) [30]. The Σ( X ) irreps are faithful with the exceptionof Σ φ (cid:39) A [30], as mentioned in Tab. 4. For the ∆-groups the conditions for groups tobe non-faithful are given in Tab. 4 as well. A faithful irrep is always provided by ( a, b ) =(1 , F ( n, , 0) and is the one used throughout this paper, e.g. inTab. 1. The important point is though that all faithful irreps have got the same image,and thus the same invariants. The same image of irreps is determined by criterion (49)in the cases of Σ(36 φ ), Σ(168), Σ(216 φ ), and Σ(360 φ ).In conclusion we have not missed anything by restricting ourselves to a particularembedding in Tab. 1. For the case of complex conjugate pairs one has to choose thecomplex conjugate invariant in order to distinguish the two cases. However, a and a ¯ are not really different in the same way as anti-matter is not really different from matter.31roup number not faithful not in SU (3) same image remainΣ(60) 2 0 0 1 1Σ(36 φ ) (4,4) 0 (3,3) 1 1Σ(168) (1,1) 0 0 1 1Σ(72 φ ) (4,4) 0 (3,3) 0 (1,1)Σ(216 φ ) (4,4) +1 1 (3,3) 1 1Σ(360 φ ) (2,2) 0 0 (1,1) (1,1)∆(3 n ) , n / ∈ Z n − , a,b gcd( a [ b ] , n ) > n ) , n ∈ Z n − , a,b idem 0 idem 1∆(6 n ) 2( n − , a n/a ∈ , .. n half of them idem 1Table 4: ( n, n ) stands for n pairs of complex conjugate representations. The subtraction ofthe third, fourth, and fifth columns from the second column results in the last column. Theirreps which are not in SU (3) do not satisfy the unit determinant criteria; they are irreps of U (3) rather than SU (3). The only non-faithful irrep is Σ φ (cid:39) A . The same image criteriais discussed around Eq. (48). The acronym “gcd” stands for greatest common divisor, and a [ b ]stands for a and/or b . Since T n [ a ] ⊂ ∆(3 n ) and the latter has only 3-dimensional irreps the T n [ a ] In this work we have been studying the breaking of SU (3) into its proper finite subgroups F , from the viewpoints of explicit breaking and SSB. These two approaches are linkedby the complex spherical harmonics, the representation functions of SU (3), as explainedin Sec. 2.3 for SO (3) and illustrated for SU (3) in Sec. 4.4.In the explicit breaking approach a field φ transforming under the fundamental irrep = (1 , 0) is considered. The crucial question is which term(s) have to be added to an SU (3)-invariant Lagrangian in order to break to F : L SU (3) →F = L SU (3) ( φ , φ , φ ) + L F ( φ , φ , φ ) . (51)In retrospect of Sec. 4 we may say that such terms, with the exception of a few smallgroups like A , T , and Σ(168), lead to potentials which are not renormalizable by powercounting, as their polynomial degrees exceed four. In four space-time dimensions a term in the Lagrangian is powercounting renormalizable if its massdimension is equal to or below four. A scalar field has mass dimension one in four space-time dimensions.For instance to enforce SU (3) → ∆(75), an explicit term δ L = c Λ ( φ + φ + φ ) would serve the purposeaccording to Fig. 4. Restricting oneself to terms up to dimension four with symmetry ∆(75), only δ L = c (cid:48) Λ φ φ φ would remain but would lead, according to Fig. 4, to an accidentally larger symmetry SU (3) → U (1) × U (1) (cid:111)Z [ ⊃ ∆(75)], reminiscent of the baryon number conservation in the renormalizable 32n the approach of SSB, a field ˜ φ in an irrep ( p, q ) of SU (3) is considered. Theassociation of a VEV to this field, singling out a direction, breaks the symmetry:( p, q ) SU (3) → | (p,q) | F = F + ... (52)The full relation, including the omitted terms, is called the branching rule . In the case athand the branching rule necessarily contains the trivial irrep, as indicated. The branchingrules can be computed with our program SUtree by the formalism of the generatingfunctions. This is outlined in App. C and exemplified in Sec. 4.4 for our package SUtree .It is straightforward to find structures of invariant polynomials by virtue of theReynolds operator (15), and thus VEVs which leave the group structure F invariant.They are linked by the complex spherical harmonics, and their degrees and dimensionsare related as follows:( p, q ) = (deg φ i I [ F ] , deg φ ∗ i I [ F ]) ↔ v [ F ] ∈ C | ( p,q ) | , (53)where | ( p, q ) | = ( p + 1)( q + 1)( p + q + 2) is the dimension of the ( p, q )-irrep, and for thesake of clarity, ( p, q ) = (4 , 5) if for example I [ F ] = φ φ φ ∗ ( φ ∗ ) .The non-trivial issue is to find sufficient conditions, since a supergroup always sharescommon invariants with its subgroups. We have provided solutions for all crystallographicgroups Σ( X ) and for the series of trihedral groups T n [ a ] , ∆(3 n ), and ∆(6 n ) in Sec. 4 forrepresentations of the ( p, SU (3) subgroups and showing that the results areindependent of the particular embedding Tab. 1. We wish to emphasize once more thatthe criterion for breaking into faithful irrep can be seen as an alternative definition of thegroup. This is close, but not identical, to the original classification of SU(3) subgroups[18].The reason we are restricted to the ( p, φ ∗ , transforming as ¯ = (0 , p, 0) isnot very restrictive, as (0 , q )-fields and other ( p (cid:48) , Nevertheless the ( p, q )-case is more generic and doable with thisformalism through the tensor generating function. We leave such a possibility to futurework.Further to that we have computed all primary and secondary invariants, and thus the SM. For explicit breaking, an inclusion of φ ∗ might be necessary depending on the charges of the field.For particle physics model-building the SSB approach is more important, as it is the model-builders’ goalto explain symmetry patterns dynamically rather than to work in a framework where the symmetry isbroken explicitly. A and S under a deformation of the Euclidianmetric. We have computed the same data for the remaining groups in the databaseas given in Tab. 5. This information is stored in the software package and database SUtree . Further to that Molien functions, tensor generating functions, branching rules,translations from invariants to VEVs and back, character tables, Kronecker products, andfurther things can be found in the example notebook.Let us end by emphasizing an interesting nuance: Whereas there is a one-to-one linkbetween the degree of explicit terms and the dimension of the irrep in the SSB sce-nario (53) for SU (3) → F , we are not aware of a relation to the form of the potential U ( φ i ) enforcing SSB, in particular to the degrees of terms needed. As the explicit termstend to be non-renormalizable, as discussed above, it is thus an interesting question ofwhether they could be renormalizable in the SSB approach. Low dimensional irreps whichlead to power counting renormalizable potentials have been analyzed in [32, 39, 40, 42].Possibly one or the other counterexample already exists in the literature. Note added: Shortly after this paper was finished, the preprint [41] on discrete groupsappeared, which is more directed to the practical aspects used in model building. Thatpaper is accompanied by the software package Discrete and it is a very useful addendumto our work. Acknowledgments We are grateful to Maximilian Albrecht, Claudia Hagedorn, Gareth Jones, Ron King,Patrick Ludl, Christoph Luhn, and Tim Morris for useful discussions and/or commentson the manuscript. We are indebted to Thomas Fischbacher for collaboration in the earlystages of the project and for discussions on embedding theory. RZ is grateful to ThomasMannel for his sincere interest in the subject. The work of AM is supported by the G¨oranGustafsson foundation. RZ gratefully acknowledges the support of an advanced STFCfellowship. A Mini group theory compendium In this appendix, for the reader’s convenience, we state a few definitions, facts, andtheorems (frequently) used throughout our work.34 Branching rule: Let g be an irrep of G and h i be irreps of H where H ⊂ G . Thenthe restriction of G to H leads to g | H → (cid:88) i a h i h i . (A.1)The positive number a h i counts how many times the irrep h i is contained in g . • Dimensionality theorem: The order of a group is equal to the sum of squares ofthe dimensions of all its irreps, | H | = irreps (cid:88) i |R i ( H ) | . (A.2) • Center of a group : The center C of a group G is the set of elements that commutewith all group elements, C := { g (cid:48) ∈ G : ∀ g ∈ G : gg (cid:48) = g (cid:48) g } . • A version of Schur’s lemma : If R ( G ) is a d -dimensional irrep of G and A R ( G ) = R ( G ) A for some matrix A , then A can only be a multiple of the d -dimensionalidentity matrix. • Lagrange’s theorem: Let H be a subgroup of the finite group G . Then | G | / | H | is an integer. • Semidirect product : The semidirect product G (cid:111) H ≡ G (cid:111) φ H between two groups G and H is defined as the operation mapping ( g , h ) and ( g , h ), with g , ∈ G and h , ∈ H , onto ( g φ h ( g ) , h h ), where φ h is a homomorphic mapping H → G . • Theorem II.2 [17]: Let G be a finite group with m -dimensional faithful irrepand c the order of the center, then G × Z n has an m -dimensional faithful irrep ⇔ gcd( n, c ) = 1. (The acronym “gcd” stands for the greatest common divisor.) • Multiplicity : Writing the Kronecker product of two irreps as R ( G ) × R ( G ) = n R ( G ) + ... , the positive number n is the multiplicity. It is computed via the scalar product n = (cid:104)R R , R (cid:105) , where (cid:104)R i , R j (cid:105) ≡ | G | − (cid:80) g ∈ G χ i [ g ] χ j [ g ] ∗ with character χ i [ g ] =tr[ R i ( g )]. 35 otation: • (cid:39) isomorphic • (cid:44) → group embeddingTo this end let us mention that representations of a group H are generically denotedby R ( H ), but when a very specific group is considered often the dimension of the rep-resentation is boldfaced, as in , which is not unambiguous and often results in writinga second 3-dimensional irrep by (cid:48) for instance. In the cases of SO (3) and SU (3) it iscommon to refer to an irrep by ( l ) and ( p, q ), respectively. The latter are unambiguousand partly discussed in App. G.1. We switch between these notations throughout thiswork always adopting to the most convenient one. B The group database In this appendix additional useful information can be found on the group database whichis listed in Tab. 5 and described in the main Sec. 3. B.1 Generators The generators needed for the groups in Tab. 1 are given by [17]: E = , F ( n, a, b ) = η a η b 00 0 η − a − b , G ( d, r, s ) = δ r δ s − δ − r − s ,H = 12 − µ − µ + µ − µ + − µ + − µ − , J = ω 00 0 ω , K = 1 √ i ω ω ω ω ,L = 1 √ i ω ω ωω ω , M = β β 00 0 β , N = i √ β − β β − β β − β β − β β − β β − β β − β β − β β − β ,P = (cid:15) (cid:15) 00 0 (cid:15)ω , Q = − − ω − ω . (B.1)36o. 〚 g, j 〛 c Names01 〚 , 〛 · ), A , T 〚 , 〛 C (7 , , T 〚 , 〛 · ), S , O 〚 , 〛 · )05 〚 , 〛 C (13 , , T 〚 , 〛 · )07 〚 , 〛 · )08 〚 , 〛 C (19 , , T 〚 , 〛 A , Σ(60), I , Y 〚 , 〛 · )11 〚 , 〛 C (9 , , 〚 , 〛 C (14 , , 〚 , 〛 C (31 , , T 〚 , 〛 · )15 〚 , 〛 φ )16 〚 , 〛 · )17 〚 , 〛 C (37 , , T 〚 , 〛 C (43 , , T 〚 , 〛 C (49 , , 〚 , 〛 · )21 〚 , 〛 · )22 〚 , 〛 C (26 , , 〚 , 〛 D (9 , , 1; 2 , , 〚 , 〛 P SL (2 , 〚 , 〛 C (61 , , T 〚 , 〛 C (21 , , 〚 , 〛 · )28 〚 , 〛 C (67 , , T 〚 , 〛 φ )30 〚 , 〛 · ) 31 〚 , 〛 C (73 , , T 〚 , 〛 C (38 , , 〚 , 〛 C (79 , , T 〚 , 〛 · )35 〚 , 〛 C (91 , , T 〚 , 〛 C (91 , , T 〚 , 〛 C (97 , , T 〚 , 〛 · )39 〚 , 〛 · )40 〚 , 〛 C (103 , , T 〚 , 〛 C (18 , , 〚 , 〛 C (109 , , T 〚 , 〛 C (28 , , 〚 , 〛 C (39 , , 〚 , 〛 · )46 〚 , 〛 C (62 , , 〚 , 〛 C (127 , , T 〚 , 〛 · )49 〚 , 〛 C (133 , , T 〚 , 〛 C (133 , , T 〚 , 〛 C (139 , , T 〚 , 〛 · )53 〚 , 〛 C (74 , , 〚 , 〛 C (151 , , T 〚 , 〛 C (157 , , T 〚 , 〛 · )57 〚 , 〛 C (163 , , T 〚 , 〛 C (169 , , T 〚 , 〛 · )60 〚 , 〛 φ )61 〚 , 〛 φ )Table 5: The groups contained in our database, together with their group numbers and GAPnumbers [25, 26], while c = ord( C ) is the order of the center of the respective group, which canonly be 1 or 3 by theorem II.2 stated in App. A. Note that, in some cases, it might not work outto describe the T n [ a ] groups by the number n only, as different choices for the second parameter a might be possible, due to the definition of these groups as C ( n, , a ) with a + a + 1 = 0 mod n . g g g g g g g g E √ − √ √ F a − b ) n √ a + b ) n G d r (cid:16) π ( r +2 s ) d (cid:17) cos (cid:16) π ( r +2 s ) d (cid:17) √ n rH √ √ − √ − √ √ K − √ − √ − √ √ − L − √ − √ 34 12 √ − − √ − N − √ − √ − √ √ − Table 6: Gell-Mann vector components for all generators, with F = F ( n, a, b ) and G = G ( d, r, s ). Any generator G in Eq. (B.1) can be displayed as G = exp( iπ(cid:126)g [ G ] · (cid:126)T ) =exp( iπ (cid:80) a =1 g [ G ] a T a ). T a for a = 1 , .., SUtree , satisfying the commutation relations as givenin App. G.1 in Eq. (G.10). The generators J , M , P , and Q can be obtained through Eq. (B.2). .Here, we have used the abbreviations η ≡ e πi/n , δ ≡ e πi/d , µ ± ≡ (cid:16) − ± √ (cid:17) , ω ≡ e πi/ , β ≡ e πi/ , (cid:15) ≡ e πi/ . Further to that note that the generators J, M, P, Q can be expressed as follows: J = F (3 , , , M = F (7 , , , P = F (9 , , , Q = G (6 , , . (B.2)The orders of the generators, X o = , are:Generator X E F ( n, a, b ) G ( d, r, s ) H J K L M N P Qo n gcd( n,a,b ) d gcd( d,r,s ) Z o ∈ F . The acronym “gcd” stands for greatest common divisor. B.2 The subgroup tree within the group database The subgroup structure within our choice of groups can be found in Tab. 7 and in Fig. 5.It has been obtained with the help of GAP [25, 26] and the generator basis given in Tab. 1.More precisely, we have first searched for the subgroup structure within our basis and thentested the remaining possibilities, allowed by Lagrange’s theorem, with GAP. Note thatthe highest order groups that can be deduced from this table are not necessarily maximal38 ( , , ; , , ) T [ ] S A Δ ( × ) T [ ] Δ ( × ) Δ ( × ) T [ ] Δ ( × ) C ( , , ) C ( , , ) T [ ] Δ ( × ) Σ ( ϕ ) Σ ( ) Δ ( × ) C ( , , ) Δ ( × ) Δ ( × ) C ( , , ) C ( , , ) Δ ( × ) Σ ( ϕ ) Δ ( × ) T [ ] C ( , , ) Δ ( × ) T [ ] T [ ] Δ ( × ) Δ ( × ) C ( , , ) C ( , , ) C ( , , ) C ( , , ) Δ ( × ) T [ ] T [ ] Δ ( × ) C ( , , ) Δ ( × ) T [ ] Δ ( × ) Σ ( ϕ ) Σ ( ϕ ) Σ ( ) Figure 5: Subgroup tree within our database in Tab. 5. Black (bold faced) groups denotemaximal subgroups of SU (3). Gray (Italic) groups denote largest groups within our tree (withina branch). Note that ten subgroup relations do not follow directly from the generators as inTab. 1. The similarity transformations relating the generators are described in App. B.2. A = T = ∆(3 · ) S = O = ∆(6 · ), ∆(3 · ), A = Σ(60) = I = Y ,∆(3 · ), C (26 , , C (38 , , · ), C (62 , , C (74 , , T = C (7 , , C (14 , , C (49 , , · ), P SL (2 , 7) = Σ(168), C (21 , , T = C (91 , , T = C (91 , , T = C (133 , , T = C (133 , , S = O = ∆(6 · ) ∆(6 · ), P SL (2 , 7) = Σ(168), ∆(6 · ), Σ(360 φ )04 ∆(3 · ) ∆(6 · ), C (9 , , · ), C (21 , , C (39 , , T = C (13 , , C (26 , , T = C (91 , , T = C (91 , , C (39 , , T = C (169 , , · )06 ∆(3 · ) ∆(6 · ), ∆(3 · ), C (28 , , · )07 ∆(6 · ) Σ(36 φ ), D (9 , , 1; 2 , , · )08 T = C (19 , , C (38 , , T = C (133 , , T = C (133 , , A = Σ(60) = I = Y Σ(360 φ )10 ∆(3 · ) ∆(6 · ), ∆(3 · )11 C (9 , , D (9 , , 1; 2 , , · ), C (18 , , C (14 , , C (28 , , T = C (31 , , C (62 , , · ) ∆(6 · )15 Σ(36 φ ) Σ(72 φ ), Σ(360 φ )16 ∆(3 · ) ∆(6 · ), C (18 , , · )17 T = C (37 , , C (74 , , · ) ∆(6 · )23 D (9 , , 1; 2 , , 1) ∆(6 · ), Σ(216 φ )27 ∆(3 · ) ∆(6 · )29 Σ(72 φ ) Σ(216 φ )34 ∆(3 · ) ∆(6 · )Table 7: The subgroup structure among the 61 groups under consideration. Note that, forgroups with numbers greater than 34, the number of elements is larger than 512 ÷ C -and D -type groups. largest groups . As arguedin Sec. 4.2, Σ(216 φ ), Σ(360 φ ), and Σ(168) are the only maximal subgroups of SU (3) inthat list.Not all subgroup relations in Fig. 5 and Tab. 7 follow from the specific embeddingin Tab. 1. In fact there are ten cases, which we shall discuss to various degrees of de-tail according to importance and feasibility. For all cases there ought to be similaritytransformations, g (cid:48) i = Ag i A − , A − = A † , (B.3)where g i are the group generators and A can be written as a unitary matrix as discussedin Sec. 5.1. There are a total of ten cases, which we make explicit below, in the groupdatabase which necessitate the transformation (B.3) in order to make the subgroup rela-tion apparent.4) Four cases include T n [ a ] -relations: T ⊂ T ⊃ T , T ⊃ T ⊂ T . (B.4)For the subgroups, the similarity transformation is given by: F ( n, a, 1) = AF ( n, , a ) A − , E = AEA − , A = G (2 , , . (B.5)Essentially we are saying here that T n [ a ] can be generated either by { E, F ( n, , a ) } or by { E, F (1 , n, a ) } .1) Σ(36 φ ) as a subgroup of Σ(360 φ ) is crucial for the criteria given in Sec. 4.2. Theembedding of Σ(360 φ ) is chosen such that the subgroup relation with Σ(60) is mosttransparent. According to Eq. (B.1), Σ(36 φ ) is generated by { E, J, K } . First wenote that E = K J K . Furthermore we have verified that Σ(36 φ ) is also generatedby { F , F } , where F = EJ E and F = K , or equivalently, through the followingalgebraic relations:Σ(36 φ ) = (cid:28) F , F | F = F = ( F F ) = F F F F F F F F = 1 (cid:29) , (B.6)where (cid:28) · · · (cid:29) denotes the generating relations. Relations such as the one aboveare called presentations . We are not aware of such a result in the literature. Thisresult was achieved by working out a number of relations with explicit generators, The former is equivalent to the central extension of the A [24], and the latter is isomorphic to A . φ ) in the SmallGroups library [25, 26]. Moreover, coming back to the mainpoint, we have verified that, among the Σ(360 φ )-generators, F (cid:48) = F E and F (cid:48) = E F QHQF H with F = F (2 , , 1) are related to F and F as quoted above by: F (cid:48) = AF A − , F (cid:48) = AF A − , where A = a ( √ − i ) √ i √ (cid:0) √ − i √ (cid:0) √ − (cid:1)(cid:1) (cid:0) − i √ (cid:1) − − (cid:0) i √ (cid:1) (cid:0) − i √ (cid:1) . (B.7)The parameter a is not constrained, unless that one should have a (cid:54) = 0 in order for A to remain invertible, but the choice a = √ e iφ with φ = − arccot (cid:0) √ (cid:1) leads to A ∈ SU (3).5) For the five remaining cases are A , S ⊂ Σ(168) and S , ∆(3 · ) , ∆(6 · ) ⊂ Σ(360 φ ). The generators E remain the same in both representations. This fixesthe basis only partly. In fact A from Eq. (B.3) is a matrix that commutes with E .By going to a diagonal basis this matrix is readily found to be the two-parametermatrix: A → A ( a, b ) = f ( a, b ) f ( a, b ) f ( a, b ) f ( a, b ) f ( a, b ) f ( a, b ) f ( a, b ) f ( a, b ) f ( a, b ) , [ E, A ( a, b )] = 0 , (B.8)with f ( a, b ) = 13 (cid:0) e iπ ( a + b ) + e − iπa + e − iπb (cid:1) ,f ( a, b ) = 13 e iπ ( a + b ) (cid:0) e − iπ (2 a + b ) + ρ e − iπ ( a +2 b ) − ρ (cid:1) ,f ( a, b ) = 13 e iπ ( a + b ) (cid:0) e − iπ (2 a + b ) − ρ e − iπ ( a +2 b ) + ρ (cid:1) , (B.9)where here and further below we use the notation: ρ x ≡ exp(2 πi/x ). We shalldenote the generators of the subgroup by F i and the ones of the supergroup by F (cid:48) i and the common generator E is chosen to be F = F (cid:48) = E . In this notation (B.3)reads: F (cid:48) i = A ( a, b ) F i A ( a, b ) − , A ( a, b ) − = A ( a, b ) † = A ( − a, − b ) (B.10)42e would like to add that no attempt is made to choose the optimal representationi.e. find the presentation where the transformation matrices are simplest. – A as subgroup of Σ(168): A presentation of A is given by (cid:28) F , F | F = F = ( F F ) = F F F F F F F F = (cid:29) , with generators as given in Tab. 1. For A and Σ(168), { F = F (2 , , } A , { F (cid:48) = M N M } Σ(168) . We find f ( a , b ) = α − / , f ( a , b ) = ρ β / , f ( a , b ) = β / , (B.11)where α is the first root of 175616 − x + 86240 x − x + 756 x − x + x and β i is the i -th root of 1 − x + 756 x − x + 86240 x − x + 175616 x . The ordering of the root is proportional to the real partof the root. Numerically, it turns out that ( a , b ) = (2 . , . – S as subgroup of Σ(168): A presentation of S is given by (cid:28) F , F , F | F = F = ( F F ) = ( F F ) = ( F F ) = F F F F F F F F = F F F F F F F = F F F F F F F F = (cid:29) , with generators as given in Tab. 1. For S and Σ(168), { F = F = F (2 , , , F = G = G (2 , , } S , { F = EF E , F = G } Σ(168) . We find f ( a , b ) = γ , f ( a , b ) = 2 − ρ − ρ + ρ + ρ ρ γ , (B.12) f ( a , b ) = 1 + 3 ρ − ρ − ρ + 5 ρ − ρ + ρ − ρ + 2 ρ + 2 ρ ) γ ,γ = 7 − / (1 − ρ + 2 ρ + 2 ρ + 2 ρ )(339 − ρ + 351 ρ − ρ + 333 ρ − ρ + 349 ρ ) / . Numerically, it turns out that ( a , b ) = (1 . , . S as subgroup of Σ(360 φ ): The presentation of S is given as above and withgenerators as before we find, { F (cid:48) = F QEF HEF HEF Q , F (cid:48) = EF EF QE F EHQHQ } Σ(360 φ ) . It then turns out that f ( a , b ) = 2 δ , f ( a , b ) = ρ δ (cid:16) √ (cid:17) , f ( a , b ) = 12 , (B.13)where δ = (cid:112) − √ 5. Numerically, we obtain ( a , b ) = (1 . , . – ∆(3 · ) as subgroup of Σ(360 φ ): A presentation of ∆(3 · ) is given by (cid:28) F , F | F = F = ( F F ) = F F F F F F F F = (cid:29) , with generators as given in Tab. 1. For ∆(3 · ) and Σ(360 φ ), { F = F (3 , , } ∆(3 · ) , { F (cid:48) = QH |} Σ(360 φ ) . We find f ( a , b ) = 4 (cid:112) − i √ , f ( a , b ) = (cid:118)(cid:117)(cid:117)(cid:116) i (cid:113) , f ( a , b ) = ρ f ( a , b ) . (B.14)Numerically, it turns out that ( a , b ) = (2 . , . – ∆(6 · ) as subgroup of Σ(360 φ ): A presentation of ∆(6 · ) is given by (cid:28) F , F | F = F = ( F F ) = F F F F F F F F = (cid:29) , with generators as given in Tab. 1. For ∆(6 · ) and Σ(360 φ ) and { F = EF E , F = G ) } ∆(6 · ) , { F (cid:48) = ( EH ) E ( HQ ) , F = H ) } Σ(360 φ ) . 44e find: f ( a , b ) = 1 (cid:113) (cid:0) i √ √ i √ (cid:1) ,f ( a , b ) = ρ · / (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) − √ − i (cid:114) (cid:16) − √ (cid:17)(cid:33) ,f ( a , b ) = ρ (cid:118)(cid:117)(cid:117)(cid:116) √ − i (cid:115) − √ . (B.15)Numerically, it turns out that ( a , b ) = (2 . , . C Tensor generating function The aim of this appendix is to present the generating function for counting covarianttensors in our language. From the latter the branching rules can be obtained as shown inSec. C.2. For a summary on the generating function related to other problems in grouptheory the reader is referred to [11].The Molien function (11), by virtue of Molien’s theorem, counts the number of invari-ants of a group in a certain representation R f ( H ) of a finite group H . It is a natural toask whether this can be generalized to count the number of covariants . By covariants wemean tensors under a certain representation R c ( H ).The answer is given by the (tensor-) generating function [6]: M H ( c , f ; P ) = 1 |R f ( h )) | (cid:88) h ∈ H χ c [ h ] ∗ det( − P R f ( h )) = (cid:88) n ≥ c n P n , (C.1)where P is a real number and χ c [ h ], given by χ c [ h ] = tr[ R c ( h )] , (C.2)is the character of h in the representation c . It should be emphasized that f and c areirreps. The generating function ought to reduce to the Molien function (11) in the case In this language the Molien function is the invariant-generating function . c is the trivial irrep, M H ( P ) = M ( , R f ( h ); P ) , (C.3)and does so since χ [ h ] = 1. The generalization of the Molien theorem states that thepositive coefficients c n count the number of linearly independent R c ( H )-tensors whosecomponents transform under R f ( H ). The generating function can be written in thefollowing way, M H ( c , f ; P ) = (cid:80) i a cn i · P n i (1 − P m )(1 − P m )(1 − P m ) , (C.4)in analogy to the form of the Molien function (13). We shall quote here a few facts, assum-ing that the reader has digested some of the material on the Molien function presentedin the main text: • There are a cn i linearly independent R c -tensors of degree n i , denoted by E ( n i ) ( f, c ).The entire set { E ( n i ) } , for all irreps c and f , is known as the integrity basis . • The denominator is the same as for the Molien function (13), and thus correspondsto the degrees of the primary invariants rather than covariants. To appreciate thelatter statement, in connection with the generalization of the Molien theorem, onehas to note that a tensor times an invariant is a tensor of the same degree, or thata tensor times a tensor corresponds to a tensor of a higher degree. • Knowing the degrees of the tensors, one can compute the various tensors by ataking a suitable polynomial ansatz for the c -tensor and then demand that itscomponent elements transform as f -tensors under the generators [11]. An exampleof an integrity basis element is given in the next section for the sake of clarity. • The generalization of (14) is [46], (cid:88) i a cn i = | c | · m · m · m | H | , (C.5)where | c | is the order of the irrep c . We note that the numerator has no 1 since, forrepresentations other than the trivial one, the identity is not an R c -tensor. • The coefficients a cn i satisfy the following symmetry property [46]: a cn i = a ¯ cn (cid:48) i , for n i + n (cid:48) i = m + m + m − | f | . (C.6)46 The composition laws are as follows [44]: M ( c + c , f ; P ) = M ( c , f ; P ) · M ( c , f ; P ) ,M ( c , f + f ; P ) = (cid:88) ii (cid:48) n cii (cid:48) · M ( i , f ; P ) · M ( i (cid:48) , f ; P ) , (C.7)where the sum runs over all irreps c and c (cid:48) of the finite group, and n cii (cid:48) is the numberof times the irrep c appears in the Kronecker product i × i (cid:48) , which is easily computedfrom the character table, c.f. App. A.Being aware that all of this is rather heavy to digest for the reader we pass on to ourguinea pig S of Sec. 2, where some of the properties mentioned above can be verifiedexplicitly. C.1 S as an example The group S = ∆(6 · ) has irreps denoted by { , (cid:48) , , , (cid:48) } . The tensor generatingfunctions are easily computed, using formula Eq. (C.1): M S ( , ; P ) = 11 − P ,M S ( , (cid:48) ; P ) = 11 − P ,M S ( , ; P ) = 1 + P (1 − P )(1 − P )(1 − P ) = M S ( P ) | Eq . (17) ,M S ( , ; P ) = P + P + P + P + P + P (1 − P )(1 − P )(1 − P ) . (C.8)The first two generating functions are concerned with invariants of a one-dimensionalrepresentation space, I ( , )[ S ] = x , I ( , (cid:48) )[ S ] = x , (C.9)which we have taken to be x ∈ R . Note that (cid:48) acts as x → − x . The third generatingfunction is the Molien function (17) for S , as discussed in the main text, and thus we donot need to repeat the discussion here. The fourth one is new and we in particular see thatthe degrees of the primary invariants remain the same, as previously stated. Furthermore,47he property from Eq. (C.5) is verified. Somewhat arbitrarily we quote, from Ref. [44],out of the six the three -tensors of lowest degree: E (1) ( , ) = xyz , E (3) ( , ) = x y z , E (4) ( , ) = ( y − z ) yz ( z − x ) zx ( x − y ) xy , (C.10)for the sake of clarity through an example. As stated previously, it would be no problemto compute them with a suitable ansatz and the Reynolds operator. C.2 Branching rules for SO (3) → F and SU (3) → F The branching rules , also known as correlation tables , can be computed using the charactergenerator [11], but here we shall use the method of tensor generating functions presentedin [46]. The problem is the following: We would like to know how many times the irrep x is contained in the representation ( l ) or ( p, q ), respectively, when restrained to thesubgroups H SO (3) ⊂ SO (3) and H SU (3) ⊂ SU (3), respectively, Branching rules: ( l ) SO (3) → ( r x l x + ... ) H SO (3) , ( p, q ) SU (3) → ( r x p,q x + ... ) SU (3) . (C.11)This follows from the tensor generating functions, see e.g. [44, 46], B ( x ; l ) = (1 − L ) M ( x , , L ) = (cid:88) l r x l L l ,B ( x ; P, Q ) = (1 − P Q ) (cid:88) c , c (cid:48) n mcc (cid:48) M ( c , , P ) M ( c (cid:48) , ¯ , Q ) = (cid:88) p,q r x p,q P p Q q , (C.12)where the prefactors (1 − L ) and (1 − P Q ) correspond to the O (3) and U (3) conditionsthat x + y + z = constant and xx ∗ + yy ∗ + zz ∗ = constant, respectively. In the secondequation in (C.12), use of the second composition law in (C.7) has been made. It is worthto note that M ( c , ¯ , Q ) = M (¯ c , , Q ), since the generating function is real. Moreover,since the sum extends over all irreps, one may effectively replace M ( c (cid:48) , ¯ , Q ) → M ( c (cid:48) , , Q )in the sum in Eq. (C.12). The positive coefficients r x l and r x p,q give the numbers of linearlyindependent x -tensors whose components transform under l SO (3) and ( p, q ) SU (3) irreps,respectively. Thus they correspond to the multiplicity of the branching in Eq. (C.11). The function B has been computed in the literature [46] for Σ(168), Σ(216 φ ), and Σ(360 φ ). More precisely, here, linear independence is understood over the ring of denominator scalars [46]. for the groups in our database can be obtained from our package SUtree . Below we shall illustrate the formalism, once more, through SO (3) → S . To thisend we would like to add that the functions B ( x ; l ) and B ( x ; P, Q ) can be brought into aform where there are two and five factors in the denominator [46], which corresponds thethe two and five parameters that characterize the corresponding representation vector,c.f. App. G.1, Tab. 8. C.2.1 Examples of branching rules for SO (3) → S The branching rules of SO (3) → S can be obtained by first identifying the SO (3) → S .The additional necessary generating functions to (C.8) are M S ( (cid:48) , ; P ) = P + P (1 − P )(1 − P )(1 − P ) ,M S ( , ; P ) = P + P + P + P (1 − P )(1 − P )(1 − P ) ,M S ( (cid:48) , ; P ) = P + P + P + P + P + P (1 − P )(1 − P )(1 − P ) . (C.13)Let us consider l = 2 ↔ SO (3) . The only quadratic powers in the Taylor expansions of(1 − P ) M S ( c , ; P ) are B ( , l ) = l + ... and B ( (cid:48) , l ) = l + ... , and thus 5 SO (3) → ( + (cid:48) ) S .Let us quote a few more branching rules so that the reader can assure him- or herself: l = 1 : SO (3) → S ,l = 2 : SO (3) → ( + (cid:48) ) S ,l = 3 : SO (3) → ( (cid:48) + + (cid:48) ) S ,l = 4 : SO (3) → ( + + + (cid:48) ) S . (C.14)For the branching rules for SU (3) → S we refer the reader to our package SUtree . Except for SU (3) → T and SU (3) → T . From the Molien function to invariants in practice Ideally we would like infer from the Molien function as given in (11) to the degrees ofprimary and secondary invariants. Unfortunately this works only the other way around,as depicted in Eq. (13). In the case where the degrees are not too degenerate, one can getthe invariants, check their algebraic independence with the Jacobian criterion, and thendetermine the syzygies (12), to be certain that one has obtained the right primary andsecondary invariants. We shall discuss this in more detail below and first point towardsan ambiguity of the Molien function. D.1 A manageable ambiguity of the Molien function We simply note that a Molien function of the form (13), can be multiplied by (1 + P m ) / (1 + P m ), which leads to M H ( ) ( P ) = 1 + (cid:80) i a n i P n i (1 − P m )(1 − P m )(1 − P m ) = (1 + (cid:80) i a n i P n i )(1 + P m )(1 − P m )(1 − P m )(1 − P m ) , (D.1)from where we one could be tempted to infer that the number of secondary invariantschanges by a factor of 2, and the product of degrees of primary invariants by a factorof m . Supposing the first form was correct, then the second one would only satisfy theproposition (14) in the case where m = 2. So one has to pay special attention only tothis case and for our list of subgroups the only invariant of degree two is the Euclidiandistance (2). Indeed, a rather manageable ambiguity. D.2 Degeneracies Let us first note the rules for adding primary and secondary invariants, denoted by I and I respectively,1. I + I is primary,2. I + I is not secondary (not primary either),3. I + I is secondary.It is silently assumed that the degrees match. These rules follow from the definitionsof the primary and secondary invariants, c.f. in particular (12). We further discuss twoexamples below to make these issues more transparent, of which D.2.1 is of the first typeand D.2.2 concerns types one and three mentioned in the list.50 .2.1 Degeneracies of invariants of lower degrees. From the Molien function (17) we know that there is an invariant polynomial of degreefour, for example I [ S ] given in (18). The trial function f ( x, y, z ) ∈ { x , y , z } , usingthe Reynolds operator (15), will lead to this invariant. A generic trial function leads toan invariant I (cid:48) = a I + b I . The choice of ( a, b ) ∈ C corresponds to the choice of a basisand is arbitrary. We have made the particular choice ( a, b ) = (0 , D.2.2 Degeneracies of invariants of the same degree A prime example is the case of ∆(6 × ). Note that in practice this example is doable asthe degeneracies for ∆(6 n ) | n (cid:54) =3 are lifted, and we may guess the primary and secondaryinvariants on ground of “analytic continuation” in n , as discussed in Sec. 3.2.We shall discuss it without this trick for the sake of the example. A Molien functionof the following form can be found: M ∆(6 × ) ( P ) = 1 + P + P + P (1 − P ) . (D.2)Accordingly, we would expect 4 invariants of degree 6. Possible choices are: I a = ( xyz ) , I b = x y + y z + z x , I c = x + y + z , I d = xyz ( x + y + z ) . (D.3)Using, e.g., the Jacobian criterion (25), the algebraic independence of any three of themis readily verified. In order to find primary and secondary invariants, the syzygies (12)have to be found. For this question we can disregard the invariants of higher degree forthe moment, since their degrees are too high to play a role, 9 + 6 = 15 > · {I a , I b , I c } are primary invariants. It may be instructive to seewhy or how {I a , I b , I d } fail to be primary invariants: The left-hand side of the syzygy I c has got a term of the form x , but this term can never be obtained by multiplyingany two of the invariants {I a , I b , I d } . We hope that this example is useful to the readerand the practitioner. E Multiple representations Our setup could be generalized to include multiple spin-0 fields ϕ i and φ j , possibly carryingdifferent representations R ( SU (3)). In models with flavour symmetries such fields are51eferred to as flavons [43]. In our view two new features arise as opposed to a singlespin-0 field. Consider the interaction of the SM with the flavon sector, L = F a O a SM , (E.1)where summation over repeated indices is understood, and O a SM consists of SM fieldsonly. The index a is an index of a representation of the flavour symmetry group SU (3).In the case where we intended to be more complete we should also sum over all irrepsof the flavour group in the equation above. In Eq. (E.1), the complexity remains in thecomposite fields F b , which can be written as follows: F a = (cid:88) n,m c mn Λ n + m − T i ..i n j ..j m a ϕ i ..ϕ i n φ j ..φ j m , (E.2)where Λ is some generic suppression scale and c nm are coefficients of order one. Thenew elements are first that more a -covariant objects in F b can be formed, since twoantisymmetric indices do not vanish under contraction of ϕ and φ and second that therelative direction of the VEV of the two fields does matter. In connection with the latter,suppose the two fields were in irreps which are complex conjugate to each other. Then L = m ϕ a φ a = m ϕ · φ is not invariant to separate rotations of the fields ϕ and φ . Onespeaks of vacuum alignment . Thus, by combining the two fields in one potential, onecan enforce rich patterns of flavour symmetry breaking, which have the potential to shinelight on the hierarchies in the flavour sector. We would like to add to this end that thegenerating function as discussed in App. C constitutes a powerful tool in tackling thisproblem in the most general way. F Conjectures concerning the T n [ a ] -groups Contrary to ∆(3 n ) and ∆(6 n ), we have not been able to find the first and second primaryinvariants of the T n [ a ] groups in full generality. As for the latter we can start to guess theprimary invariants on grounds of the examples in our database. Our guesses are: I = xyz , I a +1 = x a +1 y a + y a +1 z a + z a +1 x a , I n = x n + y n + z n . (F.1)52hey are invariant under the T n [ a ] generators E and F ( n, , a ) , and they are also alge-braically independent.If we assume that (F.1) are the correct primary invariants then, by virtue of proposi-tion (14), the number of secondary invariants is given by:3 · n · (2 a + 1)3 n = 2 a + 1 . (F.2)Furthermore, from the examples in our database we are led to conjecture the followingpatterns:1. If the secondary invariants are put in ascending order of their respective degrees,then the degree of the (2 a )-th invariant isdeg (cid:16) I (2 a ) (cid:17) = n + 2 a + 1 = deg( I n ) + deg( I a +1 ) . (F.3)Note that we denote the k -th secondary invariant by I ( k ) , starting with I (0) = 1.2. The degree of the a -th invariant isdeg (cid:16) I ( a ) (cid:17) = 12 ( n + 2 a + 1) = 12 [deg( I n ) + deg( I a +1 )] . (F.4)3. If m is the degree of the first invariant, then the degree of the (2 a − n + 2 a + 1 − m ).4. For k = 1 , , ..., a − (cid:16) I ( a ) (cid:17) − deg (cid:16) I ( k ) (cid:17) = deg (cid:16) I ( a + k ) (cid:17) − deg (cid:16) I ( a ) (cid:17) . (F.5)We hope that these observations may help to solve out this problem in future studies. G SU (3) G.1 The complex spherical harmonics A widely used method to construct irreps is the method of highest weights, as advocatedin many textbooks [50]. For our purposes it is more convenient to work in an explicit Recall that n are the primes out of 3 k + 1 where k is an integer. SO (3), explicit representations in terms of spherical harmonics Y l,m are wellknown as the representations of the Lie algebra elements directly relate to coordinate andmomentum representations in quantum mechanics. What are the spherical harmonicsof SU (3)? It appears that this question was explicitly studied in the late sixties, inconnection with the eightfold way [49], and the corresponding representation functionsare known as the complex spherical harmonics . A thorough mathematical treatment ofso-called solid SU ( n ) harmonics can be found in the book of Louck [47]. In this appendixwe shall present the material in a rudimentary way, relying on the analogy to SO (3) andthe spherical harmonics.The spherical harmonics Y l,m are the solutions of the Laplace equation on the two-sphere S . This can be seen to originate from the quotient of SO (3) with the stabilizerof a representive vector, which is SO (2). In analogy one gets, S (cid:39) SO (3) /SO (2) , S (cid:39) SU (3) /SU (2) , (G.1)the group manifold for the complex spherical harmonics. The five-sphere can be embeddedinto C . Thus there will be five parameters as opposed to two, m and l , for Y l,m .The same conclusions can be reached in a way which parallels the introduction ofthe spherical harmonics in quantum mechanics and possibly justifies the name complexspherical harmonics best [48]. Consider complex coordinates ( z , z , z ) ∈ C , and theLaplace equation: (cid:18) ∂ ∂z ∂ ¯ z + ∂ ∂z ∂ ¯ z + ∂ ∂z ∂ ¯ z (cid:19) f = 0 . (G.2)Let f ( p,q ) be a polynomial solution of degree p and q in z i and ¯ z i , respectively, then f ( p,q ) ( z, ¯ z ) = ρ ( p + q ) h ( p,q ) ( z, ¯ z ) , ρ ≡ ¯ z z + ¯ z z + ¯ z z , where h ( p,q ) ( z, ¯ z ) is a complex spherical harmonic of order ( p, q ). The number of linearlyindependent h ( p,q ) is ( p + 1)( q + 1)( p + q + 2). All SU (3) irreps can be generated inthis way. If real coordinates are chosen, (cid:126)z ∈ R , the discussion reduces to the sphericalharmonics of SO (3). In this section only, honouring the standard notation of complex analysis, we use ¯ to denote thecomplex conjugate instead of the ∗ -symbol. 54n orthogonal basis can be obtained from the following generating function: G ( a , a , b ) = (¯ z − a ¯ z ) − q − (¯ z − a ¯ z − a ¯ z ) p +1 × ( b ( a z + z )(¯ z − a ¯ z − a ¯ z ) + z (¯ z − a ¯ z ) + a ( z ¯ z + z ¯ z )) q = q (cid:88) r =0 p + q +1 (cid:88) s =0 ∞ (cid:88) t =0 h rst ( p,q ) a t a s b r . (G.3) h rst ( p,q ) is an orthogonal basis for a ( p, q )-representation whose states are characterized bythe labels ( r, s, t ) ranging from: r = 0 ..q , s = 0 ..p , t = 0 .. ( p + r − s ) . (G.4)It is readily verified that r, s, t sums over ( p + 1)( q + 1)( p + q + 2) elements. Theparameters p, q, r, s, t correspond to the five parameters of the five-sphere (G.1). In ourwork we adapt the phase convention which follows from (G.3). An alternative conventionbased on isospin has been suggested in Ref. [49]. For the readers convenience, we give asummary of some basic facts in Tab. 8, in comparison of SO (3) and SU (3).group SO (3) SU (3)rank 1 ↔ l ↔ ( p, q )repres. fct. Y l,m h rst ( p,q ) fct. on manifold SO (3) /SO (2) (cid:39) S SU (3) /SU (2) (cid:39) S embedding (cid:44) → R with x + y + z = r (cid:44) → C with z ¯ z + z ¯ z + z ¯ z = ρ labelling irrep ( l ) ∈ N ( p, q ) ∈ N dim(irrep) (2 l + 1) ( p +1)( q +1)( p + q +2) / m = − l..l r = 0 ..q , s = 0 ..p , t = 0 .. ( p + r − s )Table 8: Comparison of SO (3) vs. SU (3) data. The acronym “fct” stands for function. G.2 Construction of explicit ( p, q ) representations G.2.1 Polyomial basis We have stated that h ( p,q ) are polynomials of degree p and q in the variables z i and ¯ z i .We shall denote such a space by H ( p,q ) . As an example let us quote x y z ∈ H (6 , . (G.5)55he dual vector is given by( x y z ) † ≡ ( | (cid:105) ) † = (cid:104) | ≡ ∂ x ∂ y ∂ z , (G.6)where the association with bra and ket should be obvious. The normalization then follows: (cid:104) | (cid:105) = 2!3!1! ⇒ | abcdef (cid:105) N = 1 √ a ! b ! c ! d ! e ! f ! | abcdef (cid:105) . (G.7)The entire space is spanned by H = ⊕ p ≥ ,q ≥ H ( p,q ) , and the identity on H ( p,q ) is repre-sented as ( p,q ) = p + q ) ( x∂ x + y∂ y + z∂ z + ¯ x∂ ¯ x + ¯ y∂ ¯ y + ¯ z∂ ¯ z ). G.2.2 Gell-Mann basis on polynomial space Noting that the fundamental representation space (1 , 0) in the polynomial basis is givenby { x, y, z } , the Gell-Mann operators of the SU (3) Lie-algebra are readily read off: B (1 , GM = { T , T , ..., T } = (G.8)12 { ( y∂ x + x∂ y ) , i ( y∂ x − x∂ y ) , T , i ( z∂ y − y∂ z )( z∂ x + x∂ z ) , ( z∂ y + y∂ z ) , i ( z∂ x − x∂ z ) , T } , with the Cartan sub algebra, T = 12 ( x∂ x − y∂ y ) , T = 12 1 √ x∂ x + y∂ y − z∂ z ) . (G.9)The Gell-Mann matrices satisfy the SU (3) Lie-algebra relations:[ T a , T b ] = if abc T c , (G.10)with f = 1 for example. The basis B (1 , GM works on H ( p, space, but it does not act on H (0 ,q ) space. We must therefore construct B (0 , GM . This follows by complex conjugation, B (0 , GM = − ( B (1 , GM ) ∗ , (G.11)where the extra minus sign stems from the fact that an extra factor of i comes in whenthe representation is exponentiated, exp( iv a T a ). Then, B GM = B (1 , GM + B (0 , GM (G.12)56s a basis that gives all ( p, q ) representations:[( T i ) ( p,q ) ] kl = (cid:104) k | T i | l (cid:105) , (G.13)where T i ∈ B GM , and | l (cid:105) corresponds to | ab ( p − a − b ) de ( q − d − e ) (cid:105) N ∈ H ( p,q ) and isunderstood to be an orthonormal basis. Note that we have taken into account the degreeof the polynomial state, which constrains the third and sixth entries with p and q . Wehave verified this construction for many examples, and we have also verified the Dynkinindex, Tr[( T a ) ( p,q ) ( T b ) ( p,q ) ] = k ( p,q ) δ ab , (G.14)which can be computed using Racah’s formula [51]. 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