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Conjugation properties of tensor product multiplicities

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product lambda x mu of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for the fusion product of the affine algebra of su(3) at finite levels, but this is not investigated in the present paper.


Foreword and Outline
In a recent paper [1] we have proved the following theorem Theorem 1 Let N ν λµ denote the multiplicity in the tensor product decomposition λ ⊗ µ → ν. Then the total multiplicities in λ ⊗ µ and λ ⊗μ are equal This theorem applies to the tensor product category of finite dimensional representations of simple Lie groups or algebras, and also to the fusion category of simple affine Lie algebra at finite level k.
In the present note we present a refined version of this conjugation property.
In view of the two sum rules (1)(2), one could think that the lists of multiplicities in the tensor (or fusion) products λ ⊗ µ and λ ⊗μ might be identical (up to permutations). It turns out that this is not true in general, except for the case of SU(3).

Theorem 2
The lists of multiplicities of SU(3) h.w. appearing in the decomposition into irreps of λ ⊗ µ and λ ⊗μ are the same (up to permutation).
That the theorem fails in general for other simple Lie algebras may be seen by explicit counterexamples, see sect. 5.
A first proof in sect. 2 relies on "well known" facts on multiplicities in the decomposition of tensor products of irreps of SU(3), combined with Theorem 1. Fifty years ago, Mandel'tsveig [2], motivated by the physics of hadrons and what is called nowadays "flavor SU(3)", gave a detailed discussion of irreps of SU (3), including the decomposition rules of tensor products. We shall see in sect. 2 that using either his results, or Wesslen's more recent discussion of the same issue [3], we can derive Theorem 2 from Theorem 1. Another proof (sect. 3) is more detailed and technical, since it gives a description of a mapping from the representations (ν, α) appearing in λ ⊗ µ, with α a multiplicity index, to those (ν , α ) appearing in λ ⊗μ. We find it convenient to adopt the combinatorial picture where the multiplicity α is regarded as an integral index satisfying a set of inequalities. Several such pictures have been proposed in the literature (B-Z triangles, K-T honeycombs, hives or puzzles, O-blades, etc), some of which we review briefly in sect. 4 for the convenience of the reader. The mapping (ν, α) → (ν , α ) that we propose (a priori it is non unique) turns out to be piecewise linear. In our final section 5, we comment briefly on what goes wrong in higher rank cases.

About intertwiners
An intertwiner of the group G is an element of (V 1 ⊗ V 2 ⊗ V 3 ) G , i.e., a G-invariant vector of this tensor product, where the V j are representations spaces for G. Such an element can be considered as an equivariant map V 1 ⊗V 2 → V 3 where the last term denotes the dual of V 3 (with the action of G given by the contragredient representation). If the V j are irreducible representations, the dimension d of this space of intertwiners is called the multiplicity of V 3 in the decomposition of V 1 ⊗ V 2 into irreducible representations (irreps in short). We say that V 3 occurs in Given the action of G on V 1 ⊗ V 2 , we may consider its centralizer algebra (Schur), or, equivalently, the commutant of G in the space of endomorphisms End(V 1 ⊗V 2 ). If the multiplicity of V 3 is d, the centralizer algebra of G in this space of endomorphisms contains a simple block d × d.
Let G be a semi-simple Lie group. In this framework the numbers m ν λµ giving the multiplicity of V ν in the tensor product V λ ⊗ V µ are called Littlewood-Richardson coefficients for G. As irreps of G are highest weight representations we can characterize the three vector spaces V λ , V µ , V ν by a triple (λ, µ, ν) of (integral) dominant weights. The notation λ ⊗ µ → ν or, equivalently, λ ⊗ µ ⊗ ν → l 1, will refer to the corresponding space of intertwiners.
The "3J-operators" of physicists are intertwiners, thought of as linear maps from V λ ⊗ V µ ⊗ V ν to the scalars. Once evaluated on a triple of vectors, such a 3J operator becomes a 3J symbol (a number), also called a Clebsh-Gordan coefficient.
Combinatorial models typically associate to a space of intertwiners of dimension d, and therefore to a given triple 1 (λ, µ, ν), a set of d combinatorial and graphical objects, that we call generically pictographs 2 . Examples are provided by the Knutson-Tao (KT) honeycombs. There are other models, whose definition is recalled in section 4, that can be used in the same way as KT-honeycombs, but the proof of the permutation property that we shall present in section 3 will use the latter. For our purposes, we do not need to establish a precise correspondance between specific intertwiners and 1 The order matters: despite the trivial symmetry m ν λµ = m ν µλ coming from a representation theory interpretation, the two families of objects associated with the intertwiners spaces describing λ ⊗ µ → ν and µ ⊗ λ → ν are different. 2 Webster says : "pictograph. 1 : an ancient or prehistoric drawing or painting on a rock wall ; 2 : one of the symbols belonging to a pictorial graphic system; etc.". Here we are more concerned with the pictorial graphic system than with the prehistoric graffiti. . . pictographs, it is enough to remember that one can associate, with the (ordered) triple of weights labelling a space of intertwiners, a collection of d distinct pictographs if this space of intertwiners is of dimension d.

Notations
In the present paper, we shall be mainly dealing with the SL(3) and SU(3) Lie groups and algebras. Their irreducible finite dimensional representations are labelled by their dominant integral highest weights λ = λ 1 ω 1 + λ 2 ω 2 , where {ω 1 , ω 2 } are the fundamental weights, and λ may be given by its Dynkin labels (λ 1 , λ 2 ), where 0 ≤ λ 1 , λ 2 ∈ Z. Occasionally we shall make use of the three weights i , i = 1, 2, 3, of the ω 1 representation (they are non-independent, i = 0) and of the simple roots In terms of the 's we have and thus In SU(3) there is a Z 3 grading τ ("triality") on irreps, stemming from the fact that this discrete group is the center of SU(3); we set τ (ω 1 ) = 1, τ (ω 2 ) = 2 mod 3 and more generally This triality is conserved (mod 3) in the decomposition of tensor products : N ν λµ = 0 implies

Miscellaneous comments
We should stress the fact that the property of multiplicities described by theorem 2 is discussed only in the framework of the Lie group SU(3). In particular, we do not discuss the analogous property for the affine algebra su(3) k , although we conjecture that it should hold as well. As already mentioned, the higher rank case, for simple Lie groups or for the corresponding affine algebras, is briefly commented in one paragraph of our final section (in those cases theorem 1 holds but theorem 2 does not).
There are many references in the literature about fusion rules and tensor product multiplicities, see for instance [4] and references therein. We should probably mention at that point that, in the mathematical literature, there is a result about tensor multiplicities known as the PRV conjecture [5]. This result, which is usually expressed in terms of the geometry of the Weyl group, gives a sufficient condition, for a dominant weight, to appear as the highest weight of an irreducible representation contained in the tensor product of two given irreps. This conjecture became a theorem when it was independently proved in [6] and [7]. It was later generalized in [8]. In the framework of affine algebras, a similar result for the fusion product was obtained in [9]; actually, it was only after the work of [10] that the result of [9] could be recognized as a generalization of the PRV theorem to the affine case. Our result (theorem 2), although leading to severely contrived conditions on the irreps that may occur in a tensor product of two given irreducible representations, should not be considered as a generalization of the PRV theorem. Indeed, although it is, in a sense, stronger than the latter result, it only applies to the Lie group SU(3).
2 A proof based on "well known" facts on SU(3) multiplicities

A second proof of the permutation property.
On the one hand, the formula (9) giving the number of terms with prescribed multiplicity, strictly smaller than the maximal one, is obviously symmetric if we interchange µ and µ (i.e., µ 1 and µ 2 ). The maximal multiplicity (and hence the list of possible multiplicities), given by equation (8) stays also the same for the same reason. On the other hand, the total multiplicity is also the same, because of theorem 1, for λ ⊗ µ and λ ⊗ µ. We conclude that the number of distinct h.w. with maximal multiplicity is also the same in both cases, or equivalently that the M in (7) and (9) is the same for λ ⊗ µ and λ ⊗ µ. From these two results we deduce that, for SU(3), the lists of multiplicities remain the same under the "pseudo-conjugation" λ ⊗ µ → λ ⊗ µ.
In Fig. 1 (a) as expected, the two polygons are symmetric with respect to the diagonal of the first quadrant. In Fig. 1 (b), the second polygon can be obtained from the first by a reflection. This last property is not a generic feature as it can be seen for instance from Fig. 2 that displays the tensor polygons λ ⊗ µ and λ ⊗ µ, for λ = (5, 3) and µ = (4, 2): indeed, the sides of the outer shells (convex hulls) do not contain the same number of integer points. A detailed analysis of this last example shows that using a reflection to map the first figure to the next would send one of its vertices outside of the main chamber, so that such a reflection has to be amended. As we shall see in section 3.7, the transformation mapping λ ⊗ µ to λ ⊗ µ, is, in the general case, only piecewise linear.     Figure 3 gives a protruded 3d version of the previously displayed (9, 5) ⊗ (6, 2) tensor polygon (the two dark heptagons in Fig. 1 (a) and 1 (b)) where the "degeneracy" of irreps stemming from the non-trivial multiplicities (1, 2, 3) is lifted by the introduction of a hive parameter discussed below in section 3.5.

A proof based on KT-honeycombs and hives
The determination of tensor products multiplicities, aka Littlewood-Richardson coefficients, is a fascinating subject, with many far reaching implications in various fields of mathematics and physics. As such it has attracted a lot of attention and has been -and still is-the object of much ingenuity in the development of combinatorial and graphical tools. To quote a few, skew Young tableaux [11], Berenstein-Zelevinsky triangles [12], honeycombs or their dual hives [13], puzzles [14], "O-blades" [15], Littelmann paths [16], etc. Each of those has its own merits and may be best suited for a specific problem. The discussion that follows can be carried out with various combinatorial models (various pictographs), but for definiteness, in this section, we present a proof of our result about the conjugation properties of tensor products by using the honeycombs, or hives, of Knutson and Tao. We shall indicate, in section 4, how the discussion should be modified if we were using other combinatorial models.
We then compute the coordinates of vertices, i.e., the labels of edges, of such a KT honeycomb. There is only one free parameter α in the central hexagon, see Fig. 4 (b) for an example, and Fig.  6 for the generic case. Note that an SU(3) honeycomb depends in fine on 7 parameters, the 3 × 2 independent components of λ, µ and ν and the α parameter.
For instance, in our favorite example, λ 1 = 21, λ 2 = 6, µ 1 = 17, µ 2 = 16, we know (from the discussion at the end of section 2.1) that multiplicities range from 1 to 7, and that there are 44 possible ν for which the multiplicity λ ⊗ µ → ν is 5. This is the case of ν 1 = 12, ν 2 = 8, hence ν 3 = 18, for which the parameter α may take 5 values, from 60 to 64, as it will be clear in the next section, see Fig. 4 Figure 5: The three edge patterns

Wesslen's inequalities (SU(3))
In this subsection we write more explicitly in terms of external variables, i.e., of components of the weights, the inequalities that honeycombs must satisfy. In so doing, we make contact with former work by Wesslen [3]. The afore-mentioned positivity constraint is expressed in the case of SU(3) by the 9 inequalities with the notation Σ := ν 1 + ν 2 + ν 3 . In terms of the triality τ defined above, we see that Σ = Both Σ and ν 3 are integers thanks to (6).
Rephrasing these inequalities as 6 lower and 3 upper bounds on α, a i ≤ α ≤ b j , i = 1, · · · , 6 and 1 ≤ j ≤ 3, we find 3 × 6 consistency relations a i ≤ b j , namely The inequalities λ 1 , λ 2 , µ 1 , µ 2 ≥ 0 are satisfied by definition. We are left with the following inequalities on ν 1 and ν 2 These 8 inequalities are equivalent to those given by Wesslen [3], and are written in a way that is explicitly symmetric in the exchange λ ↔ µ or in the conjugation of representations ( 1 ↔ − 3 , 2 ↔ − 2 ). Note that by adding the second and fourth lines, one gets 3ν which was another of Wesslen's inequalities, clearly not independent of the former inequalities; likewise Finally adding the second and third lines yields ν 1 + ν 2 ≤ λ 1 + λ 2 + µ 1 + µ 2 , which expresses that the level of ν is bounded by the level of the "highest highest weight" λ + µ (recall that for SU (3), the level of an irrep is the sum of its Dynkin labels.)

Multiplicity of ν
This subsection is devoted to writing explicit expressions for the multiplicity of ν in the tensor product λ ⊗ µ. We shall need such expressions in our subsequent comparison of multiplicities in λ ⊗ µ and λ ⊗μ.
with Σ and ν 3 given above as functions of , and using (6), we find the formula which is made out of the arguments of the inequalities (13) (up to factors 1 3 ) and which is symmetric under S 3 permutations of (λ 1 , λ 2 ), (µ 1 , µ 2 ), (ν 2 , ν 1 ) and complex conjugation (λ 1 , µ 1 , ν 2 ) ↔ (λ 2 , µ 2 , ν 1 ). Alternative expressions making use of the weights i of the fundamental representation read or equivalently where w I denotes one of the three weights λ, µ,ν and w II , w III the two others. This expression is explicitly symmetric under S 3 permutations of λ, µ,ν, and under complex conjugation where Note that one recovers the "well known" fact that if any of the Dynkin labels λ 1 , λ 2 , µ 1 , µ 2 , ν 1 , ν 2 vanishes, the multiplicity is 0 or 1: this is Pieri's rule, see also (8). Either of these formulae enables one to set up the plot of multiplicities, see Fig. 7 for an example.

Convexity and tensor polygons
Recall that in R d , the points x i satisfying a set of inequalities H j (x i ) ≥ 0, where H j = 0, j = 1, 2, · · · p, are equations of p hyperplanes, form a convex polytope, bounded or not bounded.
Here, in SU (3), we may apply this argument either to the set of points (ν, α) which live in R 2+1 and are subject to inequalities (12), or to points α (in the finite segment (16)), or to the resulting ν ∈ R 2 , cf (13) or (14). Notice that the fact that ν should obey the triality constraint (6) means that ν belongs to the root lattice Q shifted by some weight of λ ⊗ µ, for example the "highest highest weight" λ + µ ν ∈ Q + λ + µ .
Moreover it is clear that the number of points, in either of the previous contexts, either (ν, α), or α, or ν, is finite, for given λ and µ. Thus the previous statement should be rephrased as ν belongs to a convex bounded polytope drawn on the lattice Q + λ + µ, thus reaches all integer points interior to or on the boundary of that polytope. This is a particular case of a convexity property proved by Knutson and Tao for general GL(N ).
This convex polygon has been charted by Wesslen [3]: it has v ≤ 8 vertices as depicted in Fig.  8, where we have assumed that λ 1 = max(λ 1 , λ 2 , µ 1 , µ 2 ). Two vertices play a special rôle: H = (λ 1 + µ 1 , λ 2 + µ 2 ) is the "highest highest weight" (h.h.w.), while the "lowest highest weight" h is defined by The coordinates of the other vertices may be readily computed from the data of Fig. 8, which gives the components of the edge vectors in the (ω 1 , ω 2 ) basis. Note that some edges may "degenerate" (i.e., be of length 0) and the octogon reduce to an heptagon, an hexagon, . . . , or a single point.
From the formula (17) for the multiplicity of a weight ν, it follows that the polygons of increasing multiplicity have a "matriochka" pattern. Let C m the set of ν's of multiplicity ≥ m. If H j , j = 1, · · · 18 denote the 18 arguments of min in (17), the points in C m+1 satisfy H j ≥ m, hence a fortiori H j ≥ m − 1, thus C m+1 ⊂ C m .
In the hive picture, the inequalities (12) express that for each rhombus made of two adjacent equilateral triangles, the sum of weights on the short diagonal is larger than or equal to that along the long one. (In (12), the three successive lines correspond to rhombi with a short diagonal respectively in −, \ and / directions.)
Given a pair (ν, α) satisfying the inequalities (12), we may define its multiplicity index by Clearly m takes values in {1, · · · , mult(λ, µ; ν)}. We shall make use of that notion in the following section.

A piecewise linear map (ν, α) → (ν , α )
We now turn to the issue of writing a mapping between h.w. of irreps appearing in λ ⊗ µ and in λ ⊗μ. In this section, for the sake of simplicity of notations (and to avoid an overflow of min and max), we assume that λ 1 is the supremum of all Dynkin labels of λ and µ 6 We shall show below that there exists a piecewise linear mapping acting on pairs (ν, a) where the latter equation expresses that the multiplicity indices of (ν, α) and of (ν , α ) as defined in (25) are the same.
We stress that a given h.w. ν coming in λ ⊗ µ with some multiplicity may be sent onto several distinct h.w. ν , depending on its multiplicity index (or equivalently on the value of α). The map (27) is, however, one-to-one between the sets of allowed (ν, α) and (ν , α ). Therefore it maps all allowed weights ν appearing in λ ⊗ µ, with their multiplicity, onto all allowed weights ν appearing in λ ⊗μ, with their multiplicity. This is thus a new proof of Theorem 2, namely that the lists of multiplicities in λ ⊗ µ and λ ⊗μ are the same up to permutations.
We show below that for λ and µ "deep enough" in the fundamental Weyl chamber, in a sense to be made precise, the transformation t 1 maps all ν appearing in λ ⊗ µ onto all ν appearing in λ ⊗μ while preserving their multiplicity. For generic λ and µ, however, the transformation t 1 may map some of the points inside (or on the boundary of) the polygon P (1) (λ, µ) outside the polygon P (1) (λ,μ), in other words, the transformed weight ν violates some of the inequalities (15), in which case we have to modify the transformation. This is apparent in Fig. 2 where the two polygons are obtained by reflection of one another . . . but for some extremal vertices that would fall out of the dominant Weyl chamber. Three cases may occur: In that case it follows from that inequality and from the inequality ν 1 +ν 2 ≤ λ 1 +λ 2 +µ 1 +µ 2 noticed above (end of sect. 3.2) that ν 1 +ν 2 ≤ 2λ 1 +λ 2 ; also from the same inequality and from the second row of inequalities (14), it follows that 2ν 1 +ν 2 ≤ 2λ 1 +λ 2 +2µ 1 +µ 2 ≤ 5λ 1 + 4λ 2 − µ 1 − 2µ 2 and thus, in (27), we take ν = t 1 (ν) = (2λ 1 + λ 2 − ν 1 − ν 2 , ν 2 ). It is then tedious but straightforward to check that if ν satisfies the inequalities (14), then ν = t 1 (ν) satisfies the inequalities (14) with µ changed intoμ. Moreover as the transformation is involutive and thus one-to-one, it conserves the number of points: the number of ν with m = 1 (first layer of λ ⊗ µ) equals that of ν in the first layer of λ ⊗μ. See Fig. 11 for an example.
In that case, one sees that one of the inequalities (14) (in which µ has been changed intoμ) may be violated, namely ν 1 < 0 whenever ν 1 + ν 2 > 2λ 1 + λ 2 . The other inequalities are still satisfied. One thus modifies t 1 , keeping a reflection in the domain ν 1 + ν 2 ≤ 2λ 1 + λ 2 , but changing it into a translation outside that domain Again it is straightforward to verify that the resulting ν satisfy the right inequalities (14), thus lie inside or on the boundaries of P (1) (λ,μ), and that the transformation is one-to-one. 12 for an example, on which vertices below the line AB are reflected by t 1 , while those above are shifted by (−2, 2).

The other layers
To find the right transformation for the higher layers (values of m > 1 in (25)), we make use of the relation (22) between the successive polygons and of the transformation (31) on the first layer. This results in the formula (27) and completes our discussion of the piecewise linear and layer-dependent map of ν onto ν .

Pictographs as combinatorial descriptions of the spaces of intertwiners
In this section we review various types of graphical representations of intertwiners, aka pictographs.
In the first subsection we gather several comments about the notion of "fundamental pictographs". There, we use the language of KT-honeycombs, defined in sect. 3, but the concepts could be equivalently described in terms of the other types of pictographs whose definition will be recalled or introduced in the second subsection. In a third subsection, using the existence of fundamental pictographs, we shall obtain (still in the framework of SU(3)) a dictionnary between KT-honeycombs and these other pictographs. In particular this dictionnary can be used to express the proof of section 3 in terms of the other models.

Decomposition of pictographs
As we did not find any competing terminology in the literature to refer to the forthcoming concepts, we introduce the following terminology. For SU(N ), we call fundamental pictographs the combinatorial models associated with intertwiners of the type f 1 ⊗ f 2 ⊗ f 3 where the f i 's denote either a fundamental representation (such intertwiners may not exist) or the trivial one. The dimension of the corresponding spaces of intertwiners being equal to 0 or 1, there is only one such pictograph or none. For SU(N ), we call non-primitive fundamental pictographs the combinatorial models associated with the intertwiners : where f is a fundamental irrep. As the dimension of the corresponding spaces of intertwiners is equal to 1, there is no ambiguity in this definition. Clearly there are 3 × (N − 1) such pictographs, in particular six of them for SU(3). For SU(N ), we call primitive fundamental pictographs the combinatorial models associated with the intertwiners : where none of the f i is the trivial representation, and where the f i are fundamental representations. It is easy to see that, if G = SU(N ), there are 2(N − 2)(N − 1)/2 such pictographs, in particular two of them for SU (3), that are respectively associated with the cubes of (1, 0) and of (0, 1).
Typically, the external structure ("external legs", in physicists parlance) of an object like a KThoneycomb, i.e., the set of numbers specifying the three chosen irreps, for instance the Dynkin labels of the corresponding highest weights, can be deduced from its internal structure, usually a set of numbers carried by the internal edges or by other graphical elements of the model. In the case of SU(N ), honeycombs or their like obviously contain 3 × (1 + 2 + . . . + (N − 1)) = 3 × N (N − 1)/2 internal edges, in particular 9 edges for SU (3). Arbitrary elements belonging to the vector space built over the set of internal edges cannot, in general, be considered as honeycombs, since the constraints that enter the definition of the latter would not, in general, be obeyed. Nevertheless, it is clear that one can add arbitrary honeycombs by adding the values of their internal edges, and multiply them by scalars.
One could think that the previously described family of (N − 1)(N − 2) + 3(N − 1) = (N 2 − 1) fundamental pictographs (primitive and non-primitive) can be chosen as a basis for the space of pictographs, but this is not so, as we shall see below, because they are not independent. In the case of SU(3) honeycombs for instance, we have 9 internal edges, and 8 fundamental honeycombs (6 non-primitive, 2 primitive), see Fig. 14, but there is also one relation which is displayed in Fig.  15 in terms of KT-honeycombs. The number of independent fundamental honeycombs is therefore 7. Up to permutations, we can construct two particular basis, each one contains seven fundamental intertwiners: the six non-primitive ones, and a last one chosen among the two that are primitive.
For SU(N ) we have one relation for every inner hexagon (or for every inner vertex in the O-blade model, see below) i.e., (N − 2)(N − 1)/2 relations. Formally, one may consider the vector space over the set of fundamental pictographs, a space of dimension (N 2 − 1), and take its quotient by the relations. This quotient, the "space of pictographs", has dimension (N + 4)(N − 1)/2. Actually, in view of the correspondance between pictographs and intertwiner spaces, what matters are integral pictographs, but the fact that they are integral is usually understood. Going back to our problem of relating the intertwiner spaces λ ⊗ µ → ν and λ ⊗ µ → ν , it is quite natural to consider the sets of associated pictographs and decompose them on a basis of (independent) fundamental ones. Notice that the space of pictographs comes with a distinguished generating family -those that are fundamental -but not with a distinguished basis as there is no canonical way to choose half of the fundamental primitive pictographs to build a basis.

BZ-triangles
As Berenstein-Zelevinsky (BZ)-triangles are well documented in other places [12], we shall be happy with the following operational definition, adapted to the case of SU(3). A generic BZ triangle for SU(3) is displayed in Fig. 16 (a). It contains nine vertices called m 12 , m 23 , m 13 , n 12 , n 23 , n 13 , l 12 , l 23 , l 13 labelled with non-negative integers. The BZ-triangle for SU(3) has a single inner hexagon, and obeys one single constraint: the opposite sides of the hexagon should be equal (the value of a side being defined as the sum of its endpoints). This reads: l 12 + n 23 = l 23 + n 12 m 12 + l 23 = m 23 + l 12 (32) n 12 + m 23 = n 23 + m 12 .
In the case of SU(3), the BZ theorem can be expressed as follows: the dimension of the space of intertwiners associated with the branching rule λ ⊗ µ → ν (or, equivalently λ ⊗ µ ⊗ ν → 1, or equivalently, the multiplicity of the irrep ν in the tensor product λ ⊗ µ) is equal to the number of BZ-triangles obeying the above set of constraints. As this is discussed elsewhere, let us just mention that the nine integers entering a BZ-triangles are the components of the (non-unique) decomposition of the three vectors 7 σ 1 = λ + µ − ν, σ 2 = λ − µ + ν, σ 3 = −λ + µ + ν, belonging to the SU(3) root lattice, along the generating family (not a basis) of positive roots (a triplet in this case). More generally, for SU(N ), we have N (N − 1)/2 positive roots, hence 3 × N (N − 1)/2 vertices in the BZ-triangles, each inner hexagon determining the constraints that are obeyed by the integers associated with vertices.
Thought of as 9-plets, SU(3) BZ-triangles can be linearly combined, or multiplied by scalars; however, and as it was discussed in sect. 4.1, any BZ-triangle can be written -in a non-unique way -as a linear combination of 8 fundamental BZ-triangles. The non-uniqueness comes from the fact that, for SU (3), there is one linear relation between the fundamental triangles. This relation, displayed 8 in Fig. 17, can be immediately checked. One can then select, for instance, a basis (see our discussion in section 4.1) made of the six non-primitive fundamental triangles, and a last one, chosen among the two that are primitive. It is therefore enough to choose 7 arbitrary non-negative integers 9 to build arbitrary BZ-triangles obeying all the constraints. If (a, b, c, d, e, f ; g) denote those seven components, one sees immediately that an arbitrary BZ-triangle is given by Fig. 18  (a), and that, with this parametrization, the hexagon constraint (32) becomes automatic. 7 The writing of the three σi looks more symmetrical if one uses ν instead of ν in the definition. 8 It can be expressed in terms of the other combinatorial models (see section 4.1 and Fig. 15 and 20). 9 Only the integral BZ-triangles have an interpretation in terms of intertwiners.

O-blades (or oblades)
The name "oblade" that we use in the following to denote an alternative kind of combinatorial model is not supposed to be understood as a fish (Oblada melanura, a seabream) but as an Oblade, because, to the best of our knowledge, this way of displaying the BZ equations is due to A. Ocneanu [15], whom we want to thank for having shared this viewpoint with us.
A generic O-blade for SU(3) is displayed in Fig. 16 (b). It contains nine edges m 12 , m 23 , m 13 , n 12 , n 23 , n 13 , l 12 , l 23 , l 13 , labelled with non-negative integers. There is an obvious geometrical way to establish a one-to-one correspondence between this combi- Because of this correspondence, everything that was written in the previous section can be immediately translated in this other language. For instance, the O-blade for SU(3) has a single inner vertex, and obeys one single constraint: the three pairs of opposite "angles" defined by the lines intersecting at the inner vertex should be equal, the value of an angle being defined as the sum of its sides. The equations themselves read the same as in the previous case. As for BZ-triangles, the three weights λ, µ, ν follow each other cyclically, in a clockwise manner (this is our choice), so that the components of the weights appearing in the branching rule λ ⊗ µ → ν are as indicated on the pictures. The eight fundamental O-blades are given in Fig. 19 -edges carrying a "1" label have been thickened -with the corresponding KT-honeycombs already given in Fig.14 (the interested reader will have no difficulty in drawing the corresponding BZ-triangles). Notice that the two primitive fundamental O-blades appear as Y-shapes 10 (forks) with a horizontal tail to the left or to the right of the inner vertex. The unique relation between them is displayed in Fig. 20. Finally, chosing again the "left" basis (i.e., the last basis element being Y-shaped, with a horizontal tail to the left of the inner vertex), we see in Fig. 18 (b) how an arbitrary O-blade can be defined as a superposition of its seven components. In terms of O-blades, the five KT-honeycombs discussed at the beginning of sect. 3 and that are associated with the space of intertwiners specified by the branching rule (21, 6) ⊗ (17, 16) → (12,8), read as in Fig. 22.

SU(3)-honeycombs
The dual of an O-blade is obviously a honeycomb. The geometrical relation between the two should be clear from Fig. 21 which displays one of the corresponding pairs associated with the intertwiners defined by the already discussed branching rule (21, 6) ⊗ (17, 16) → (12,8). These "SU(3)-honeycombs" are not "KT-honeycombs" since the integers labels, and the constraints they obey, are different. In particular the lengths of all edges are non-negative. Also, the external labels refer to Dynkin labels of SU(3) highest weights, and are therefore also non-negative, whereas in the case of KT-honeycombs, the external indices refer to partitions labeling GL(3) irreps. The translation of BZ-triangles, or of O-blades, to this last combinatorial model is immediate, and we leave this as an exercise for the reader. As a side remark, note that one of the merits of these SU(3)-honeycombs is to be "metric", in the sense that with angles of 2π/3, their edges may be drawn with the indicated value, see Fig. 21.
The purpose of this short section was only to avoid possible misunderstandings and warn the reader that one can devise several types of honeycomb pictographs. In the present paper all calculations involving combinatorial models were done using KT-honeycombs or O-blades, but "De gustibus non est disputandum".

From BZ-triangles, O-blades and SU(3)-honeycombs to KT-honeycombs and hives: Yet another approach to the permutation property
As it was discussed in the previous section, the relation between the graphical components belonging to the first three combinatorial models appearing in the title of this section is immediate. Although slightly less obvious, the relation between the latter and KT-honeycombs or hives is not difficult to grasp. One way to obtain this relation is to decompose an arbitrary O-blade (for instance) and an arbitrary KT-honeycomb on a basis of fundamental ones, and compare the results. Indeed the matching between fundamental KT-honeycombs and O-blades is one-to-one, and it is shown explicitly in the corresponding figures 14 and 19. As before, let us assume that we have chosen some basis which contains the six non-primitive ones and one of the two that are primitive, and call (a, b, c, d, e, f ; g) the components along that basis. The values of the nine edges of the O-blade (or of the nine vertices of the BZ-triangle), in terms of (a, b, c, d, e, f ; g) are immediately obtained. The result was already given (compare Fig. 16 (b) and 18 (b). The nine edges of the KT-honeycomb (read from top to bottom, and left to right, see Fig. 6), in terms of these seven parameters, read respectively: In terms of the same components, the three external GL (3)  Finally, the "hive parameter" called α everywhere in sect. 3 (also in the figures of that section), is simply related to first parameter a, of the list (a, b, c, d, e, f, g), one finds: λ 1 + 2λ 2 + µ 1 + µ 2 − α = a. For instance, in the case of the example we found that α runs from 60 to 64, so that the corresponding a = 66 − α runs from 2 to 6. Notice that a = n 13 is the edge located at the top of the O-blade (or at the top vertex of the BZ-triangle).
Using this dictionnary, the reader will have no difficulty to re-express the 9 "Wesslen inequalities" of section 3.2, already interpreted in terms of KT-honeycombs, as positivity constraints for the 9 edges ("blades") of the O-blades, and, more generally, to recast the whole analysis carried out in the two previous sections in terms of the other combinatorial models.

Final comments.
We have observed that theorem 2, proved for SU(3), also holds for the fusion of representations of the affine algebra su(3) k at a finite level k. This, however, would require a separate proof that we actually did not complete, and that in any case would markedly increase the size of the paper. By way of contrast, the same properties do not extend, in general, to higher rank algebras or groups endowed with complex representations. We gather a few comments on the case SU(4).
For the affine algebra su(4) k , theorem 2 holds true for all weights of levels less or equal to 7, but fails for some weights at levels greater or equal to 8. For instance, both for su(4) 8 and for the classical group SU(4), a weight of multiplicity 8 appears in the decomposition of the tensor product (1, 2, 2)⊗(2, 1, 3), while all multiplicities are bounded by 7 in the decomposition of (1, 2, 2)⊗(3, 1, 2). On the other hand, theorem 2 is still satisfied by many pairs (λ, µ), but we have been unable to find a simple criterion for this to hold. It is nevertheless interesting to ask: what goes wrong in the previous proofs, when one passes from SU(3) to SU(4) ?
In the case of SU(N ), the pictographs obtained in sect. 4 for given external λ, µ and ν depend on (N −1)(N −2)/2 parameters rather than on a single one, α, as in the case of SU(3). As a result, the determination of possible values of these parameters, i.e., the multiplicity, becomes more delicate since it typically amounts to count integral points in a convex set, a notoriously difficult problem. The analysis that we could perform for SU(3) cannot therefore be extended in a straightforward manner and such a generalization would not give rise to the same conclusions.
It is also interesting to observe that the convexity property of domains, like the "layers" that appeared in SU (3), is less stringent for SU(4) and beyond : the polytopes of increasing multiplicity may touch one another. An instance is presented in Fig. 23: the weights ν appearing in the decomposition of (1, 2, 2) ⊗ (2, 2, 1) form a polytope determined by the (black) vertices