TYPE A FUSION RULES FROM

ELEMENTARY GROUP THEORY

Alex J. Feingold, Michael D. Weiner

1. Introduction

Fusion rules play a very important role in conformal field theory [Fu], in the representation theory of vertex operator algebras [FHL], and in quite a few other areas. We are interested in understanding fusion rules from a combinatorial point of view, using very elementary aspects of group theory. Several authors have also investigated fusion rules from a combinatorial point of view [BWM,BKMW,T], but we believe our approach is unique. We have used this approach successfully in [AFW] to obtain all of the (p, q)-minimal model fusion rules ([Wa]) from elementary 2-groups. The (p, q)-minimal models are a certain series of highest weight representations of the Virasoro algebra ([KR]) which also have the structure of a vertex operator algebra ([FLM]), and modules for it ([FZ]). The Goddard-Kent-Olive coset construction [GO] can be used to get the unitary (p, p + 1)-minimal models from representations of the affine Kac-Moody Lie algebra ĝ where g = sl_{2} is the finite dimensional Lie algebra of type A_{1}. Knowing that, we were not surprised to notice that the approach in [AFW] would give the fusion rules for level k representations of the affine A_{1} algebra from elementary 2-groups. It is perhaps a bit more surprising that the technique worked for all of the (p, q)-minimal models, not just the unitary ones coming from affine A_{1} coset constructions. Until now our approach has only given fusion rules which have the special property that the fusion coefficients N_{ij}^{k} are in {0, 1}. In order for our approach to be of greater significance, we had to find a way to incorporate the higher multiplicities which can occur even in the case of affine A_{2}. In this paper we achieve that goal in such a way as to see how the simpler special cases only have multiplicity one or zero. We cannot yet give the complete answer for all affine type A algebras, but we present here for all levels, our approach for ranks N = 1 and N = 2.

The inspiration for this work on fusion rules comes from certain explicit constructions of vertex operator algebras (VOAs), their modules, and spaces of intertwining operators ([Fe,FFR,FRW,We]) which can all be unified into one large algebraic system. When the VOA and its modules are indexed by a finite abelian group, G, and certain technical conditions are satisfied, we call such a system a vertex operator para-algebra (VOPA). A somewhat more general structure, called an abelian intertwining algebra, was defined in [DL]. In those systems the fusion rules are given

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 17B67, 17B65, 81T40; Secondary 81R10, 05E10.

We wish to thank Matthias Beck, Jürgen Fuchs and Thomas Zaslavsky for valuable advice.

precisely by the group algebra of G. This is the situation for level one modules of an untwisted affine Kac-Moody Lie algebra ĝ, where the group G is the quotient of the weight lattice by the root lattice of the underlying finite dimensional Lie algebra, g. For example, in [FFR] the spinor construction of the four level one modules for the orthogonal affine Kac-Moody Lie algebra of type D_{4}^{(1)} was used to construct a vertex operator para-algebra with G = Z_{2} ×Z_{2}. A similar phenomenon occurred in [We], where a vertex operator para-algebra was constructed from the four level −\frac{1}{2} modules of the symplectic affine Kac-Moody Lie algebras of type C_{n}^{(1)}, but the group G could be taken as either Z_{4} or Z_{2} × Z_{2}. We were searching for the appropriate generalization of this idea which would include the VOAs and their modules coming from higher level representations of Kac-Moody Lie algebras ([FZ,TK]) and the (p, q)-minimal models. A key aspect of the generalization would have to incorporate the fusion rules which can come from spaces of intertwining operators which are of dimension greater than one. We first looked at the spinor construction of the (3, 4)-minimal model in [FRW], that is, the representations of the Virasoro algebra with central element c = \frac{1}{2} and with h = 0, h = \frac{1}{2} and h = \frac{1}{16}, which naturally has two copies of the h = \frac{1}{16} module. Let us denote those modules by [0], [\frac{1}{2}], [\frac{1}{16}]_{1} and [\frac{1}{16}]_{2}, respectively. Having two copies of [\frac{1}{16}] allows the usual Ising model fusion rules to be replaced by the group Z_{4} or by Z_{2} × Z_{2} because there are intertwining operators labelled by independent vectors which act on the two [\frac{1}{16}] modules, sending them to either [0] or [\frac{1}{2}]. This is different from the usual situation, where one intertwining operator labelled by one vector from the one h = \frac{1}{16} module stands for a two-dimensional space of operators sending that [\frac{1}{16}] module to a linear combination of [0] and [\frac{1}{2}]. This paper and [AFW] are the result of looking for other situations where having more copies of each irreducible VOA module, each copy labelled by an element of a finite abelian group, allows the fusion rules to be “pulled apart”, showing an underlying group algebra.

One suspects that such an explanation of fusion rules for the higher level Kac-Moody algebras should come from the level one para-algebra structure (or perhaps, abelian intertwining algebra structure). For example, in type A_{N−1}, where the weight lattice modulo the root lattice is Z_{N}, the direct sum of the N level one fundamental modules should form an abelian intertwining algebra, V. The k^{th} tensor power of V contains all level k modules, and is naturally an abelian intertwining algebra graded by the group Z_{N}^{k}. Then the fusion rules among those level k modules must be strongly influenced by that group Z_{N}^{k}, but a precise understanding of how is still lacking. The fact that the symmetric group S_{k} acts naturally on the k^{th} tensor power of V is certainly relevant, and plays a crucial role in our current paper.

A brief summary of our results is as follows. Let ĝ be the affine algebra of type A_{N−1}^{(1)} built from g = sl_{N}. Let λ_{1}, ⋯, λ_{N−1} denote the fundamental weights of g, and let P^{+} denote the dominant integral weights of g. The irreducible modules V̂_{Λ} for ĝ of level k ≥ 1 are indexed by a dominant integral highest weight Λ = kc + λ where

λ = ∑_{j=1}^{N−1} a_{j}λ_{j} ∈ P^{+}

satisfies the “level k condition” ∑_{j=1}^{N−1} a_{j} ≤ k. Then the irreducible modules on level k are in one-to-one correspondence with the set of N-tuples of nonnegative integers (i_{0}, i_{1}, ⋯, i_{N−1}) whose sum is k. Such an N-tuple corresponds to

Λ = kc + i_{1}λ_{1} + ⋯ + i_{N−1}λ_{N−1}.

Fix level k ≥ 1 and write the product in the fusion algebra for such modules as

[λ] × [μ] = ∑_{ν∈P^{+}} N_{[λ],[μ]}^{[ν]} [ν].

The distinguished module, [0], is an identity element for this product, and for each [λ] there is a distinguished conjugate [λ^{+}] such that N_{[λ],[μ]}^{[0]} = δ_{μ,λ^{+}}. A knowledge

of the fusion coefficients, N_{[λ],[μ]}^{[ν]}, is equivalent to a knowledge of the completely symmetric coefficients

N_{[λ],[μ],[ν]} = N_{[λ],[μ]}^{[ν^{+}]}.

Let G = Z_{N}^{k} and let the symmetric group S_{k} act on G by permuting the k-tuples. For a ∈ G, let [a] denote the orbit of a under this action and let O be the set of all such orbits. These orbits are precisely the subsets

P (i_{0}, i_{1}, …, i_{N−1}) = {x ∈ Z_{N}^{k} | j occurs exactly i_{j} times in x, 0 ≤ j ≤ N − 1}

where (i_{0}, i_{1}, ⋯, i_{N−1}) is any N-tuple of nonnegative integers such that i_{0} + i_{1} + ⋯ + i_{N−1} = k. We now have a bijection between O and the set of level k modules. For [a], [b], [c] ∈ O we believe the fusion coefficients N_{[a],[b]}^{[c]} have a combinatorial description in terms of the group G. The conjugate of [c] is [−c] and we prefer to study the symmetric coefficients

N_{[a],[b],[c]} = N_{[a],[b]}^{[−c]}.

The combinatorial question which we were led to investigate is as follows. For any [a], [b], [c] ∈ O, the group S_{k} acts on

T ([a], [b], [c]) = {(x, y, z) ∈ [a] × [b] × [c] | x + y + z = 0}

which decomposes into a finite number of orbits under that action. Let the number of such orbits be denoted by M([a], [b], [c]). For ranks N = 2 and N = 3 we determine M([a], [b], [c]) and show how it is related to N_{[a],[b],[c]}. In future work we will try to give the relationship for all ranks. We have the following results.

Theorem. For N = 2, for any integral level k ≥ 1, with notation as above, we have

M([a], [b], [c]) = N_{[a],[b],[c]}.

Theorem. For N = 3, for any integral level k ≥ 1, with notation as above, we have

M([a], [b], [c]) = \binom{N_{[a],[b],[c]} + 1}{2}.

2. Background

We use the definition of fusion algebra given in [Fu]. A fusion algebra F is a finite dimensional commutative associative algebra over Q with some basis B = {x_{a} | a ∈ A} such that the structure constants N_{a,b}^{c} defined by

x_{a} · x_{b} = ∑_{c∈A} N_{a,b}^{c}x_{c}

are non-negative integers. It is also required that there be a distinguished index Ω ∈ A with the following properties. Define a matrix C = [C_{a,b}] by C_{a,b} = N_{a,b}^{Ω} and define an associated “conjugation” map C : F → F by

C(x_{a}) = ∑_{b∈A} C_{a,b}x_{b}.

It is required that C be an involutive automorphism of F. This implies that C^{2} = I_{F} is the identity on F, so C^{2} = I is the identity matrix. Because 0 ≤ N_{a,b}^{c} ∈ Z, we get that either C = I or C must be an order 2 permutation matrix, that is, there is a permutation σ : A → A with σ^{2} = 1 and

C_{a,b} = δ_{a,σ(b)}.

Since C is an automorphism, we must also have C(x_{a}) · C(x_{b}) = C(x_{a} · x_{b}), that is,

x_{σ(a)} · x_{σ(b)} = ∑_{c∈A} N_{a,b}^{c}x_{σ(c)}

which means that N_{σ(a),σ(b)}^{σ(c)} = N_{a,b}^{c}. Sometimes we may write σ(a) = a^{+} and call x_{a^{+}} the conjugate of x_{a}. One may use it to define the non-negative integers

N_{a,b,c} = N_{a,b}^{c^{+}}

which, by commutativity and associativity of the structure constants, are completely symmetric in a, b and c. Furthermore, one finds that

N_{Ω,b}^{c} = N_{Ω,b,c^{+}} = N_{b,c^{+},Ω}
= N_{b,c^{+}}^{Ω^{+}} = N_{b^{+},c}^{Ω}
= C_{b^{+},c} = δ_{b,c}

which means that x_{Ω} is an identity element for multiplication in F, usually written x_{Ω} = 1. It also follows that Ω^{+} = Ω.

In [AFW] we introduced the idea of covering a fusion algebra by a finite abelian group and proved that the (p, q)-minimal model fusion algebra, which comes from the discrete series of 0 < c < 1 representations of the Virasoro algebra, can be covered by the group Z_{2}^{p+q−5}. The basic idea, which is only set up to handle fusion algebras whose fusion coefficients N_{ij}^{k} are in {0, 1}, is as follows.

Definition. Let (G, +, 0) be a finite abelian group and let G = P_{0}∪P_{1}∪...∪P_{N} be a partition into N disjoint subsets with P_{0} = {0}. Let W be an N-dimensional vector space over Q with basis P = {P_{0}, P_{1}, ..., P_{N}} and define a bilinear multiplication on W by the formula

P_{i} ∗ P_{j} = ∑_{k∈T (i,j)} P_{k}

where T (i, j) = {k | ∃a ∈ P_{i}, ∃b ∈ P_{j}, a + b ∈ P_{k}}. We say that such a partition is associative if the product ∗ is associative. We say that a group G covers a fusion algebra if there is an associative partition P of G and a bijection Φ between A and P which gives an algebra isomorphism between F and W such that Φ(Ω) = P_{0}.

There are certainly examples of groups with partitions which are not associative. Perhaps it is surprising that there are any interesting examples of groups with associative partitions. As an example of a nonassociative partition, let G = Z_{5}, and let P_{0} = {0}, P_{1} = {1, 2} and P_{2} = {3, 4}. The multiplication table for the corresponding W is as follows.

Table 1: A nonassociative partition for Z_{5}.

P_{i} ∗ P_{j} P_{0}  P_{1}  P_{2}
P_{0}  P_{0}  P_{1}  P_{2}
P_{1}  P_{1}  P_{1} + P_{2}  P_{0} + P_{1} + P_{2}
P_{2}  P_{2} P_{0} + P_{1} + P_{2}  P_{1} + P_{2}

We then compute

(P_{1} ∗P_{2})∗P_{2} = (P_{0} +P_{1} +P_{2})∗P_{2} = P_{2} +(P_{0} +P_{1} +P_{2})+(P_{1} +P_{2}) = P_{0} +2P_{1} +3P_{2}

and

P_{1} ∗ (P_{2} ∗ P_{2}) = P_{1} ∗ (P_{1} + P_{2}) = (P_{1} + P_{2}) + (P_{0} + P_{1} + P_{2}) = P_{0} + 2P_{1} + 2P_{2}

so (P_{1} ∗ P_{2}) ∗ P_{2} ≠ P_{1} ∗ (P_{2} ∗ P_{2}).

The fusion rule tables presented here were produced by the computer program of Bert Schellekens, called “Kac”, available from his webpage [http://norma.nikhef.nl/ ∼t58/]. As an example of a nontrivial fusion algebra which can be covered by a group, let the W algebra coming from the coset construction of

\frac{SU(N)_{r} ⊗ SU(N)_{s}}{SU(N)_{r+s}}

be denoted by W_{N}(r, s). The fusion rules for W_{3}(1, 1) are as follows.

Table 2: Fusion rules for W_{3}(1, 1).

[a]×[b] [0]  [1]  [2]  [3]  [4]  [5]
[0]  [0]  [1]  [2]  [3]  [4]  [5]
[1]  [0]+[1] [3] [2]+[3] [5] [4]+[5]
[2]  [4]  [5]  [0]  [1]
[3]  [4]+[5] [1] [0]+[1]
[4]  [2]  [3]
[5]  [2]+[3]

Note that {[0], [2], [4]} forms a subgroup isomorphic to Z_{3}. We find that Z_{3}^{2} covers these fusion rules as follows:

\begin{matrix}\{(0, 0)\} & ↔ [0]  \{(1, 2), (2, 1)\} ↔ [1]  \{(1, 1)\} ↔ [2] \\ \{(0, 2), (2, 0)\} & ↔ [3]  \{(2, 2)\} ↔ [4]  \{(1, 0), (0, 1)\} ↔ [5] \end{matrix}

A multiplication table which illustrates exactly how Z_{3}^{2} covers the W_{3}(1, 1) fusion rules is as follows.

Table 3: Z_{3}^{2} covering the W_{3}(1, 1) fusion rules.

[a]×[b] (0,0) (1,2),(2,1) (1,1) (0,2),(2,0) (2,2) (1,0),(0,1)
(0,0) (0,0) (1,2),(2,1) (1,1) (0,2),(2,0) (2,2) (1,0),(0,1)
{(1,2) \atop (2,1)}  {(2,1),(0,0) \atop (0,0),(1,2)} {(2,0) \atop (0,2)} {(1,1),(2,0) \atop (0,2),(1,1)} {(0,1) \atop (1,0)} {(2,2),(0,1) \atop (1,0),(2,2)}
(1,1)  (2,2) (1,0),(0,1) (0,0) (2,1),(1,2)
{(0,2) \atop (2,0)}  {(0,1),(2,2) \atop (2,2),(1,0)} {(2,1) \atop (1,2)} {(1,2),(0,0) \atop (0,0),(2,1)}
(2,2)  (1,1) (0,2),(2,0)
{(1,0) \atop (0,1)}  {(2,0),(1,1) \atop (1,1),(0,2)}

The simplest series of fusion algebras comes from the affine algebra ĝ of type A_{1}^{(1)} built on g = sl_{2}. There are k + 1 modules V̂_{a} for ĝ of level k ≥ 1 which can be indexed by “spin” a ∈ \frac{1}{2}Z with 0 ≤ a ≤ \frac{k}{2} . The conformal weight of the highest weight vector in V̂_{a} is

Δ_{a} = \frac{a(a + 1)}{k + 2}.

The g-submodule V_{a} of V̂_{a} generated by a highest weight vector is of dimension 2a + 1. There is a simple formula for the decomposition of the tensor product V_{a} ⊗ V_{b} into the direct sum of g-modules V_{c},

V_{a} ⊗ V_{b} = ∑_{|a−b|≤c≤a+b} V_{c}

where the sum is only taken over those c ∈ \frac{1}{2}Z such that a + b + c ∈ Z. This decomposition is independent of the level k and is part of the basic representation theory of sl_{2}. It is remarkable that the fusion rules for level k are obtained simply by truncating the above summation. The following formula for the A_{1}^{(1)} level k fusion rules is well-known ([TK]):

N_{a,b}^{c} = 1 if |a − b| ≤ c ≤ a + b,  a + b + c ∈ Z,  a + b + c ≤ k,

and N_{a,b}^{c} = 0 otherwise. The conditions above imply that c ≤ \frac{k}{2} .

We can also express these fusion rules in the following simple way. Re-index the k + 1 modules V̂_{a} on level k by m = 2a + 1 ∈ Z with 1 ≤ m ≤ k + 1. Then m = dim(V_{a}) and we write V̂_{a} = V̂ (2a + 1) = V̂ (m). Let p = k + 2.

Definition. For integer p ≥ 2, we say that the triple of integers (m, m′, m′′) is p-admissible when 0 < m, m′, m′′ < p, the sum m + m′ + m′′ < 2p is odd, and the “triangle” inequalities m < m′ + m′′, m′ < m + m′′, and m′′ < m + m′ are satisfied.

Then the fusion rules are given by N_{m,m′}^{m′′} = 1 if (m, m′, m′′) is p-admissible, N_{m,m′}^{m′′} = 0 otherwise.

The proof in [AFW] is easily modified to show that the A_{1}^{(1)} fusion ring for level k is covered by the group Z_{2}^{k}.

Theorem 1. The level k fusion rules for ĝ of type A_{1}^{(1)} define a fusion algebra F with A = {m ∈ Z | 1 ≤ m ≤ k + 1} with distinguished element Ω being m = 1 and the conjugate of m is m. The the fusion algebra F is covered by the elementary abelian 2-group G = Z_{2}^{k} with partition given by

P_{i} = {g ∈ G | exactly i coordinates of g are 1} for 0 ≤ i ≤ k.

We illustrate how this theorem works in the following tables. We have used a labelling scheme where on level k the k +1 generators of the fusion ring are denoted by [i] for 0 ≤ i ≤ k and the distinguished element Ω is [0]. Following each fusion rule table is a table showing how the group Z_{2}^{k} covers it.

Table 4: Fusion Table for A_{1} of level k = 2.

[i]×[j] [0] [1]  [2]
[0] [0] [1]  [2]
[1]  [0]  [2]
[2]  [0]+[1]

Table 5: Group Z_{2}^{2} covering the Fusion Table for A_{1} of level k = 2.

[i]×[j]  (0,0) (1,1) (1,0),(0,1)
(0,0)  (0,0) (1,1) (1,0),(0,1)
(1,1)  (0,0) (0,1),(1,0)
(1,0),(0,1)  {(0,0),(1,1) \atop (1,1),(0,0)}

Table 6: Fusion Table for A_{1} of level k = 3.

[i]×[j] [0] [1]  [2]  [3]
[0] [0] [1]  [2]  [3]
[1]  [0]  [3]  [2]
[2]  [0]+[2] [1]+[3]
[3]  [0]+[2]

Table 7: Group Z_{2}^{3} covering the Fusion Table for A_{1} of level k = 3.

[i]×[j] (0,0,0) (1,1,1) (0,1,1),(1,0,1),(1,1,0) (1,0,0),(0,1,0),(0,0,1)
(0,0,0) (0,0,0) (1,1,1) (1,0,0),(0,1,0),(0,0,1) (0,1,1),(1,0,1),(1,1,0)
(1,1,1)  (0,0,0) (0,1,1),(1,0,1),(1,1,0) (1,0,0),(0,1,0),(0,0,1)

(0,1,1) (1,0,1) (1,1,0)  (0,0,0),(1,1,0),(1,0,1) (1,1,0),(0,0,0),(0,1,1) (1,0,1),(0,1,1),(0,0,0) (1,1,1),(0,0,1),(0,1,0) (0,0,1),(1,1,1),(1,0,0) (1,0,0),(1,0,0),(1,1,1)

(1,0,0) (0,1,0) (0,0,1)  (0,0,0),(1,1,0),(1,0,1) (1,1,0),(0,0,0),(0,1,1) (1,0,1),(0,1,1),(0,0,0)

The next simplest series of fusion algebras comes from the affine algebra ĝ of type A_{2}^{(1)} built from g = sl_{3}. There are (k + 1)(k + 2)/2 irreducible modules V̂_{Λ} for ĝ of level k ≥ 1 which can be indexed by a dominant integral highest weight Λ = kc + λ where λ = a_{1}λ_{1} + a_{2}λ_{2} ∈ P^{+} is a dominant integral weight for the finite dimensional algebra g satisfying the “level k condition” a_{1} + a_{2} ≤ k for 0 ≤ a_{i} ∈ Z. The irreducible g-submodule V_{λ} of V̂_{Λ} generated by a highest weight vector is of dimension (a_{1} + 1)(a_{2} + 1)(a_{1} + a_{2} + 2)/2. The decomposition of the tensor product of irreducible g-modules

V_{λ} ⊗ V_{μ} = ∑_{ν∈P^{+}} Mult_{λ,μ}^{ν}V_{ν}

into the direct sum of irreducible g-modules includes multiplicities. This decomposition is independent of the level k and is part of the basic representation theory of sl_{3}. The fusion rules for level k are obtained by a rather subtle truncation of the above summation. The tensor product multiplicity, Mult_{λ,μ}^{ν}, is equal to the multiplicity of the trivial module V_{0} in the triple tensor product V_{λ} ⊗ V_{μ} ⊗ V_{ν^{+}}, where V_{ν^{+}} is the contragredient module of V_{ν}. This means that ν^{+} is the highest weight of the dual space of V_{ν}, so ν^{+} is the negative of the lowest weight of V_{ν}. Then

Mult_{λ,μ,ν} = Mult_{λ,μ}^{ν^{+}}

is the multiplicity of V_{0} in the triple tensor product V_{λ} ⊗ V_{μ} ⊗ V_{ν}. This number is clearly symmetric in λ, μ and ν.

We have the following result from [BMW] giving the A_{2} tensor product multiplicities and the fusion rules on level k.

Theorem 2. [BMW] Let λ = a_{1}λ_{1} + a_{2}λ_{2}, μ = b_{1}λ_{1} + b_{2}λ_{2}, and ν = c_{1}λ_{1} + c_{2}λ_{2} be dominant integral weights for g = sl_{3}, and let

A = \frac{1}{3} (2(a_{1} + b_{1} + c_{1}) + a_{2} + b_{2} + c_{2}) , B = \frac{1}{3} (a_{1} + b_{1} + c_{1} + 2(a_{2} + b_{2} + c_{2})) .

Define k_{0}^{max} = min(A, B) and

k_{0}^{min} = max(a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2}, A − min(a_{1}, b_{1}, c_{1}), B − min(a_{2}, b_{2}, c_{2})).

Let δ = 1 if k_{0}^{max} ≥ k_{0}^{min} and A, B ∈ Z, and let δ = 0 otherwise. Denote the consecutive integers k_{0}^{min}, k_{0}^{min} + 1,..., k_{0}^{max} by k_{0}^{(i)}, for 1 ≤ i ≤ M. Then we have the tensor product multiplicity

M = Mult_{λ,μ,ν} = (k_{0}^{max} − k_{0}^{min} + 1)δ

and the fusion rule coefficient on level k is

N_{λ,μ,ν}^{(k)} = \{ \begin{matrix}max(i) & such that k ≥ k_{0}^{(i)} and Mult_{λ,μ,ν} ≠ 0, \\ 0 & if k < k_{0}^{(1)} or Mult_{λ,μ,ν} = 0 \end{matrix}

so if Mult_{λ,μ,ν} ≠ 0 and k_{0}^{min} ≤ k then N_{λ,μ,ν}^{(k)} = min(k_{0}^{max}, k) − k_{0}^{min} + 1.

The fusion algebra for A_{2}^{(1)} on level k = 1 is rather trivial, with three modules whose fusion rules are given exactly by the group Z_{3}, which is the weight lattice modulo the root lattice of type A_{2}. The fusion algebra for A_{2}^{(1)} on level k = 2, and its covering by the group Z_{3}^{2}, are in the next two tables.

Table 8: Fusion Table for A_{2} of level k = 2.

[i]×[j] [0] [1] [2]  [3]  [4]  [5]
[0] [0] [1] [2]  [3]  [4]  [5]
[1]  [2] [0]  [4]  [5]  [3]
[2]  [1]  [5]  [3]  [4]
[3]  [0]+[3] [1]+[4] [2]+[5]
[4]  [2]+[5] [0]+[3]
[5]  [1]+[4]

Table 9: Group Z_{3}^{2} covering the Fusion Table for A_{2} of level k = 2.

[a]×[b] (0,0) (1,1) (2,2)  {(1,2) \atop (2,1)}  {(2,0) \atop (0,2)}  {(1,0) \atop (0,1)}
(0,0) (0,0) (1,1) (2,2)  {(1,2) \atop (2,1)}  {(2,0) \atop (0,2)}  {(1,0) \atop (0,1)}
(1,1)  (2,2) (0,0)  {(2,0) \atop (0,2)}  {(0,1) \atop (1,0)}  {(2,1) \atop (1,2)}
(2,2)  (1,1)  {(0,1) \atop (1,0)}  {(1,2) \atop (2,1)}  {(0,2) \atop (2,0)}
{(1,2) \atop (2,1)}  {(2,1),(0,0) \atop (0,0),(1,2)} {(0,2),(1,1) \atop (1,1),(2,0)} {(2,2),(0,1) \atop (1,0),(2,2)}
{(2,0) \atop (0,2)}  {(1,0),(2,2) \atop (2,2),(0,1)} {(0,0),(1,2) \atop (2,1),(0,0)}
{(1,0) \atop (0,1)}  {(2,0),(1,1) \atop (1,1),(0,2)}

In the fusion table for A_{2} of level k = 3, part of which is shown below, we found a fusion coefficient greater than one, which our previous scheme could not handle. Our main problem has been to modify our method of covering fusion tables so as to account for such higher multiplicities. In doing so we found a surprising connection to a combinatorial problem which may be of some independent interest. In the next section we will set up the notation needed for this new point of view for ĝ of type A_{N−1}^{(1)}, and see how the combinatorics enters the picture.

Table 10: Partial fusion Table for A_{2} of level k = 3.

[i]×[j] [0] [1] [2]  [9]
[0] [0] [1] [2]  [9]
[1]  [2] [0]  [9]
[2]  [1]  [9]
[9]  [0]+[1]+[2]+2[9]

3. A New Approach Incorporating Higher Multiplicities

Let ĝ be the affine algebra of type A_{N−1}^{(1)} built from g = sl_{N}. Let λ_{1}, ⋯, λ_{N−1} denote the fundamental weights of g, and let

P^{+} = { ∑_{j=1}^{N−1} a_{j}λ_{j} | 0 ≤ a_{j} ∈ Z}

denote the dominant integral weights of g. The irreducible modules V̂_{Λ} for ĝ of level k ≥ 1 are indexed by a dominant integral highest weight Λ = kc + λ where

λ = ∑_{j=1}^{N−1} a_{j}λ_{j} ∈ P^{+}

satisfies the “level k condition” ∑_{j=1}^{N−1} a_{j} ≤ k. On level 1, those dominant integral weights are the fundamental weights of ĝ,

Λ_{0} = c, Λ_{1} = c + λ_{1}, ⋯, Λ_{N−1} = c + λ_{N−1}.

Then the irreducible modules on level k are in one-to-one correspondence with the set of N-tuples of nonnegative integers (i_{0}, i_{1}, ⋯, i_{N−1}) whose sum is k. Such an N-tuple corresponds to

Λ = ∑_{j=0}^{N−1} i_{j}Λ_{j}

= ( ∑_{j=0}^{N−1} i_{j}) c + i_{1}λ_{1} + ⋯ + i_{N−1}λ_{N−1}

= kc + i_{1}λ_{1} + ⋯ + i_{N−1}λ_{N−1}.

Let V_{λ} be the irreducible finite dimensional g-submodule of V̂_{Λ} generated by a highest weight vector. In the special case when Λ = kΛ_{0} = kc, that finite dimensional g-module is V_{0}, the one dimensional trivial g-module. Since g is semisimple, any finite dimensional g-module is completely reducible. Therefore, we can write the tensor product of irreducible g-modules

V_{λ} ⊗ V_{μ} = ∑_{ν∈P^{+}} Mult_{λ,μ}^{ν}V_{ν}

as the direct sum of irreducible g-modules, including multiplicities. This decomposition is independent of the level k and is part of the basic representation theory of sl_{N}. The fusion algebra for ĝ on level k has a basis in one-to-one correspondence with the irreducible ĝ-modules on level k, and its fusion rules are obtained by a rather subtle truncation of the above summation.

The fusion algebra for A_{N−1}^{(1)} on level k = 1 is rather trivial, with a basis of N vectors labelled by the highest weights of the N irreducible fundamental modules, and with fusion rules given by the group Z_{N}, which is the weight lattice modulo the root lattice of g. This means that N_{λ,μ}^{ν} = δ_{λ_{+}μ,ν} where the addition takes place in the quotient group of the weight lattice modulo the root lattice.

Let G be the group Z_{N}^{k}, and for each N-tuple of nonnegative integers

(i_{0}, i_{1}, ⋯, i_{N−1}) such that i_{0} + i_{1} + ⋯ + i_{N−1} = k,

define the subset of G

P (i_{0}, i_{1}, …, i_{N−1}) = {x ∈ Z_{N}^{k} | j occurs exactly i_{j} times in x, 0 ≤ j ≤ N − 1}.

Then the cardinality of the subset P (i_{0}, i_{1}, …, i_{N−1}) is

\binom{k}{i_{0}, i_{1}, …, i_{N−1}}

and G is the disjoint union of these subsets. For example, with N = 3 and k = 2, a bijection between these subsets and a basis of the sl_{3} fusion algebra on level 2 is:

[0] = {(0, 0)} = P (2, 0, 0)
[1] = {(1, 1)} = P (0, 2, 0)
[2] = {(2, 2)} = P (0, 0, 2)
[3] = {(1, 2), (2, 1)} = P (0, 1, 1)
[4] = {(2, 0), (0, 2)} = P (1, 0, 1)
[5] = {(1, 0), (0, 1)} = P (1, 1, 0)

We can now start to describe our new method of covering the fusion algebra F = F (g, k) for g = sl_{N} on level k by the group G = Z_{N}^{k}. First note that the symmetric group S_{k} acts on G by permuting the k-tuples. For a ∈ G, let O_{a} = [a] denote the orbit of a under this action and let O be the set of all such orbits. It is easy to see that these orbits are precisely the subsets P (i_{0}, i_{1}, …, i_{N−1}) defined above, and that each orbit contains a unique representative in the “standard” form

(0^{i_{0}}, 1^{i_{1}}, ⋯, (N − 1)^{i_{N−1}})

where the exponent indicates the number of repetitions of the base. For [a], [b], [c] ∈ O we wish to define a fusion coefficient N_{[a],[b]}^{[c]} in terms of the group G so that the resulting product

[a] · [b] = ∑_{[c]∈O} N_{[a],[b]}^{[c]}[c]

defines a fusion algebra isomorphic to the fusion algebra F. It is clear that the distinguished element Ω of the fusion algebra should correspond to [0] and that the conjugate of [c] should be [−c]. Then we can understand the fusion coefficients by studying the completely symmetric coefficients

N_{[a],[b],[c]} = N_{[a],[b]}^{[−c]}.

We may assume that

\begin{aligned}a & = (0^{α_{0}}, 1^{α_{1}}, ⋯, (N − 1)^{α_{N−1}}) \\ b & = (0^{β_{0}}, 1^{β_{1}}, ⋯, (N − 1)^{β_{N−1}}) \\ c & = (0^{γ_{0}}, 1^{γ_{1}}, ⋯, (N − 1)^{γ_{N−1}}) \end{aligned}

are in standard form. The exponents are nonnegative integers satisfying

∑_{i=0}^{N−1} α_{i} = ∑_{i=0}^{N−1} β_{i} = ∑_{i=0}^{N−1} γ_{i} = k.

The group S_{k} acts on

T ([a], [b], [c]) = {(x, y, z) ∈ [a] × [b] × [c] | x + y + z = 0}

which decomposes into a finite number of orbits under that action. We wish to determine that number of orbits, and show how it is related to the fusion coefficient N_{[a],[b],[c]}. We will show how to do this for N = 2 and for N = 3 in the next two sections, and we hope that future work will give the relationship for all ranks.

Definition. Let M([a], [b], [c]) be the number of orbits of T ([a], [b], [c]) under the action of S_{k}.

Each orbit of T ([a], [b], [c]) under S_{k} has representatives whose third component is in the standard form, and in fact, we may assume that z = c is in that standard form. This makes it clear that the above counting problem is equivalent to counting the number of orbits of

T_{c}([a], [b]) = {(x, y) ∈ [a] × [b] | x + y + c = 0}

under the action of the stabilizer of c in S_{k}, Stab_{S_{k}}(c) = {σ ∈ S_{k} | σ(c) = c}. The structure of Stab_{S_{k}}(c) is obviously the product of symmetric groups S_{γ_{0}} × S_{γ_{1}} × ⋯ × S_{γ_{N−1}} since each subset of γ_{j} occurences of j can be freely permuted without changing c. We now write

x = (x_{0}, x_{1}, …, x_{N−1}) and y = (y_{0}, y_{1}, …, y_{N−1})

where x_{i} is the part of x in the positions where c has i’s and where y_{i} is the part of y in the positions where c has i’s. By the action of the stabilizer of c, we can find a unique representative of the orbit of that stabilizer so that each

x_{i} = (0^{k_{i0}}, 1^{k_{i1}}, …, (N − 1)^{k_{i,N−1}})

is in standard form. This defines the nonnegative integers k_{ij} for 0 ≤ i, j ≤ N − 1, which we use to form the matrix K = [k_{ij}]. Then by the sum property, x_{i}+y_{i}+c_{i} = (0^{γ_{i}}), we must have

y_{i} = ((N − i)^{k_{i0}}, (N − 1 − i)^{k_{i1}}, …, (1 − i)^{k_{i,N−1}}).

The x and y that have these forms are a system of distinct representatives of the orbits, thus the number of such pairs (x, y) is the number M([a], [b], [c]). Below is a picture of the three vectors x_{i}, y_{i} and c_{i}.

\begin{matrix}x_{i} = ( & 0^{k_{i0}} & , & 1^{k_{i1}} & , …, & (N − 1)^{k_{i,N−1}} & ) \\ y_{i} = ( & (N − i)^{k_{i0}} & , & (N − 1 − i)^{k_{i1}} & , …, & (1 − i)^{k_{i,N−1}} & ) \\ c_{i} = ( & i^{k_{i0}} & , & i^{k_{i1}} & , …, & i^{k_{i,N−1}} & ) \end{matrix}

The 3N equations which must be satisfied by the N^{2} unknowns, 0 ≤ k_{ij} ∈ Z, are

∑_{j=0}^{N−1} k_{ij} = γ_{i}, for 0 ≤ i ≤ N − 1

∑_{i=0}^{N−1} k_{ij} = α_{j}, for 0 ≤ j ≤ N − 1

∑_{i=0}^{N−1} k_{i,l−i} = β_{N−l}, for 1 ≤ l ≤ N.

The first two equations are easy to understand from the definitions above as specifications of the sums in each row and column. But the third one, which specifies the sum of each upright diagonal (wrapped as if the matrix were toroidal), may need a little more consideration. When we write out the entries

y_{0} =  (N)^{k_{00}} (N − 1)^{k_{01}} …  (1)^{k_{0,N−1}}
y_{1} = (N − 1)^{k_{10}} (N − 2)^{k_{11}} …  (0)^{k_{1,N−1}}
y_{2} = (N − 2)^{k_{20}} (N − 3)^{k_{21}} … (N − 1)^{k_{2,N−1}}
⋮
y_{N−1} = (1)^{k_{N−1,0}} (0)^{k_{N−1,1}} … (2)^{k_{N−1,N−1}}

we see that the base of the exponent k_{ij} is N − (i + j) modulo N, so the total number of occurences of base N − l in y is that upright diagonal sum.

There are N^{2} variables, and there are 3N equations, but the consistency of sums of different types means that 2 equations (of different types, say one column and one diagonal equation) are redundant. Therefore the dimension of the solution space (over Q) is N^{2} − (3N − 2) = (N − 1)(N − 2), which is 0 for N = 2, 2 for N = 3, and 6 for N = 4.

4. The case of N = 2.

For N = 2 with

a = (0^{α_{0}}, 1^{α_{1}})  b = (0^{β_{0}}, 1^{β_{1}})  c = (0^{γ_{0}}, 1^{γ_{1}})

and

α_{0} + α_{1} = β_{0} + β_{1} = γ_{0} + γ_{1} = k

we have the equations

\begin{matrix}k_{00} + k_{01} = γ_{0} & k_{10} + k_{11} = γ_{1} \\ k_{00} + k_{10} = α_{0} & k_{01} + k_{11} = α_{1} \\ k_{01} + k_{10} = β_{1} & k_{00} + k_{11} = β_{0} \end{matrix} .

Three of these involving k_{00} allow us to write

K = [ \begin{matrix}k_{00} & k_{01} \\ k_{10} & k_{11} \end{matrix} ] = [ \begin{matrix}k_{00} & γ_{0} − k_{00} \\ α_{0} − k_{00} & β_{0} − k_{00} \end{matrix} ] ,

so if a solution exists, then k_{00} determines the matrix uniquely. Each of the other equations is equivalent to

2k_{00} = α_{0} + β_{0} + γ_{0} − k = 2k − (α_{1} + β_{1} + γ_{1})

so

0 ≤ k_{00} = k − \frac{1}{2}(α_{1} + β_{1} + γ_{1}) ∈ Z

imposes the requirement that

\frac{1}{2}(α_{1} + β_{1} + γ_{1}) ∈ Z.

Since all entries of the matrix are nonnegative, we have 0 ≤ k_{00} ≤ min(α_{0}, β_{0}, γ_{0}) so

0 ≤ α_{0} + β_{0} + γ_{0} − k ≤ 2 min(α_{0}, β_{0}, γ_{0})

which is equivalent to

max(2α_{1}, 2β_{1}, 2γ_{1}) ≤ α_{1} + β_{1} + γ_{1} ≤ 2k.

Rewriting this as

max(α_{1} − β_{1}, β_{1} − α_{1}, 2γ_{1} − (α_{1} + β_{1})) ≤ γ_{1} ≤ 2k − (α_{1} + β_{1})

and using 2γ_{1} − (α_{1} + β_{1}) ≤ γ_{1} iff γ_{1} ≤ α_{1} + β_{1}, we get the condition

max(α_{1} − β_{1}, β_{1} − α_{1}) ≤ γ_{1} ≤ min(2k − (α_{1} + β_{1}), α_{1} + β_{1})

which is equivalent to

|α_{1} − β_{1}| ≤ γ_{1} ≤ α_{1} + β_{1}  and  α_{1} + β_{1} + γ_{1} ≤ 2k.

Theorem 3. With notation as above, for 1 ≤ k ∈ Z, we have M([a], [b], [c]) = 1 if

|α_{1} − β_{1}| ≤ γ_{1} ≤ α_{1} + β_{1}, α_{1} + β_{1} + γ_{1} ≤ 2k and \frac{1}{2}(α_{1} + β_{1} + γ_{1}) ∈ Z,

M([a], [b], [c]) = 0 otherwise.

Corollary. When N = 2, with notation as above, and with a_{1} = α_{1}/2, b_{1} = β_{1}/2 and c_{1} = γ_{1}/2, we have M([a], [b], [c]) = N_{[a_{1}],[b_{1}],[c_{1}]}.

Proof. The element a = (0^{α_{0}}, 1^{α_{1}}) ∈ Z_{2}^{k} corresponds to a level k highest weight Λ = α_{0}Λ_{0} + α_{1}Λ_{1} = kc + α_{1}λ_{1}, for 0 ≤ α_{1} ≤ k integral. The associated irreducible highest weight sl_{2}-module is then V_{α_{1}/2} if we use the spin a_{1} = α_{1}/2 to label the module. The labels for the sl_{2}-modules associated with b and c are b_{1} = β_{1}/2 and c_{1} = γ_{1}/2. Since c = −c ∈ Z_{2}^{k}, N_{[a_{1}],[b_{1}],[c_{1}]} = N_{[a_{1}],[b_{1}]}^{[c_{1}]}, and the conditions in the theorem above translate exactly into the conditions of the fusion rules for sl_{2}.

5. The case of N = 3.

For N = 3, after using two column sums, two row sums and one diagonal sum to rewrite the entries in the last row and column of K, we have

K = [ \begin{matrix}k_{00} & k_{01} & γ_{0} − k_{00} − k_{01} \\ k_{10} & k_{11} & γ_{1} − k_{10} − k_{11} \\ α_{0} − k_{00} − k_{10} & α_{1} − k_{01} − k_{11} & β_{2} − k_{01} − k_{10} \end{matrix} ] .

This leaves 4 more equations:

\begin{aligned}[α_{0} − k_{00} − k_{10}] + [α_{1} − k_{01} − k_{11}] + [β_{2} − k_{01} − k_{10}] & = γ_{2}, \\ [γ_{0} − k_{00} − k_{01}] + [γ_{1} − k_{10} − k_{11}] + [β_{2} − k_{01} − k_{10}] & = α_{2}, \\ k_{00} + [γ_{1} − k_{10} − k_{11}] + [α_{1} − k_{01} − k_{11}] & = β_{0}, \\ [γ_{0} − k_{00} − k_{01}] + k_{11} + [α_{0} − k_{00} − k_{10}] & = β_{1}. \end{aligned}

Using the relation

k = α_{0} + α_{1} + α_{2} = β_{0} + β_{1} + β_{2} = γ_{0} + γ_{1} + γ_{2}

we find that two of the four equations are redundant, and the system is equivalent to

\begin{aligned}k_{00} + 2k_{01} + 2k_{10} + k_{11} & = k − α_{2} + β_{2} − γ_{2}, \\ 3(k_{00} − k_{11}) & = (α_{0} + β_{0} + γ_{0}) − (α_{1} + β_{1} + γ_{1}). \end{aligned}

Letting

A = \frac{1}{3}[(α_{0} + β_{0} + γ_{0}) − (α_{1} + β_{1} + γ_{1})]

the solutions to the linear system above are

k_{00} = k_{11} + A k_{01} = −k_{10} − k_{11} + A + α_{1} − β_{0} + γ_{1}

where k_{10} and k_{11} are the “free variables”. In fact, because the entries of the matrix K are nonnegative integers, we have A ∈ Z and the following nine inequalities:

0 ≤ k_{00} 0 ≤ k_{01} 0 ≤ k_{10} 0 ≤ k_{11} k_{01} + k_{10} ≤ β_{2}
k_{00} + k_{10} ≤ α_{0} k_{01} + k_{11} ≤ α_{1} k_{00} + k_{01} ≤ γ_{0} k_{10} + k_{11} ≤ γ_{1}.

These translate into the following inequalities on the free variables k_{10} and k_{11}:

−A ≤ k_{11}  k_{10} + k_{11} ≤ A + α_{1} − β_{0} + γ_{1}  0 ≤ k_{10}  0 ≤ k_{11}
k_{10} + k_{11} ≤ α_{0} − A  A − β_{0} + γ_{1} ≤ k_{10}  2A + α_{1} − β_{0} + γ_{1} − γ_{0} ≤ k_{10}
k_{10} + k_{11} ≤ γ_{1} A + α_{1} − β_{0} − β_{2} + γ_{1} ≤ k_{11}.

Combining these, we can say that

C = max(0, A − β_{0} + γ_{1}, 2A + α_{1} − β_{0} + γ_{1} − γ_{0}) ≤ k_{10}
D = max(−A, 0, A + α_{1} − β_{0} − β_{2} + γ_{1}) ≤ k_{11}
k_{10} + k_{11} ≤ min(A + α_{1} − β_{0} + γ_{1}, α_{0} − A, γ_{1}) = E.

It is clear that when E − C ≥ D and A ∈ Z, these constraints define a “45-45-90” triangle in the first quadrant of the k_{10}k_{11}-plane whose vertices have integral coordinates. The number M([a], [b], [c]) is exactly the number of integral points within that triangle, including its boundary. The number of points on one leg of the triangle is clearly E − D − C + 1 so that M([a], [b], [c]) is a triangular number.

Theorem 4. With notation as above, for 1 ≤ k ∈ Z, we have

M([a], [b], [c]) = \frac{(E − D − C + 1)(E − D − C + 2)}{2} = \binom{E − D − C + 2}{2}.

We will now show that E − D − C + 1 is equal to the fusion rule coefficient N_{[a],[b],[c]} by using the theorem of [BMW]. The level k highest weights, λ, μ and ν, referred to in the [BMW] theorem, correspond to a, b and c in Z_{3}^{k} so that

λ = α_{1}λ_{1} + α_{2}λ_{2} ,  μ = β_{1}λ_{1} + β_{2}λ_{2} ,  ν = γ_{1}λ_{1} + γ_{2}λ_{2}.

We have

A = \frac{1}{3} [2(α_{1} + β_{1} + γ_{1}) + (α_{2} + β_{2} + γ_{2})] ,

B = \frac{1}{3} [(α_{1} + β_{1} + γ_{1}) + 2(α_{2} + β_{2} + γ_{2})] ,

so that

A = k − A ,  2A − B = α_{1} + β_{1} + γ_{1},  2B − A = α_{2} + β_{2} + γ_{2},

and

A + B = (α_{1} + β_{1} + γ_{1}) + (α_{2} + β_{2} + γ_{2}).

The variables used in the [BMW] theorem are

k_{0}^{max} = min(A, B)
k_{0}^{min} = max(α_{1} + α_{2}, β_{1} + β_{2}, γ_{1} + γ_{2}, A − min(α_{1}, β_{1}, γ_{1}), B − min(α_{2}, β_{2}, γ_{2}))
= max(k − min(α_{0}, β_{0}, γ_{0}), A − min(α_{1}, β_{1}, γ_{1}), B − min(α_{2}, β_{2}, γ_{2})).

The [BMW] theorem says that the fusion coefficient

N_{λ,μ,ν}^{(k)} = min(k_{0}^{max}, k) − k_{0}^{min} + 1

if the conditions

k_{0}^{min} ≤ k_{0}^{max} , k_{0}^{min} ≤ k and A, B ∈ Z

are satisfied, and the coefficient is zero otherwise. Using the formulas above, it is straightforward to check that

C = max(0, β_{1} + β_{2} + γ_{1} − A, B − A − α_{2} + γ_{1}),
D = A − min(A, B, k) = A − min(k_{0}^{max}, k),
E = A − max(α_{1} + α_{2}, A − γ_{1}, B − β_{2}).

Two identities useful for checking this are

2A + α_{1} − β_{0} + γ_{1} − γ_{0} = B − A − α_{2} + γ_{1},
A + α_{1} − β_{0} − β_{2} + γ_{1} = B − (α_{2} + β_{2} + γ_{2}).

From these expressions for C, D and E we can now compute

E − D − C + 1 = min(k_{0}^{max}, k) + 1−

{max(0, β_{1} + β_{2} − A + γ_{1}, B − A − α_{2} + γ_{1}) + max(α_{1} + α_{2}, A − γ_{1}, B − β_{2})}

= min(k_{0}^{max}, k) + 1−

max(α_{1} + α_{2}, A − γ_{1}, B − β_{2}, B − γ_{2}, β_{1} + β_{2}, A − α_{1}, A − β_{1}, B − α_{2}, γ_{1} + γ_{2})

= min(k_{0}^{max}, k) + 1 − k_{0}^{min}.

We also have

E − C = A − k_{0}^{min}

so the condition E − C ≥ D is equivalent to the condition

min(k_{0}^{max}, k) ≥ k_{0}^{min},

that is,

k_{0}^{max} ≥ k_{0}^{min} and k ≥ k_{0}^{min}.

It is clear that A ∈ Z iff A ∈ Z iff B ∈ Z since A = k − A and A + B ∈ Z. We have shown that the conditions for the fusion coefficient to be nonzero are equivalent to the conditions for M([a], [b], [c]) ≠ 0. We have now proved our main result.

Theorem 5. For N = 3, the number of orbits of T ([a], [b], [c]) under the action of S_{k} is related to the fusion coefficient N_{[a],[b],[c]} by

M([a], [b], [c]) = \binom{N_{[a],[b],[c]} + 1}{2}.

Another form of the conditions and number of orbits can be given as follows. For i, j, l, r ∈ {0, 1, 2} taken modulo 3, let

s_{ijl} = α_{i} + β_{j} + γ_{l},

and

D_{ij}^{r} = s_{i,i+r,i+2r} − s_{j,j+r,j+2r}.

Then, using the fact that for (i, j, l) any permutation of (0, 1, 2),

s_{i,i+r,i+2r} + s_{l,l+r,l+2r} = 3k − s_{j,j+r,j+2r},

we get that

D_{ij}^{r} − D_{jl}^{r} = 3(k − s_{j,j+r,j+2r})

so

D_{ij}^{r} ≡ D_{jl}^{r} (mod 3).

In fact, we have that D_{01}^{0} ≡ D_{12}^{0} ≡ D_{20}^{0} ≡ D_{01}^{1} ≡ D_{12}^{1} ≡ D_{20}^{1} ≡ D_{01}^{2} ≡ D_{12}^{2} ≡ D_{20}^{2} (mod 3). This makes the necessary condition written below (3|D) for existence of a solution, independent of which D_{ij}^{r} one checks for divisibility by 3.

Theorem 6. [Zaslavsky] Let D be any D_{i,i+1}^{r} defined above and define m by

3m = min(s_{000}, s_{111}, s_{222}) + min(s_{012}, s_{120}, s_{201}) + min(s_{021}, s_{102}, s_{210}) − 2k.

If 3 | D and m ≥ 0 then we have m ∈ Z and M([a], [b], [c]) = \binom{m+2}{2}. Otherwise, M([a], [b], [c]) = 0.

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Dept. of Math. Sci., The State University of New York, Binghamton, New York 13902-6000

E-mail address: alex@math.binghamton.edu

Math. Dept., Pennsylvania State University, Altoona, Pennsylvania 16601 E-mail address: mdw8@psu.edu