Tensor product decomposition of ̂𝔰𝔩(𝔫) modules and identities

Abstract

We decompose the sl(n)-module V (Λ 0 ) ⊗ V (Λ i ) and give generating function identities for the outer multiplicities. In the process we discover some seemingly new partition identities for n = 3, 4.

Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

“…Thus, once a description of Γ Φ for Φ of level 2 is given, the steps of the proof of Proposition 2.6.2 also lead to a formula for outer multiplicities of tensor products of the form V (Λ i ) ⊗ V (Λ j ) in terms of partitions with bounded parts. As mentioned above, such outer multiplicities were computed in [25] using very different methods. Moreover, we have seen that, even in the sl 2 case, the two methods lead to different expressions in terms of partitions.…”

“…By comparing the expressions from [24,25] with Proposition 2.6.2, we obtain the partition identities (2.6.4). We believe that the results of [25] will also be helpful for carrying out the algorithm of [12] to obtain, in the spirit of [26], expressions for the multiplicities of the level-2 Demazure flags of Weyl modules for type A in terms of partitions associated to colored Young diagrams. We remark that, even in the sl 2 -case, this may lead to different expressions than those given in [6].…”

“…However, it is well-known that more direct descriptions lead to deep connections with combinatorics, number theory, and mathematical physics. For instance, in the case of affine Lie algebras this leads to proofs of Rogers-Ramanujan-type identities as well as partition identities (see [8,20,24,25] and references therein).…”

“…In particular, the subset of D(W ) which contributes to the summation is explicitly characterized. It is interesting to observe that, also for ℓ = 1, the outer multiplicities were recently calculated in [24,25] as the cardinality of a different set of partitions (for g of type A). By comparing the expressions from [24,25] with Proposition 2.6.2, we obtain the partition identities (2.6.4).…”

“…It is interesting to observe that, also for ℓ = 1, the outer multiplicities were recently calculated in [24,25] as the cardinality of a different set of partitions (for g of type A). By comparing the expressions from [24,25] with Proposition 2.6.2, we obtain the partition identities (2.6.4). We believe that the results of [25] will also be helpful for carrying out the algorithm of [12] to obtain, in the spirit of [26], expressions for the multiplicities of the level-2 Demazure flags of Weyl modules for type A in terms of partitions associated to colored Young diagrams.…”