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

Abstract

Abstract. We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for sl2 in terms of partitions with bounded parts, which subsequently lead to certain partition identities.

Highest-weight vectors in tensor products of Verma modules for U_q(sl_2)

“…The study of tensor products of representations of (classical and quantum) Kac-Moody algebras is particularly relevant as it leads to deep connections with combinatorics and has important applications to mathematical physics (see [1,5,8,9] and references therein). For instance, the study of highest-weight vectors in tensor products of two finite-dimensional irreducible modules for U q (sl 2 ) can be used to compute the celebrated Clebsch-Gordan coefficients which are numbers that arise in angular momentum coupling in quantum mechanics (see for instance [9,Sections 3.4 and 3.5]).…”

Combinatorial aspects of affine Weyl groups and orbits of dominant weights

“…The main result of (JAKELI Ć; MOURA, 2018) explores the connection between two kinds of multiplicity problems, establishing a formula for computing one of them in terms of the other. On one hand, there is the problem of computing the so-called outer multiplicities for tensor products of simple modules in the category of integrable weight modules for an affine Kac-Moody algebra.…”

“…In addition to the result about Demazure flags from (CHARI et al, 2014), the other main ingredient used in (JAKELI Ć;MOURA, 2018) to obtain such expression for outer multiplicities was an explicit description of the elements of the affine Weyl group or, more importantly, the characterization of the elements of the orbit of a given dominant affine weight whose projections onto the weight lattice of g is dominant. More precisely, let Ŵ denote the affine Weyl group, P the affine weight lattice, P the underlying finite type weight lattice so that P = CΛ 0 ⊕ P ⊕ Cδ, where Λ 0 is the highest weight of the basic representation of the affine Kac-Moody algebra and δ is the generator of imaginary roots, and let π ∶ P → P be the associated projection.…”

“…More precisely, let Ŵ denote the affine Weyl group, P the affine weight lattice, P the underlying finite type weight lattice so that P = CΛ 0 ⊕ P ⊕ Cδ, where Λ 0 is the highest weight of the basic representation of the affine Kac-Moody algebra and δ is the generator of imaginary roots, and let π ∶ P → P be the associated projection. The computation in (JAKELI Ć;MOURA, 2018) relies on an explicit description of the set Γ Λ ∶= {wΛ ∶ π(wΛ) is dominant} for every dominant affine weight Λ. While such computations for type Â1 are straightforward, moving to higher rank they are much more challenging, even for type Â2 .…”