On products of sln characters and support containment

Abstract

Let λ, μ, ν and ρ be dominant weights of sl n satisfying λ + μ = ν + ρ. Let V λ denote the highest weight module corresponding to λ. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for V λ ⊗ V μ to be contained in V ν ⊗ V ρ as sl n -modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible sl n -module which appears in the decomposition of V λ ⊗ V μ does appear in the decomposition of V ν ⊗ V ρ .

Schur Positivity and Kirillov-Reshetikhin Modules

“…We should remark here that a similar result on Schur positivity on tensor products of simple g-modules of arbitrary highest weight λ was conjectured in [7] and [3] (see Section 3.5 for more details). Our result suggests, that this generalized Schur positivity may hold along the partial order on tensor products of q → 1 limits of minimal affinizations of V (λ) (the "minimal" module of the quantum affine algebra having a simple quotient whose limit is isomorphic to V (λ), see [5] for more details).…”

“…In the sl n -case, the restriction to sl n of the limit of such a minimal affinization is nothing but the simple sl n -module V (λ), so this is the conjecture of Schur positivity by Lam, Postnikov and Pylyavskyy, cited in [7]. For other types, the limit of a minimal affinization is not a simple g-module in general, for example Kirillov-Reshetikhin modules are minimal affinizations of V (mω i ).…”

“…As the characters are also known as Schur functions, this property of the left hand side is also known as Schur positivity. A generalization of this order to sl n (C) and further to a simple finite-dimensional Lie algebra g of arbitrary type was investigated in [7] (resp. [3,10]).…”

Untitled

“…The row-shuffle conjecture was proved in [Lam et al 2007], showing that our conjecture is true in the case of sl n+1 for the pair λ, µ where µ is the maximal element in the poset. A completely different approach taken in [Dobrovolska and Pylyavskyy 2007] also gives some evidence for our conjecture in the case of sl n+1 .…”

“…Suppose g is of type A n ; then Lam, Postnikov and Pylyavskyy stated the conjecture in 2.3 in an unpublished work (we may refer here to [Dobrovolska and Pylyavskyy 2007]). The following first step in proving this conjecture in the A n -case has been taken in [Dobrovolska and Pylyavskyy 2007]. It is shown there that for (µ 1 , µ 2 ) (λ 1 , λ 2 ) ∈ P + (λ, 2) and ν…”