Posets, Tensor Products and Schur positivity

Abstract

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight λ, we define a preorder on the set $P(λ, k)$ of k-tuples of dominant weights which add up to λ. Let $P(λ, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if λis a multiple of a fundamental weight (and k is general) or if k=2 (and λis general), then $P(λ, k)/\sim$ coincides with the set of S_k-orbits in $P(λ,k)$, where S_k acts on $P(λ, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row shuffle defined by Fomin et al is the unique maximal element in the poset. Given an element of $P(λ, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, λ, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when λis a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (λis general), there exists an inclusion of tensor products of g-modules along with the partial order. In particular, if g is of type A_n, this means that the difference of the characters is Schur positive.