Let be a complex finite-dimensional simple Lie algebra. Given a positive integer and a dominant weight , we define a preorder on the set of -tuples of dominant weights which add up to . Let be the equivalence relation defined by the preorder and be the corresponding poset of equivalence classes. We show that if is a multiple of a fundamental weight (and is general) or if (and is general), then coincides with the set of -orbits in , where acts on as the permutations of components. If is of type and , we show that the -orbit of the row shuffle defined by Fomin et al. (2005) is the unique maximal element in the poset.
Given an element of , consider the tensor product of the corresponding simple finite-dimensional -modules. We show that (for general , , and ) the dimension of this tensor product increases along . We also show that in the case when is a multiple of a fundamental minuscule weight ( and are general) or if is of type and ( is general), there exists an inclusion of tensor products along with the partial order on . In particular, if is of type , this means that the difference of the characters is Schur positive.
"Posets, tensor products and Schur positivity." Algebra Number Theory 8 (4) 933 - 961, 2014. https://doi.org/10.2140/ant.2014.8.933