Posets, tensor products and Schur positivity

Abstract

Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra. Given a positive integer $k$ and a dominant weight $\lambda$, we define a preorder $\preccurlyeq$ on the set $P^{+}\left(\lambda ,k\right)$ of $k$-tuples of dominant weights which add up to $\lambda$. Let $\sim$ be the equivalence relation defined by the preorder and $P^{+}\left(\lambda ,k\right)∕\sim$ be the corresponding poset of equivalence classes. We show that if $\lambda$ is a multiple of a fundamental weight (and $k$ is general) or if $k=2$ (and $\lambda$ is general), then $P^{+}\left(\lambda ,k\right)∕\sim$ coincides with the set of $S_k$-orbits in $P^{+}\left(\lambda ,k\right)$, where $S_k$ acts on $P^{+}\left(\lambda ,k\right)$ as the permutations of components. If $\mathfrak{g}$ is of type $A_n$ and $k=2$, we show that the $S_2$-orbit of the row shuffle defined by Fomin et al. (2005) is the unique maximal element in the poset.

Given an element of $P^{+}\left(\lambda ,k\right)$, consider the tensor product of the corresponding simple finite-dimensional $\mathfrak{g}$-modules. We show that (for general $\mathfrak{g}$, $\lambda$, and $k$) the dimension of this tensor product increases along $\preccurlyeq$. We also show that in the case when $\lambda$ is a multiple of a fundamental minuscule weight ($\mathfrak{g}$ and $k$ are general) or if $\mathfrak{g}$ is of type $A_2$ and $k=2$ ($\lambda$ is general), there exists an inclusion of tensor products along with the partial order $\preccurlyeq$ on $P^{+}\left(\lambda ,k\right)∕\sim$. In particular, if $\mathfrak{g}$ is of type $A_n$, this means that the difference of the characters is Schur positive.