#### Vol. 8, No. 4, 2014

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Posets, tensor products and Schur positivity

### Vyjayanthi Chari, Ghislain Fourier and Daisuke Sagaki

Vol. 8 (2014), No. 4, 933–961
##### 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 ${P}^{+}\left(\lambda ,k\right)∕\phantom{\rule{0.3em}{0ex}}\sim$ 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 ${P}^{+}\left(\lambda ,k\right)∕\phantom{\rule{0.3em}{0ex}}\sim$ 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 ${P}^{+}\left(\lambda ,k\right)∕\phantom{\rule{0.3em}{0ex}}\sim$. In particular, if $\mathfrak{g}$ is of type ${A}_{n}$, this means that the difference of the characters is Schur positive.

##### Keywords
Schur positivity, tensor products, posets, Lie algebras
Primary: 17B67
##### Milestones
Received: 12 April 2013
Accepted: 15 August 2013
Published: 10 August 2014
##### Authors
 Vyjayanthi Chari Department of Mathematics University of California Riverside, CA 92521 United States Ghislain Fourier Mathematisches Institut Universität zu Köln Germany Daisuke Sagaki Department of Mathematics University of Tsukuba Japan