On Kostant's conjecture for components of $V(\rho )\otimes V(\rho )$

For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $\rho$ be the half sum of positive roots and $P(2\rho )\subset {\mathbb{R}}^r$ be the convex hull of all dominant weights $\lambda$ of the form $\lambda =2\rho -{\sum }_{i=1}^r{a}_i{\alpha }_i$ with $a_i\in {\mathbb{Z}}_{\ge 0}$ for $1\le i\le r$. We show that if $\lambda$ is a vertex of $P(2\rho )$, then $V(\lambda )$ appears in $V(\rho )\otimes V(\rho )$ with multiplicity one, proving partially (for the vertices of $P(2\rho )$) a conjecture of Kostant describing components of $V(\rho )\otimes V(\rho )$. This result allows us to give an alternative proof for a weaker form of the conjecture (up to saturation factor) for any $\mathfrak{g}$. Further, using works of Knutson-Tau on the saturation property of $\mathfrak{s}{\mathfrak{l}}_{\mathfrak{r}+\mathfrak{1}}$, our results give an alternative proof of Kostant’s conjecture in the particular case $\mathfrak{g}=\mathfrak{s}{\mathfrak{l}}_{\mathfrak{r}+\mathfrak{1}}$.