the sum of all ${E({\nu})}^*$ with ${V({\nu})}\subset {V({{\lambda}})}\otimes {V({\mu})}$.

As a consequence we obtain the following Lemma:

\[lem:coefficientimatriciali\] Let $\nu, \nu'$ be dominant weights, then the image of ${E({\nu})}^*
\otimes {E({\nu'})}^*$ in ${\Gamma}(M, {\mathcal M}_{\nu+\nu'})$ via the multiplication map is $$\bigoplus_{{V({\mu})}\subset {V({\nu})}\otimes {V({\nu'})}}\!\!\!\!s^{\nu+\nu'-\mu}{E({\mu})}^*.$$

Indeed let $ \pi : G {\rightarrow}M$ be the map induced by the inclusion $G_\mathrm{ad}\subset M$. Then any line bundle on $G$ can be trivialized so that the image of $\pi^*:{E({{\lambda}})}^*\subset{\Gamma}(M,{\mathcal M}_\nu){\longrightarrow}{\Bbbk}[G]$ is the image of $c_{{V({{\lambda}})}}$ and the claim follows from previous remarks.

Together with Lemma \[lem:general-normality\], this gives the following

\[prp:normalita\] The variety $X_\Sigma$ is normal if and only if, for every $\mu \in {\Lambda}^+$ such that $\mu {\leqslant}{\lambda}$, there exist $n>0$ and ${\lambda}_1,\dots,{\lambda}_n\in \Sigma$ such that $${V({\mu + (n-1){\lambda}})} \subset {V({{\lambda}_1})}\otimes \cdots \otimes {V({{\lambda}_n})} .$$

Remarks on tensor products {#ssez:prodottitensore}
--------------------------

By Proposition \[prp:normalita\], in order to establish the normality (or the non-normality) of $X_\Sigma$, we need some results on tensor product decomposition.

\[lem:riduzionelevi\] Let ${\lambda}, \mu, \nu$ be dominant weights and let ${\Delta}'\subset {\Delta}$ be such that $\operatorname{Supp}_{\Delta}({\lambda}+ \mu - \nu) \subset {\Delta}'$; let $L \subset G$ be the standard Levi subgroup associated to ${\Delta}'$. If $\pi \in {\Lambda}^+$, denote by ${V_L({\pi})}$ the 