This question is in relation to my previously asked question here Construction of an Irreducible Module as a Direct Summand.
Let $V_0$ be any arbitrary finite dimensional $sl_{\ell +1}(\mathbb{C})$ module and $V$ be the standard $(\ell+1)$ dimensional representation of $sl_{\ell +1}(\mathbb{C})$ . Then does there exist $m \in$ $\mathbb{N}$ such that $V_0$ occurs as an irreducible summand of $V^{\otimes{m}}$(the decomposition is upto isomorphism of $sl_{\ell+1}(\mathbb{C})$ modules)?
I have tried to solve this problem. Can anybody please verify if the solution is ok? Thanks for any help.
We know that $V_0$ is isomorphic to a $sl_{\ell+1}(\mathbb{C})$ module of highest weight $\lambda$, say $V(\lambda)$ where $\lambda$ is a dominant integral weight. Now if $\omega_1$,.....,$\omega_\ell$ denote the fundamental weights of the root system $A_\ell$, then we know that $\omega_i$ = $\epsilon_1$ + ....+$\epsilon_i$ where $i$ = $1,....,\ell$ and the highest weight of $(\wedge^1(V))^{\otimes{k_1}}$ $\otimes$ ......$\otimes$ $(\wedge^\ell(V))^{\otimes{k_\ell}}$ is $\lambda$ = $k_1\omega_1$ + $k_2\omega_2$ +....+$k_{\ell-1}\omega_{\ell-1}$ + $k_\ell\omega_\ell$, $k_i$ $\in$ $\mathbb{N}$ $\cup${0}.So we can define the projection map onto the wedge product given by $\phi$ : $V^{\otimes{m}}$ $\longrightarrow$ $(\wedge^1(V))^{\otimes{k_1}}$ $\otimes$ ......$\otimes$ $(\wedge^\ell(V))^{\otimes{k_\ell}}$ where $m$ = $k_1$ + $2k_2$ +........+$(\ell-1)k_{\ell-1}$ +$\ell k_{\ell}$. This is clearly a homomorphism of $sl_{\ell+1}(\mathbb{C})$ module and onto. So, we have $V^{\otimes{m}}$ $\simeq$ $ker\phi$ $\oplus$ $(\wedge^1(V))^{\otimes{k_1}}$ $\otimes$ ......$\otimes$ $(\wedge^\ell(V))^{\otimes{k_\ell}}$ by complete reducibility ($\because$ $sl_{\ell+1}(\mathbb{C})$ is semisimple) where we know that $V(\lambda)$ occurs as a direct summand in $(\wedge^1(V))^{\otimes{k_1}}$ $\otimes$ ......$\otimes$ $(\wedge^\ell(V))^{\otimes{k_\ell}}$. Thus we have our required $m\in$ $\mathbb{N}$. If anyone has got some other proof or suggestions or if this can be argument can be further simplified, you are most welcome to provide me.
The following more general fact is true: if $G$ is a compact Lie group and $V$ is a faithful representation of it, then any finite-dimensional representation of $G$ occurs as a direct summand of $V^{\otimes n} \otimes V^{\ast \otimes m}$ for some $n, m$. For a proof see this MO question.
For application to this question take $G = SU(\ell+1)$, which has the same finite-dimensional representation theory (over $\mathbb{C}$) as $\mathfrak{sl}_{\ell+1}$. The standard representation $V$ is faithful, and moreover it has the property that $V^{\ast}$ occurs as a direct summand of a tensor power of $V$. In fact we have that $\wedge^k V$ is a direct summand of $V^{\otimes k}$, and
$$\wedge^{\ell} V \cong V^{\ast}$$
(because $\wedge^{\ell+1} V$ is trivial).