Let $P\_{\lambda}$ be a Young symmetriser associated to the following tableau $(a\_1 a\_2 a\_3 b\_3 ; b\_1 b\_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take $a\_i,b\_i \in V $ basis vectors and I define the action of Young projectors on tensor products in the obvious way. 


Let us construct the following tensor,


$w\_1 = P\_{\lambda} (a\_1 \otimes a\_2 \otimes a\_3 \otimes b\_1 \otimes b\_2 \otimes b\_3)$


I can define the following homomorphism $F$, 


$F:a\_1 \wedge a\_2 \wedge a\_3 \otimes b\_1 \wedge b\_2 \wedge b\_3 \to A\_a A\_b w\_1$


Here $A\_a A\_b$ antisymmetrises $a\_i,b\_i$ respectively.


**QUESTION 1)** Am I right in thinking that a necessary condition of the fact that $(2^2,1^2)$ (which corresponds to our tableau $\lambda$)is an irreducible of $(1^3 \otimes 1^3)$ is that THERE EXISTS A NON-TRIVIAL HOMOMORPHISM LIKE THE ONE I DEFINED ABOVE. Why is it the case?


**QUESTION 2)** Is this homomorphism unique? I suspect it is unique in this case but is it true in general that one can define only one such homomorphism?



Homomorphisms from irreducible spaces to reducible spaces
I assume you are talking about representations of $S\_n$ over $\mathbb{C}$. In this case, $\mathbb{C}S\_n$ is left semi-simple by Maschke's Theorem, so every representation of $S\_n$ is a direct sum of irreducible representations.


Next is Schur's Lemma: Let $V$ and $W$ be irreducible representations of $S\_n$. Then any homormorphism $\phi:V\to W$ is either 0 or an isomorphism.


proof: $\phi(V)\subset W$ and $\ker(\phi)\subset V$ are $S\_n$ submodules. QED


Corollary: The map $\phi$ above is scalar multiplication by a constant.


proof: $V$ is a $\mathbb{C}$-vector space, hence $\phi$ has an eigenvector $v\in V$ with $\phi(v) = cv$ for some $c\in \mathbb{C}$. Then $v\in\mathrm{ker}(\phi-cId\_V)\neq 0$ so $\phi=cId\_V$ by Schur's lemma. QED


Now, let $V$ be irreducible and $W$ be any submodule. Write $W=\bigoplus\_i V\_i^{\oplus m\_i}$ for non-isomorphic $V\_i$. By the previous two statements, any nonzero map $\phi:V\to W$ must factor as
$$\phi:V\cong V\_i\hookrightarrow W$$
for some $i$. This proves Q1. Oh, and Q2 too (after Abdelmalek Abdesselam's correction).


