Homomorphisms from irreducible spaces to reducible spaces

Question

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?

Answer 1

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).