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 $F$,

$F:w_1 \to A_aA_b w_1 \in \bigwedge^3 V \otimes \bigwedge^3 V$

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 $(2^2,1^4)$ (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?

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?

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 $F$,

$F:A_a A_b w_1 \to (a_1 \wedge a_2 \wedge a_3) \otimes (b_1 \wedge b_2 \wedge b_3)$

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 $(2^2,1^4)$ (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?

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 $F$,

$F:w_1 \to A_aA_b w_1 \in \bigwedge^3 V \otimes \bigwedge^3 V$

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 $(2^2,1^4)$ (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?