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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?

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  • $\begingroup$ Q1: yes. Q2: the number of linearly independent morphisms of this sort is precisely the multiplicity of the irreducible inside the reducible space. Such a homomorphism is also what invariant theorists in the 1850's and following years called covariants or mixed concomitants. $\endgroup$ Oct 13, 2014 at 18:32
  • $\begingroup$ Thanks a lot for your answer. How does one show that the answer to the Q1 is Yes? and prove the answer to the q2 is multiplicity? Can you please let me know some references where in I can learn about these stuff? and see how to concretely define these hom's in general case.I am a physicist and I am not an expert on these. Kind regards and thanks again for your reply. $\endgroup$
    – vishmay
    Oct 13, 2014 at 18:59
  • $\begingroup$ I would look up the book "Young Tableaux" by William Fulton and especially Chapter 8 on the representations of the general linear group. $\endgroup$ Oct 13, 2014 at 19:20

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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).

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  • $\begingroup$ Many thanks for your answer. For the corollary does'nt $W = V$ because W can be an equivalent but different representation? If $W \neq V$ then is this Isormophism unique? $\endgroup$
    – vishmay
    Oct 20, 2014 at 14:38

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