My questions may turn out to be related to Schur functors.


If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{g}$, I am interested in restricting the representation to an $\mathfrak{sl}(2)$ subalgebra of $\mathfrak{g}$. For example, if $\mathfrak{g} = \mathfrak{sl}(n)$, then I am interested in $V = \mathbb{C}^n$ being the standard representation. If $\mathfrak{g} = \mathfrak{so(n)}$, I am interested in $V = \mathbb{C}^n$, i.e. in the vector representation. Similarly if $\mathfrak{g} = Sp(m, \mathbb{C})$, I am interested in $V = \mathbb{C}^{2m}$ being its standard representation. I am also interested in the $5$ exceptional cases and their fundamental representations.


As for the $\mathfrak{sl}(2)$ subalgebra of $\mathfrak{g}$, I am mostly interested in "the" so-called "regular" $\mathfrak{sl}(2)$ subalgebra (it is unique up to conjugation in $\mathfrak{g}$), which was studied for instance by Kostant and others.


I can now formulate my questions. From now on, consider the representation $V$, but view it as a representation of a fixed $\mathfrak{sl}(2)$ subalgebra of $\mathfrak{g}$ (for the sake of this post, one may for instance assume it to be a regular $\mathfrak{sl}(2)$ subalgebra).



> 
> **Question 1:** Does there always exist a positive integer $m$, and a $\mathfrak{sl}(2)$ intertwining (meaning the corresponding notion at
> the Lie algebra level) linear map
> 
> 
> $$ \mathbb{C}^2 \otimes \cdots \otimes \mathbb{C}^2 \to V, $$
> 
> 
> which is onto, where the domain is the tensor product of $m$ copies of
> $\mathbb{C}^2$?
> 
> 
> 



> 
> **Question 2:** Assuming the answer to question 1 is yes, can one write down such a linear map using weights and roots? For example,
> what would $m$ be in terms of Lie-theoretic data, etc?
> 
> 
> 


*Note:* I did edit my post quite a bit, so some of the comments may not make sense now to a new reader, because of my edits. The new reader should keep this in mind. However, these are now the precise questions that I would like to see answered. I apologize for the little mess that I have made, with my latest edits. I think though that my questions are more focused now, in this version.



Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sym{Sym}$**Question:** "Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?"


**Answer:** If $k$ is the field of complex numbers and if $W:=k\{e\_1,e\_2\}$ it follows that any finite dimensional irreducible $\SL(W)$-module $V$ decomposes as
$$V \cong \bigoplus\_i \Sym\_k^{m\_i}(W),$$
where $m\_i\geq 1$ are integers. There is for every $i$ a surjection
$$\pi\_i: W^{\otimes m\_i} \rightarrow \Sym\_k^{m\_i}(W)$$
hence you get a surjection
$$\pi: U:=\bigoplus\_i W^{\otimes m\_i} \rightarrow V \cong \bigoplus\_i \Sym\_k^{m\_i}(W).$$
Hence every module $V$ is a quotient of a direct sum of tensor powers of $W$.
I'm unsure if you can choose $U$ to be a tensor product of $W$ — you may need to take direct sums.


In Fulton–Harris's book "Representation theory - a first course", §15.3, they give an elementary construction of all finite dimensional irreducible $\SL(n,k)$-modules as submodules of the tensor product
$$\Sym^{a\_1}(V)\otimes \Sym^{a\_2}({\bigwedge}^2 V) \otimes \dotsb \otimes \Sym^{a\_{n-1}}({\bigwedge}^{n-1} V)$$
where $V:=k^n$, by giving an explicit construction of the highest weight vector. In the general case you must use tensor and exterior products to get all representations.


**Note 1:** For $\SL(2,k)$ it follows $W \cong W^\*$ is an isomorphism, hence
you get isomorphisms of $\SL(2,k)$-modules
$$W^{\otimes m} \cong \operatorname{End}\_k(W)^{\otimes n}$$
if $m=2n$ and
$$W^{\otimes m} \cong \operatorname{End}\_k(W)^{\otimes n}\otimes W$$
if $m=2n+1$.


**Note 2:** On page 472 in FH you find the following construction in characteristic zero: There is for any $n,d\geq 1$ a split surjection
$$ V^{\otimes dn} \xrightarrow p \Sym^n({\bigwedge}^d V)$$
and if $G:=\SL(V)$ it follows $p$ is a split surjective map of $G$-modules.
Using this construction you get for any integers $a\_1,\dotsc,a\_{n-1}$ a split surjection of $G$-modules
$$ V^{\otimes \sum\_i ia\_i} \rightarrow \bigotimes\_i \Sym^{a\_i}({\bigwedge}^i V)$$
and since any finite dimensional irreducible $G$-module $W$ is a submodule of
$$\bigotimes\_i \Sym^{a\_i}({\bigwedge}^i V)$$
it follows you get a split surjection of $G$-modules
$$V^{\otimes \sum\_i ia\_i} \rightarrow W$$
for any finite dimensional irreducible $G$-module $W$. Hence in characteristic zero any finite dimensional irreducible $W$ may be realized as a submodule (or quotient module) of $V^{\otimes n}$ for some integer $n$. Hence if you want to give an explicit and elementary construction of all finite dimensional irreducible $\SL(V)$-modules, you may use the tensor powers $V^{\otimes n}$ for all $n\geq 1$. The above argument gives a construction of a highest weight vector


$$ v\_{\lambda} \in V^{\otimes n}$$


for any finite dimensional irreducible $SL(V)$-module $V(\lambda)$.


**Example:** Given $W\_1\subseteq V^{\otimes n\_1}, W\_2 \subseteq V^{\otimes n\_2}$. Let $V:=\mathbb{C}^{n}$ and let $T:=\wedge^n V$ be the trivial module. You may construct $T \subseteq V^{\otimes n}$ as a sub-module. For $W\_1\oplus W\_2$ to be realized as a quotient/submodule of $V^{\otimes d}$ for some $d$ there are obvious "obstructions". If $n\_1,n\_2$ are even and $n$ is odd there is no common tensorpower $V^{\otimes d}$ containing the module


$$W\_1 \otimes T^{\otimes i}\text{ and }W\_2\otimes T^{\otimes j}$$


for integers $i,j$. This is similar to the case of $SL(\mathbb{C}^2)$.


"Thank you so much for the reference to that fact in Fulton/Harris. I will check it out. – Malkoun 18 hours"


The above construction gives an "elementary" construction of all finite dimensional irreducible $\SL(V)$-modules by constructing an explicit highest weight vector $v\in V^{\otimes n}$ for some $n \geq 1$. I believe one of the reasons for introducing the Schur-Weyl functors in FH is to relate this study to the study of the representations of the symmetric group and combinatorics.


"I am still reading parts of Fulton and Harris. I get that such "tensorial" constructions can produce irreducible representations for the An type Lie groups, via the works of Schur, Weyl, Young etc. I wonder if similar constructions exist for all classical groups (I suspect the answer is yes). But what about the 5 exceptional cases? Is there a known general explicit construction for the highest weight representations which includes such "tensorial" constructions, but also works for the exceptional Lie groups? If this is known, then this would probably be directly relevant to my 2 projects! – Malkoun"


Dixmiers book "Enveloping algebras" give a construction of all finite dimensional irreducible $\mathfrak{g}$-modules for any semi simple Lie algebra using the universal enveloping algebra $U(\mathfrak{g})$ and Verma modules.
Any finite dimensional irreducible $\mathfrak{g}$-module $V(\lambda)$ has a "canonical" highest weight vector $v$ and you get an exact sequence


$$0 \rightarrow I(\lambda) \rightarrow U(\mathfrak{g}) \rightarrow^{\rho\_v} V(\lambda) \rightarrow 0$$


where $v\in V(\lambda)$ is a highest weight vector for $V(\lambda)$ and the map $\rho\_v$ is defined by $\rho\_v(x):=xv$. The ideal $I(\lambda)$ is the (2-sided) "annihilator ideal" of the vector $v$. In Theorem 7.2.6 in Dixmier they describe all finite dimensional irreducible $\mathfrak{g}$-modules by giving an explicit 2-sided maximal ideal $I(\lambda) \subseteq U(\mathfrak{g})$. In Proposition 7.2.7 they give generators of $I(\lambda)$.


