Let $\mathfrak g$ be a complex semisimple Lie algebra, and $V$ the fundamental $\mathfrak g$-module. Then we can decompose $V^{\otimes n}$ into the direct sum of irreducibles.
For example, in the case of $\mathfrak{sl_2}$, $V^{\otimes 3}=V_3\oplus V_1\oplus V_1$, where $V_n=\text{Sym}^n(V)$. If I wanna calculate $\text{End}(V\otimes V)$, an immediate thought is to use Schur's lemma.
But I've heard that there is in fact an isomorphism betteen $\text{End}(V^{\otimes n})$ and the Temperley-Lieb algebra $TL_n$, which can be represented diagrammatically as the vector space of noncrossing pairings of $2n$ points on two opposite sides of a rectangle. The dimension of $TL_n$ is the Catalan number $C_n$, for example, $\dim \text{End}(V^{\otimes 3})=5$.
as @I'm Representable points out, in the case of $\mathfrak{sl_2}$, the dimension of $\text{End}(V^{\otimes n})=\bigoplus_{i\equiv n\mod 2} m(i)V_i$ is exactly $\sum_i m(i)^2$ by Schur's lemma. A straightforward computation gives that $\dim \text{End}(V^{\otimes n})=\frac{1}{n+1}\binom{2n}{n}=C_n$.
Now my question is:
(1')In the case of $\mathfrak{sl}_2$, Why $\text{End}(V^{\otimes n})\cong TL_n$. i.e. how would the crossingless matchings between these 2n points be viewed as endomorphisms over the tensor powers of $V$?
(2)What about other lie algebras? For example, in the case of $\mathfrak{sl_3}$, there are two fundamental representations $V$ and $V^*$, how to calculate $\text{End}(V^{\otimes m}\otimes ({V^*})^{\otimes n})$ for $\mathfrak{sl_3}$? (This might be related to webs and unclasped combinatorial spiders)
Feel free to give any comments or elementary reference(all references (e.g.Greg Kuperberg) I've found so far involve the topics of quantum groups which I' not familiar with)
Best Answer
To answer question (1), it is indeed a consequence of Schur's lemma and the tensor product decomposition you have.
$V^{\otimes 3}=V_3 \oplus V_1 \oplus V_1$, and dim(Hom$(V_i, V_j))=\delta_{i,j}$ (by Schur's Lemma) since $V_i$ are irreducible representations of $\mathfrak{sl}_2$. Hence \begin{align*}\text{End(}V^{\otimes 3}) &=\text{Hom}(V_3,V_3\oplus V_1\oplus V_1)\oplus \text{Hom}(V_1,V_3\oplus V_1\oplus V_1)\oplus \text{Hom}(V_1,V_3\oplus V_1 \oplus V_1) \\ &= \text{Hom}(V_3,V_3)\oplus \text{Hom}(V_1,V_1)\oplus\text{Hom}(V_1,V_1) \oplus \text{Hom}(V_1, V_1)\oplus\text{Hom}(V_1,V_1)\end{align*} using the fact that $\text{Hom}(V_3,V_1)=0=\text{Hom}(V_1,V_3)$, and each $\text{Hom}(V_i,V_i)$ is. of dimension 1 so that you get $\text{dim(End}(V^{\otimes 3}))=5$
I'm just adding my comments which answers the other questions here.
For (1'), the isomorphism actually comes as a corollary of a much more beautiful result- there's a diagrammatic category called the Temperley Lieb category which is equivalent to the full (monoidal additive) subcategory of Rep($\mathfrak{sl}_2$) containing objects $V^{\otimes n}$. One nice reference to read about this diagrammatic category would be Section 7.4 of "Introduction to Soergel bimodules" by Elias, Makisumi, Thiel and Williamson. Essentially, the result comes from the fact that the morphism spaces in this category have a basis consisting of crossingless matchings.
To answer (2), for $\mathfrak{sl}_n$, there is a diagrammatic presentation for the full monoidal subcategory generated by fundamental representation via what are called webs. If interested, you can read this paper by Cautis, Kamnitzer and Morrison where webs were introduced: arxiv.org/abs/1210.6437. Webs essentially generalize the Temperley Lieb category which is just the special case for $\mathfrak{sl}_2$.