---- > [!theorem] Theorem. ([[weights characterize any representation of sl2(C)]]) > A [[Lie algebra representation|representation]] $\rho$ of $\mathfrak{sl}_{2}(\mathbb{C})$ is completely determined (up to [[Lie algebra homomorphism|isomorphism]]) by its [[weight space for sl2(C)|multiset of weights]] — i.e., by the eigenvalues of $\rho(h)$ counted with multiplicity. ^theorem > [!basicexample] > > **1.** Suppose an $\mathfrak{sl}_{2}(\mathbb{C})$-representation $V$ has weights $5,3,3,1,1,0,-1,-1,-3,-3,-5$. We know from [[any representation of sl2(C) is completely reducible|here]] that this representation decomposes as a [[direct sum]] of [[irreducible Lie algebra representation|irreps]], and therefore this multiset of weights is composed of all the weights of those irreps put together. > > The [[weight space for sl2(C)|highest weight]] of *some* irrep is $5$. That 6-dimensional irrep also takes with it weights of $3$, $1$, $-1$, $-3$, $-5$. So we have a copy of $V(5)$. > > The leftover weights are $3,1,0, -1,-3$. This leaves a highest weight for another irrep to be $3$. It takes with it additionally $1,-1,-3$. We are left with just $0$ now. Thus we must have $V \cong V(5) \oplus V(3) \oplus V(0).$ > > **2.** Suppose $V$ has dimension $5$ and $3$ as a weight. Then we must have a $V(3)$ component, which has dimension $4$ (if you have $V(5),V(7),$ etc. the dimension is already too big). So we get $V \cong V(3) \oplus V(0)$, $V(0)$ the trivial representation. > > > **3.** To know if a $\mathfrak{sl}_{2}(\mathbb{C})$-representation is irreducible, just check its weights: it is irreducible if and only if its weights have the form $(n,n-2,\dots,-n)$. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```