Direct sum of irreducible representation of sl2

Question

please help me have a look at this problem.

How can I establish the decomposition of $Sym^n (Sym^2(\mathbb{C^2}))$ into a direct sum of irreducible representations of $\mathcal{sl_2}$: $$Sym^n (Sym^2(\mathbb{C^2})) = \bigoplus _{a=0}^{\lfloor \frac{n}{2}\rfloor} Sym^{2n-4a}(\mathbb{C^2}).$$ More precisely, how can I compute the dimension of the weight spaces to see establish it.

Thanks

Answer 1

To give a little more of the details (that might be helpful as you read Chapter $11$ of Fulton and Harris). They describe how the symmetric square $\operatorname{Sym}^2\mathbb{C}^2$ has basis $\{x^2,xy,y^2\}$. Then, in $\operatorname{Sym}^n\operatorname{Sym}^2\mathbb{C}^2$, you're counting the number of ways to get products of $n$ terms (up to symmetry) of the form $x^ky^{2n-k}$.

For example: $\operatorname{Sym}^2\operatorname{Sym}^2\mathbb{C}^2$ has:

• One way to get $x^4$, $(x^2\cdot x^2)$

• One way to get $x^3y$, $(x^2\cdot xy)$ (the other order is the same because of symmetry).

• Two ways to get $x^2y^2$ by $(x^2\cdot y^2)$ or $(xy\cdot xy)$.

This is another way of looking at the number of times each dot is circled in the book.

Answer 2

Gomez, I think the thing that you're missing is not how to compute the dimension of a representation, but how to compute the weights for a given representation.

Some examples:

(The brackets for my $\mathfrak{sl}_2 \mathbb C$ are $[H,X]=2Y$, $[H,Y]=-2X$, $[X,Y]=H$.)

(i) Take $\mathbb C^2$. Pick a basis $x, y$ for $\mathbb C^2$ such that $$H(x) = +x, \ \ \ \ H(y) = -y.$$ The weights for the representation $\mathbb C^2$ are the two eigenvalues of $H$ that we just computed, namely, $$+1, -1.$$

(ii) Now take ${\rm Sym}^2 \mathbb C^2$. A natural basis for ${\rm Sym}^2 \mathbb C^2$ is $$x\otimes x, \ \ \ \ x \otimes y + y \otimes x, \ \ \ \ y\otimes y.$$ The action of $H$ on these basis vectors is $$H(x \otimes x) = 2(x\otimes x), \ \ \ \ H(x \otimes y + y \otimes x) = 0, \ \ \ \ H(y\otimes y) = -2(y\otimes y).$$ So the weights for the representation ${\rm Sym}^2 \mathbb C^2$ are $$+2, 0, -2.$$

(iii) How about ${\rm Sym}^2({\rm Sym}^2 \mathbb C^2)$? A natural basis is $$(x \otimes x)\otimes (x\otimes x),$$ $$(x\otimes x)\otimes (x \otimes y + y \otimes x) + (x \otimes y + y \otimes x) \otimes (x\otimes x),$$ $$(x \otimes x) \otimes (y \otimes y) + (y \otimes y)\otimes (x \otimes x),$$ $$(x \otimes y + y \otimes x) \otimes (x \otimes y + y \otimes x), $$ $$(x \otimes y + y \otimes x) \otimes (y \otimes y) + (y \otimes y) \otimes (x \otimes y + y \otimes x), $$ $$(y \otimes y) \otimes (y \otimes y). $$ The weights of this representation are the eigenvalues of $H$ when acting on this basis. These are $$+4,+2,0,0,-2,-4,$$written in the same order as the order in which I wrote the basis vectors.

To decompose ${\rm Sym}^2({\rm Sym}^2 \mathbb C^2) $ into irreducible representations, note that ${\rm Sym}^4 \mathbb C^2$ is irreducible with weights $+4,+2,0,-2,-4$ (which you can show using similar methods) and ${\rm Sym}^0 \mathbb C^2$ is irreducible with a single weight of $0$. Hence $$ {\rm Sym}^2({\rm Sym}^2 \mathbb C^2) \cong {\rm Sym}^4 \mathbb C^2 \oplus {\rm Sym}^0 \mathbb C^2.$$ as $\mathfrak{sl}_2 \mathbb C$ representations.

Michael Burr's fantastic answer gives you an efficient way of writing and counting these basis vectors and keeping track of their weights. In fact, my example is the same as his, but written out in a more long-winded way.