Question

I guess this question only requires standard knowledge, but I'm a bit rusty with highest weight theory. I'm trying to catch up, but maybe I don't need the theory in full generality.

Background

Let $V$ be an Euclidean space of dimension $n$, and consider the representations of the group $G = O(V) \cong O(n)$. If I'm not wrong we can decompose

$\mathop{Sym}^2 V = W \oplus Z$,

where $Z$ is the trivial one dimensional representation and $W$ is an irreducible representation of $G$.

More background

This is how I can see the above decomposition. I don't know if it is of any use for the question itself, so feel free to skip it.

It is enough to decompose $\mathop{Sym}^2 V^{*}$, that is degree $2$ homogeneous polynomials on $V$. Degree $n$ homogeneous polynomials are a representation of $G$ via

$A.f(x) = f(Ax)$,

where $A$ is an orthogonal matrix and $f \in \mathop{Sym}^n V^{*}$.

Now for any $n$ we have a $G$-morphism $f \colon \mathop{Sym}^n V^{*} \to \mathop{Sym}^{n + 2} V^{*}$ which is given by multiplication by $x^2 = x_1^2 + \cdots + x_n^2$. This exhibits $\mathop{Sym}^n V^{*}$ as a subrepresentation of $\mathop{Sym}^{n + 2} V^{*}$.

The complement is easily found. The invariant scalar product on $\mathop{Sym}^n V^{*}$ is given by $(f, g) = f(D)g(x)$, where $D$ is the derivation operator. From this one finds that the adjoint of $f$ is the laplacian $\Delta$.

So one can decompose

$\mathop{Sym}^{n + 2} V^{*} = \mathop{Sym}^{n} V^{*} \oplus \mathcal{H}_n$

where $\mathcal{H}_n$ is the space of harmonic homogeneous polynomials of degree $n$. If I recall well, $\mathcal{H}_n$ is irreducible.

The decomposition which interests me is then

$\mathop{Sym}^{2} V^{*} = \mathcal{H}_2 \oplus \mathcal{H}_0$.

Problem

I'd like to understand how to decompose the tensor products of $W$; in particular

What is the decomposition of $W \otimes W$ and $W \otimes W \otimes W$ into irreducible representations of $G$?

Answer 1

For $SO(n)$ a calculation using LiE gives:
(using partition notation so $W$ is [2])
and assuming $n$ is not small

For $W\otimes W$, [4],[3,1],[2,2],[2],[1,1],[]
(all with multiplicity one)

and for $W\otimes W\otimes W$,
1.[6] 2.[5,1] 3.[4,2] 1.[3,3] 1.[4,1,1] 2.[3,2,1] 1.[2,2,2] 3.[4] 6.[3,1] 2.[2,2] 3.[2,1,1] 6.[2] 3.[1,1] 1.[]

The same works for $Sp(n)$ by taking conjugate partitions.

There is also a relationship with $SL(n)$.

This is taking your question at face value. If it is understanding you're after instead then the best approach is to use crystal graphs.

The notation I have used denotes a representation by a partition. I have put $m.$ in front to denote multiplicity is $m$. A partition is $[a_1,a_2,a_3,...]$ where $a_i\ge a_j$ if $i$ less than $j$. To convert to a highest weight vector add the appropriate number of $0$s to the end. Then take $[a_1-a_2,a_2-a_3,a_3-a_4,...]$. This gives a dominant integral weight. The fundamental weights are the partitions $[1,,,,1]$. If this has length $k$ this corresponds to the $k$-th exterior power of the vector representation (provided $2k-1$ less than $n$).

In particular the trivial representation is $[]$, the vector representation $V$ is $[1]$, the exterior square of $V$ is $[1,1]$, the symmetric square is $[2]+[]$.

For the $k$-th tensor power of $W$ you will see partitions of $2k-2p$ for $0\le p\le k$ only and it remains to determine the multiplicities (possibly $0$). For $SL(n)$ just take the partitions of $2k$ (with their multiplicities) and ignore the rest.