Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I am a bit puzzeled about group invariants. The main problem is, what exactly is the decomposition in irreducibles of the symmetric power of an irreducible representation

[itex]D[/itex] of a group

[itex]G[/itex] ? “Lie online” computes this decomposition for simple Lie groups. I've been typing on google this question since yesterday but nothing came up.

Here is the starting point.

Consider a vector space [itex]V[/itex] over a field and the tensor space [itex]V^{\otimes r}[/itex].

Thanks to the [itex]GL\left(V\right)[/itex] and [itex]S_{r}[/itex] groups, one is able to decompose the space [itex]V^{\otimes r}[/itex] in

[itex]V^{\otimes r}=\bigoplus_{\lambda\dashv r}\left(V_{\lambda}\otimes U_{\lambda}\right)=\bigoplus_{\lambda\dashv r}\left(a_{\lambda}U_{\lambda}\right)=U_{\left\{ r\right\} }\oplus....\oplus U_{\left\{ 1^{r}\right\} } [/itex]

where [itex]V_{\lambda}[/itex] is the irreducible representation of the partition [itex]\lambda[/itex] of [itex]S_{r}[/itex] ([itex]a_{\lambda}[/itex] its dimension), [itex]U_{\lambda}[/itex] the irreducible representation [itex]\lambda[/itex] of [itex]GL\left(V\right)[/itex] (the latter can be viewed also as a schur functor [itex]U_{\lambda}\equiv\mathbb{S}_{\lambda}V[/itex] ).

Amongst them there are two particular vector spaces [itex]U_{\left\{ r\right\} }=Sym^{r}\left(V\right)[/itex] and [itex]U_{\left\{ 1^{r}\right\} }=\bigwedge^{r}\left(V\right)[/itex] , both spaces of irreducible representations of [itex]GL\left(V\right)[/itex] .

Now, consider a group [itex]G[/itex] and an irreducible representation [itex]D:G\rightarrow GL\left(V\right)[/itex] , [itex]g\rightarrow D_{g}[/itex] . I know how one can naturally define a new representation on [itex]V^{\otimes r}[/itex] [itex]D^{\otimes r}:G\rightarrow GL\left(V^{\otimes r}\right) [/itex]

[itex]D_{g}^{\otimes r}\left(v_{1}\otimes...\otimes v_{r}\right):=D_{g}v_{1}\otimes...\otimes D_{g}v_{n}[/itex]

I also know that [itex]D^{\otimes r}[/itex] is in general reducible and I can decompose

[itex]D_{g}^{\otimes r}=\bigoplus_{i}m_{i}\left(r\right)D_{g}^{i}[/itex]

where the sum is other all irreducibles [itex]D^{i}[/itex] of [itex]G[/itex] and [itex]m_{i}[/itex] the relative multiplicity (and one of them should also be the initial D , for some i [itex]D_{i}\equiv D[/itex] ).

Now, the symmetric power representation [itex]Sym^{r}D[/itex] should be a new representation of G on [itex]Sym^{r}V[/itex]

[itex]Sym^{r}D:G\rightarrow GL\left(Sym^{r}V\right)[/itex]

How is it defined? Is it a subrepresentation of [itex]D_{g}^{\otimes r}[/itex] ? I immagine that it's reducible

[itex]Sym^{r}D_{g}=\bigoplus_{i}n_{i}\left(r\right)D_{g}^{i}[/itex]

What's the connection between [itex]n_{i}[/itex] and [itex]m_{i}[/itex] ?

I've read some time ago that if [itex]D_{g}^{1}[/itex] is the trivial representation of [itex]G[/itex] , then [itex]n_{1}\left(r\right)[/itex] is the number of invariants of [itex]G[/itex] , built from the representation [itex]G[/itex] . Is this terminology correct?

Thank you very much.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Symmetric power of a group representation

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**