# Tensor symmetries and the symmetric groups

• A

## Main Question or Discussion Point

In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##)

To discuss general properties of tensor symmetries, we shall use the representation theory of the symmetric group as expressed, for example, in Weyl (1946). We write ##[n_1,\dots, n_r]##, where ##n_1\geq n_2\geq \dots \geq n_r##, to denote the irreducible symmetry described by the Young diagram of the partition ##(n_1,n_2,\dots, n_r)##. Then if ##n\geq 1##, the equation $$m^{\lambda_1\dots \lambda_n \mu}=m^{(\lambda_1\dots\lambda_n)\mu}$$ corresponds to the reducible symmetry $$[n]\otimes [1]=[n,1]\oplus [n+1]$$ where the parts with symmetries ##[n,1]## and ##[n+1]## may be taken as $$m^{\lambda_1\dots \lambda_{n-1}[\lambda_n\mu]}\quad \text{and}\quad m^{(\lambda_1\dots \lambda_n \mu)}$$ respectively.

Now, I can't understand what he means by that. I believe it is something about the symmetric group acting on the tensor power of a vector space by permutting the factors in some way.

Why the first equation corresponds to that reducible symmetry? By the way, what he means by the word "corresponds" in this context? And what he means that the parts with symmetries ##[n,1]## and ##[n+1]## can be taken as ##m^{\lambda_1\dots\lambda_{n-1}[\lambda_n\mu]}## and ##m^{(\lambda_1\dots \lambda_n\mu)}##?

I believe this has something to do with the Young symmetrizer construction. I mean, I believe that we must let ##S_{n+1}## act on ##V^{\otimes (n+1)}## by permuting the factors, and then use this action to say how the group algebra acts on ##V^{\otimes (n+1)}## and look to the action of the symmetrizer. I'm unsure, though, since I haven't studied in depth the representation theory of the symmetric group.

What actually is the point here? How can I understand, from a representation theory point of view what is going on in this passage of the paper?