I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context:
An arbitrary rotation of the coordinate frame will transform the tensor $T$ into a tensor $T'$, whose components $T_{ij}'$ are, in the most general case, linear combinations of all the components $T_{ij }$. However, it is always possible to find certain subgroups of the components $T_{ij}$ (formed by linear combinations thereof) that transform among themselves under a rotation. These components are called the irreducible tensor components.
Can you please explain that scientifically and linguistically?
To say that a subgroup $S$ (which a mathematician would call a subset, as it's not a group in the technical sense) of tensor components transform among themselves means that, if $T_{ij}$ is any one of the components in $S$, then the corresponding $T'_{ij}$ is a linear combination of components $T_{kl}$ from $S$, as opposed to being a linear combination of just any old components of $T$. So one can write down the transformation rules for the components in $S$ without ever mentioning any other components.