What is/are the canonical notation(s) for multi-particle states? What are they mathematically?

Question

I'm trying to refresh things I learned ages ago, and I believe I either never learned the proper mathematical background for multi-particle states or I forgot the details. I intuitively write things like $\newcommand{\ket}[1]{\left|#1\right\rangle}\ket{\uparrow\downarrow}$ and $\ket{n_1,n_2}$ and remember the latter being related to multi-mode Fock states, but sometimes also tensor products $\ket 1\otimes\ket 4$ and maybe even (vectors of Hilbert) vectors $\left(\ket a\atop \ket b\right)$ (or $\ket{\left(a\atop b\right)}$?). But are all those different but equivalent notations for the same mathematical concepts and merely a matter of taste/convenience/legibility, or are there severe differences and something like $\left(\ket{a,2}\atop \ket b \otimes \ket {42}\right)$ actually has a sensible meaning?

While I should sooner or later (re)learn the details including representation theory in (or "of"?) Hilbert spaces etc, for now I'd appreciate pointers into the proper directions.

Answer 1

The first two are different notations for the same ket : $$|\uparrow\downarrow\rangle = |\uparrow \rangle \otimes |\downarrow\rangle \qquad \text{ and }|n_1,n_2\rangle = |n_1\rangle \otimes |n_2\rangle$$

I am not too sure about the last one (the vectors of ket). Mathematically, it would correspond to an element of the direct sum of the two Hilbert spaces, rather than the tensor product. If this is indeed what you mean by $\begin{pmatrix}|a\rangle \\ |b\rangle \end{pmatrix}$, this is not a multi-particle state.