Total irreducible representation of a quantum system - Slater determinant?

I'm beginning to study quantum chemistry and I became pretty confused about the total state of the molecule.


When we describe the single orbitals using irreducible representations like in the picture below, then we can get the total quantum state by direct product
$$(2a\_1)^2(1b\_2)^2(3a\_1)^2(1b\_1)^2=(A\_1)^2\otimes (B\_2)^2\otimes (A\_1)^2\otimes (B\_1)^2 =A\_1,$$


as it is described in [this answer](https://chemistry.stackexchange.com/a/87554/53653). But when I read about multi-particle wavefunctions, I realized, that the wavefunction of the whole system should be antisymmetric, so I should assemble [Slater determinant](https://en.wikipedia.org/wiki/Slater_determinant) to get the multi-particle wavefunction with the enforced antisymmetry property.


![]](https://i.stack.imgur.com/Pj4st.png)




---


Questions
---------


1) How is Slater determinant connected with the direct product above?


2) Does $(A\_1)^2$ mean $A\_1 \otimes A\_1$ or is it purely the notation for 2 electrons in the orbital?


