[abdullahkhalids]

Re: your last point, the determinant is the matrix function, which is fully anti-symmetric under permutation of rows and columns i.e. swapping a pair of rows or a pair of cols pulls out a minus sign. The definition of the determinant is related to the alternating representation of the symmetric group.

The permanant [1] is the matrix function which is fully symmetric, so permuting any rows or cols leaves it invariant. It emerges from the identity representation.

Finally, partially symmetric matrix functions are known as immanants [2], defined using the other irreps of the symmetric group.

[1] https://en.wikipedia.org/wiki/Permanent_%28mathematics%29

[2] https://en.wikipedia.org/wiki/Immanant