I am studying the character table of $S_5$ and in this YouTube video https://www.youtube.com/watch?v=Zj5PE6r_Oeo&t=427s (video will start at ~7:05), a method for finding an irreducible was used which I have not come across. Recall that $S_5$ has a 4-dimensoinal irreducible character $\chi_\sigma$ given by a specific restriction of the standard representation $\sigma$ of $S_5$.
(Forgoing conjugacy classes for the time being), the character row for $\chi_\sigma$ is given by
$$\begin{vmatrix} \chi_\sigma: & 4 & 2 & 1 & 0 & -1 & 0 & -1 \\ \end{vmatrix}$$
And then it appears that the instructor in the video found the rows for $\chi_\sigma(g)^2$ and $\chi_\sigma(g^2)$, subtracted them from each other and then divided by 2 to get a 6-dimensional irreducible character of $S_5$. I mean specifically that the instructor did the following:
$$\begin{vmatrix} \chi_\sigma(g)^2: & 16 & 4 & 1 & 0 & 1 & 0 & 1 \\ \chi_\sigma(g)^2: & 4 & 4 & 1 & 0 & -1 & 4 & 1 \\ \end{vmatrix}$$
The difference of these rows followed by a division by two is $\begin{vmatrix} 6 & 0 & 0 & 0 & 1 & -2 & 0 \end{vmatrix}$ which happens to be the row for 6-dimensional irreducible representation of $S_5$ (of course, you still should check this is irreducible)
So, in the video, the instructor refers to this as something like the character of the alternating two-tensor of $S_5$. Unfortunately, this is all I have found of this method. Does this method go by a specific name? Can anybody provide links for more information on the study of this method for finding irreducible characters? I would be interested in finding out how to properly reference this in theses.
In order to find the irreducible character table of a group $G$, the following principles are useful.
If $\chi \in Irr(G)$ and $\lambda$ is a linear (=dimension $1$) character, then also the product $\lambda\chi \in Irr(G)$. This is how the instructor produces from $\sigma$, $\pi_5$ and $sgn$, two new irreducibles $sgn \cdot \sigma$ and $sgn \cdot \pi_5$.
Then, in general, if $\chi \in Irr(G)$, then the function $\chi^{(2)}$ defined by $\chi^{(2)}(g)=\chi(g^2)$ for $g \in G$, is a class function. However, it does not have to be a character. But, it is always the difference of two characters. This is a classical result of Frobenius and Schur.
Also, products of irreducible characters are characters (not necessarily irreducible). So now it makes sense to calculate $\sigma^2 - \sigma^{(2)}$, being a difference of characters. After a calculation of the inner product, this difference turns out to be exactly twice an irreducible character. In the video this irreducible is called $\Lambda^2\sigma$. The notation with the lambda here stems from a construction via so-called exterior powers or alternating squares.
To illustrate the above (you might do the calculations yourself) you will find that
$\sigma^2=1_G + \sigma + \Lambda^2\sigma + \pi_5$ and
$\sigma^{(2)}=1_G + \sigma - \Lambda^2\sigma + \pi_5$.
The last expression is indeed a difference of the characters $1_G + \sigma + \pi_5$ and $\Lambda^2\sigma$. Hope this helps.