I am able to work out the sign representation of $S_5$ and standard representation of $S_5$.
How do I compute the product of standard and sign representation of $S_5$?
What kind of product do I need to take? For any given group element in $S_5$, do I just need to multiply the sign representation (which is a scalar value of 1 or -1) with the standard representation (which is a $4 \times 4$ matrix)?
My effort: The answer here enter link description here suggests to take the tensor product which will increase the dimension of the vector space. But I already know that the dimensions of the product of standard and sign representation of $S_5$ and the standard representation are $4$.