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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 $4$.

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Omar Shehab
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    Why do you say it will increase the dimension? The dimension of a tensor product is the product of the dimensions, so here the dimension will be $1 \cdot 4 = 4$. In terms of matrices, the tensor product corresponds to the Kronecker product. (The Kronecker product with a $1 \times 1$ matrix is just scalar multiplication.) – Alex Provost Dec 03 '15 at 19:13
  • @A.P., you are right. I was incorrect. – Omar Shehab Dec 03 '15 at 19:23

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