I was reading this paragraph from Fulton & Harris :
But I could not understand the difference between permutation representation and standard representation and how they are complement and why the standard one is two dimentional irreducible while the permutation representation is reducible, could anyone explain this for me please?
The standard representation is a subrepresentation of the permutation representation.
The permutation representation of $S_n$ contains a 1-dimensional $S_n$-invariant subspace spanned by $e_1+\cdots+e_n$. The compliment is the subspace spanned by $\{e_1-e_2,e_2-e_3,\ldots,e_{n-1}-e_n\}$ and is called the standard representation. It is $(n-1)$-dimensional and has no $S_n$-invariant subspaces.