How to Conjecture $U \otimes V$ is a **irreducible** sub-representation of $S_4$? Let's consider the symetric group $S_4$. Let's consider it's standard representation V , and it's trivial representation U.
More precisisely (wasn't enough precise): In this example , we can define P as the representation of permutation of $S_4$ and $P=U\oplus V$
A author i read writes that :
$U\otimes V$ is a irreducible sub-representation of $S_4$.
How to conjecture it ? To be more precise , what's the reasoning that may be used that i could reproduce for other groups.
$S_4$ has four representations , containing the standard , trivial , signature , + the latter , that is also called the cube isometry  representation i think. (Incorrect sentence corrected by user).
 A: $\frak{S}_4$ has five irreducible representations (and infinitely many representations) in characteristic zero.
I assume by 'standard representation' you mean the permutation representation.
Tensor of the (one-dimensional) trivial representation with any other representation, just gives that same representation back.  I suppose it's also a "subrepresentation" (an invariant subspace) but only in the sense that a set it a "subset" of itself, since the invariant subspace here is the whole space.  (It does have other invariant subspaces, since the permutation representation is not irreducible.)
If by 'standard representation' you mean the regular representation, then all irreducible representations are "subrepresentations" of it, but there would be no need for the tensor product (though it also doesn't change anything).
On seeing the edits to your question, I suspect you intended $U \oplus V$ rather than $U\otimes V$.   If $W$ is the natural permutation representation, then it is a direct sum of two irredubicle subrepresentations, one of which is the trivial representation $U$, whilst the other is a three-dimensional irreducble representation $V$.
Also, you would have an isomorphism of representations $W\otimes W\cong (U\otimes U)\oplus (U\otimes V)\oplus (V\otimes U)\oplus (V\otimes V)$, where $U\otimes V$ is a subrepresentations of $W\otimes W$ isomorphic to $V$.
A: There are two good ways I know of to prove a representation $W$ is irreducible:
(I). Show $\sum_{g\in G}|\chi_W(g)|^2=|G|$ where $\chi_W(g)=\mathrm{tr}\rho_W(g)$, where $\chi_W$ is the character (trace) of the representation $\rho_W:G\to GL(W)$. This involves knowing the character values $\chi_W(g)$. You don't need to calculate them for every group element - in general you need to find a representative $g$ for each conjugacy class, the character values $\chi_W(g)$, and the size of the conjugacy classes.
(II). This one makes more sense if you are familiar with algebras and modules. We can consider $W$ as a module over the group algebra $\mathbb{C}[G]$. It is a simple module (i.e. irreducible representation) if and only if every vector is a cyclic generator, i.e. $\{\rho_W(g)w\mid g\in G\}$ spans $W$ for every $w\in W$. In practice, it is useful to find a nice basis (or spanning set) $B$ for $W$ and then show for every nonzero $w\in W$ the span of the orbit $\{\rho_W(g)w\mid g\in G\}$ contains an element $b\in B$. Or, you can show $Gw$ is spanning for a particular vector $w\in W$, then show $w$ is contained in the span of any other orbit. This is useful for showing the standard representation of $S_n$ is always an irreducible representation.
Let $U$ be the trivial representation, $V$ the standard representation, and $P=U\oplus V=\mathbb{C}^4$ the permutation representation of $S_4$ (coming from the usual action of $S_4$ on $\{1,2,3,4\}$ by permutations). We can define $U$ to be the span of $(1,1,1,1)$ and $V$ to be the subpace of vectors whose coordinates sum to $0$ (indeed, the usual complex inner product is $S_4$-invariant, and these two reps are orthogonal complements). If we apply the symmetrizer $\sum_{g\in S_3}\rho_{\small P}(g)$ to any element $(a,b,c,d)\in V$ we end up with a vector whose first three coordinates are the same and whose last coordinate is still $d$, i.e. $(\frac{1}{3}d,\frac{1}{3}d,\frac{1}{3}d,d)$. So we know every orbit contains the vector $w=(1,1,1,3)$. If you can then show $w$'s orbit, the four permutations of $(1,1,1,3)$, form a basis, then you're done.
Or you can compute the character values $\chi_{\small P}(g)$ using Burnside's lemma (it just counts the number of fixed points of $g$). From here you can determine $\chi_V(g)=\chi_{\small P}(g)-\chi_{\small U}(g)$, knowing $\chi_U\equiv1$ identically. Then you can verify $\sum_{g\in S_4}|\chi_U(g)|^2=24$ by direct calculation. Can you do this?
Note that if $U$ is the trivial representation of a group $G$ and $V$ is any other representation, then $U\otimes V\cong V$. So talking about $U\otimes V$ being irreducible amounts to just verifying $V$ is irreducible.
