About a representation of $A_5$ over $\mathbb{F}_2$ Let $V$ be a $5$-dimensional vector space over $\mathbb{F}_2$ with basis $\{e_1,\cdots,e_5\}$. The group $A_5$ acts on this vector by permutation of basis elements. Let $W$ be the subspace of $V$ generated by 
$$e_1-e_2,\,\,\,\, e_2-e_3,\,\,\,\, e_3-e_4,\,\,\,\, e_4-e_5.$$
Then $A_5$ also acts on this subspace (or $W$ is $A_5$-invariant subspace of $V$).
I wanted to know whether this $4$-dimensional representation is irreducible. 

What I know about this is that if we had a field of characteristic not dividing order of $A_5$ then over that field, this $4$-dimensional representation would have been irreducible; but here field has characteristic $2$.
 A: I think the representation is irreducible:
every vector in $W$ is of the form $\lambda_1e_1+\cdots + \lambda_5e_5$ with $\lambda_i\in \{0,1\}$ and $\lambda_1+\cdots + \lambda_5=0$.
The condition $\lambda_i\in \{0,1\}$ and $\lambda_1+\cdots + \lambda_5=0$ implies that 
(1) either all $\lambda_i$'s are zero (trivial vector);
(2) exactly two $\lambda_i$'s have value $1$;
(3) exactly four $\lambda_i$'s have value $1$.


*

*In $W$ consider an arbitrary vector $\lambda_1e_1+\cdots + \lambda_5e_5$ with $\lambda_i$'s in type (2), say $e_1+e_2$. Then by action of $A_5$ we can obtain vectors 
$$e_2+e_3,\,\,\,\, e_3+e_4,\,\,\,\, e_4+e_5.$$
(for example, apply $(123)$ on $e_1+e_2$ to get $e_2+e_3$).


This means if an $A_5$-invariant subspace of $W$ contains $e_1+e_2$, then it contains the basis $\{e_1+e_2,\cdots e_4+e_5\}$, i.e. that invatiant subspace should be $W$.


*

*In $W$ suppose there is a vector $\lambda_1e_1+\cdots + \lambda_5e_5$ with $\lambda_i$'s in type (3). For example, let it be 
$$e_1+e_2+e_3+e_4.$$
Then by applying permutations ($5$-cycles) to this vector, we can obtain 
$$e_1+e_2+e_3+e_4,\,\,\,\, e_2+e_3+e_4+e_5, \,\,\,\,e_3+e_4+e_5+e_1, \,\,\,\,e_4+e_5+e_1+e_2,$$
and these four vectors are independent in $W$, so the $A_5$-invariant subspace of $W$ should be $W$ itself. 


Thus the representation $W$ is irreducible.
