What is the span of the symmetric group $S_{5}$?

Question

My professor while writing the character table of $A_{5}$, uses that the span of $S_{5}$ is $\{e_{1} - e_{2}, e_{2} - e_{3}, e_{3} - e_{4}, e_{4} - e_{5} \}$ but I do not know why this is the span of $S_{5}$ or $A_{5}$, could anyone explain this for me please?

Answer 1

The terminology here isn't correct: It's meaningless to talk about the span of elements of a generic group.

What's probably going on here is this: The group $S_5$ acts faithfully on $\Bbb C^5$ by permutation of coordinates, so that $$\sigma \cdot (z_1, \ldots, z_5) = (z_{\sigma(1)}, \ldots, z_{\sigma(5)}) .$$ This representation is not irreducible: If we denote the standard basis of $\Bbb C^5$ by $(e_1, \ldots, e_5)$, then the action of $S_5$ fixes the line $\{(z, z, z, z, z) : z \in \Bbb C^5\}$, which is spanned by $e_1 + \cdots + e_5$, as well as the complementary hyperplane $\{(z_1, \ldots, z_5) : z_1 + \cdots + z_5 = 0\}$, which is spanned by $e_1 - e_2, \ldots, e_4 - e_5$, and these two subspaces are irreducible subrepresentations.