What does the notations like $\mathbb{C}(e_1,e_2)$, $\mathbb{C}(e_1+e_2+e_3)$ mean here in https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups ?
I understands that these are subspaces of $\mathbb{C}^n$ and here $e_i, \ i=1,2,...,n$ are standard vectors. These symbolic notations makes me confused and I have no other access to ask this question. I am trying to learn at my own effort, so please answer my question
Actually, all I found in the above link is a example of group representation of the permutation group $Per(3)$ over the set $\{1,2,3 \}$. The representation $\rho: Per(3) \to GL_5(\mathbb{C})$ be a linear representation of $Per(3)$. (please see the example in the above given link). Also $ \eta: Per(3) \to GL_2(\mathbb{C})$ and $1$, the identity representation and $\tau$ be another subrepresentation so that
A decomposition of $(\rho, \mathbb{C}^5)$ in irreducible subrepresentations is: $\rho=\eta \oplus \tau \oplus 1,$ and $$\mathbb{C}^5= \mathbb{C}(e_1,e_2) \oplus \mathbb{C}(e_3-e_4,e_3+e_4-2e_5) \oplus \mathbb{C}(e_3+e_4+e_5)$$ is the the decomposition of the representation space.
Does the symbol $\mathbb{C}(e_1,e_2)$ mean the subspace of $\mathbb{C}^5$ generated by $e_1,e_2$ ?
Does the symbol $\mathbb{C}(e_1+e_2+e_3)$ mean the subspace of $\mathbb{C}^5$ generated by linear combination of $e_1,e_2,e_3$?
Can you please explain the symbols above in the context of representation theory as well as linear algebra? At least please attach link of a book of linear algebra having the above symbols.