Question about group representation

[Tio Barnabe]

After reading some books on Group Theory, I have two questions on group representations (Using matrix representation) with the second related to the first one:

1 - Can we always find a diagonal generator of a group? I mean, suppose we find a set of generators for a group. Is it always possible to have at least one of them being a diagonal matrix?

2 - If so, then the eigenvectors of this diagonal operator can be used as a basis for the vector space where the representation takes place. My question is if the eigenvectors can be used or if they must be used as the basis?

 

[andrewkirk]

Consider the two-element group of 2 x 2 matrices consisting of two elements,
\begin{pmatrix}
1&0\\
0&1
\end{pmatrix}
and
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
The second of these is not diagonal, and it cannot be generated by the other elements, since there is only one other element and that is the identity.

I'm not sure I fully understand your question, so I don't know whether this answers it.

 

[Tio Barnabe]

Thanks for your reply, but

I don't know whether this answers it

unfortunately, it doesn't.

I will try to be more clear. Suppose we have a group $G$ with generators $\{g_1,g_2,... \} \equiv \{g_i \}$ and a representation $R$, such that $\{R(g_1),R(g_2),... \}$ are matrices representing the group $G$.

My first question is, For any group $G$, is it always possible to find a representation $R$ such that there is at least one element $g$ of $\{g_i \}$ such that $R(g)$ is a diagonal matrix? (End of the first question.)

The representation of $G$, i.e. the matrices $R(\{g_i \})$ will act on vectors belonging to a vector space. I read of cases where one of the $R(\{g_i \})$ was a diagonal matrix. In that case, the author has chosen the corresponding eigen-vectors as the basis vectors of the vector space where the transformations $R(\{g_i \})$ act.

My second question is, Do we need always to choose the basis vectors as being the eigen-vectors of one of the $R(\{g_i \})$? (That's the reason I said in post #1 that the second question is related to the first question.)

 

[Infrared]

Thanks for your reply, but
My first question is, For any group $G$, is it always possible to find a representation $R$ such that there is at least one element $g$ of $\{g_i \}$ such that $R(g)$ is a diagonal matrix? (End of the first question.)

Yes, take $R$ to be the trivial representation, for example. This is the only such representation if $G$ is simple and nonabelian (for example, $A_5$) and the identity is not one of your chosen generators since then any representation would either be faithful or trivial (kernel must be a normal subgroup), but a faithful representation would exhibit an isomorphism from $G$ to a group with a nontrivial center, which is impossible (nonabelian simple groups have trivial center since the center of a group is normal).