I am learning group representation theory, and I have troubles understanding the applications of group representation. Could you please help me to give explain the following question?
My question is:$$ $$ For infinite dihedral group: $$ D_∞ = <r,s|s^2=1,srs=r^{-1}> $$ Then define a map $$ \phi^α (r) = \begin{bmatrix} 0 & 1 \\ -1 & α \end{bmatrix} $$ $$ \phi^α (s) = \begin{bmatrix} 1 & 0 \\ α & -1 \end{bmatrix} $$ Then the question need me to show that for what value of α, $$\phi^α$$will
- extend to a representation of D∞
- $\phi^α decomposable$
- $\phi^α reducible$
- $\phi^α faithful$
- $\phi^α "isomorphic "to" \phi^β $ if and only if α=β
My major problem is that I can not figure out how to show it is a homomorphism because be definition of homomorphism, $$\phi^α(rs)=\phi^α(r)\phi^α(s)$$ How can I know the matrix of $\phi(rs)$ based on the information?
Also I only have a clue for part 3, I only need to show the above to matrices have common eigenvectors.
Could anyone help me with the rest of questions?
Thank you in advance!
New Edit I total get how to do part 1 now. Could anyone help me with the rest of them?
Hint: $\psi:F_2\rightarrow GL(2)$ where $F_2$ is the free group generated by $r$ and $s$ defined by
$\psi(r)=\pmatrix{0&1\cr -1&\alpha}, \psi(s)=\pmatrix{1&0\cr \alpha&-1}$
It is enough to show that $\psi(s)^2=Id, \psi(s)\psi(r)\psi(s)=\psi(r)^{-1}$to deduce that $\psi$ induces a morphism $\phi^{\alpha}$ on $D_{\infty}$ with the desired properties.