the representation is irreducible because and have no common eigenvectors. see the theorem in my blog.
you should do the second part of the problem yourself. note that and
Show that the matrix representation of the dihedral group D4 by M is irreducible.
You are given that all of the elements of a matrix group M can be generated
from the following two elements,
A=
|0 -1|
|1 0|
B=
|1 0|
|0 -1|
in the sense that all other elements can be written A^n B^m for integer m, n >or= 0.
Find the remaining elements in M.
the representation is irreducible because and have no common eigenvectors. see the theorem in my blog.
you should do the second part of the problem yourself. note that and
Alternatively, you could do the second part first (which is trivial) and then (assuming you are talking about -representations) appeal to the common theorem that a representation is irreducible if and only if the inner product of the character with itself is one, i.e. if .