Show that the matrix representation of the dihedral group D4 by M is irreducible.

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.

Re: Show that the matrix representation of the dihedral group D4 by M is irreducible.

Quote:

Originally Posted by

**blueyellow** 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 $\displaystyle A$ and $\displaystyle B$ have no common eigenvectors. see the theorem in my blog.

you should do the second part of the problem yourself. note that $\displaystyle 0 \leq m \leq 1$ and $\displaystyle 0 \leq n \leq 3.$

Re: Show that the matrix representation of the dihedral group D4 by M is irreducible.

Quote:

Originally Posted by

**blueyellow** 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.

Alternatively, you could do the second part first (which is trivial) and then (assuming you are talking about $\displaystyle \mathbb{C}$-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 $\displaystyle \displaystyle \frac{1}{|G|}\sum_{g\in G}|\chi_M(g)|^2=1$.