- #1
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1. Homework Statement [/b]
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.
I tried reading through the notes and they say:
An n-dimensional matrix REP M(G) of a finite group G is reducible if there exists a similarity transformation S such that
S^(-1) M (g) S=
|M(subscript 1) (g) 0 |
|0 M(subscript 2) (g)|
for each g (is an element of G)
but I do not know how I would go about starting with trying to find a similarity transformation
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 Attempt at a Solution
I tried reading through the notes and they say:
An n-dimensional matrix REP M(G) of a finite group G is reducible if there exists a similarity transformation S such that
S^(-1) M (g) S=
|M(subscript 1) (g) 0 |
|0 M(subscript 2) (g)|
for each g (is an element of G)
but I do not know how I would go about starting with trying to find a similarity transformation