Math 220C
May 9, 2008
Further examples of group representations
In class, we discussed several additional examples of group representations.
(1) Present
Z
n
as
x

x
n
= 1 , and let
ξ
∈
C
be a primitive
n
th
root of unity.
Consider the matrix representation
ˆ
T
:
Z
n
→
GL(2
,
C
) defined by
ˆ
T
(
x
) =
ξ
0
0
ξ

1
.
This is a direct sum of two onedimensional representations.
(2) Consider a group
BD
2
m
presented as
BD
2
m
=
x, y

x
2
m
= 1
, y
2
=
x
m
, y

1
xy
=
x

1
.
(Note that the subgroup generated by
x
is isomorphic to
Z
2
m
.) Let
ξ
∈
C
be a
primitive 2
m
th
root of unity, and define a matrix representation
ˆ
T
:
BD
2
m
→
GL(2
,
C
) by
ˆ
T
(
x
) =
ξ
0
0
ξ

1
ˆ
T
(
y
) =
0
1

1
0
.
(Note that when restricted to the subgroup
Z
2
m
generated by
x
, this is the
same representation defined in example 1.
(3) Consider a group
BT
presented as
BT
=
x, y, z

x
4
= 1
, y
2
=
x
2
, z
3

1
, y

1
xy
=
x

1
, z

1
xz
=
xy, z

1
yz
=
x

1
.
(Note that the subgroup generated by
x
and
y
is isomorphic to
BD
4
.) Let
ε
∈
C
be a primitive 8
th
root of unity.
Define a matrix representation
ˆ
T
:
BT
→
GL(2
,
C
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 Spring '08
 MORRISON
 Algebra, Addition, Matrix representation

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