Posted: November 11th, 2015

Mathematics

Mathematics

1) Let G be the group of order 16 defined in terms of generators and
relations
G =< x, y : x
4 = 1 = y
4
, yx = x
3
y >
Find all 1-dimensional representations of G.
2) i) Prove that there is a unique representation
?1 : D8 ? GL2(C)
for which
?1(a) =
?
?
v
2
2 –
v
2
2
v
2
2
v
2
2
?
?
and
?1(b) = 
1 0
0 -1

Here
D6 = {1, a, a2
, a3
, a4
, a5
, a6
, a7
, b, ab, a2
b, a3
b, a4
b, a5
b, a6
b, a7
b}
is the dihedral group, where a
8 = 1 = b
2 and bab = a
-1
. Is ?1 an
irreducible representation? Compute the character of ?1.
ii) Let
? =
v
2
2
(1 + i) and ? =
v
2
2
(1 – i).
Prove that there is a unique representation
?2 : D4 ? GL2(C)
1
2 PROBLEM SHEET 1
for which ?2(a) = C and ?2(b) = D, where
C =

? 0
0 ?

, D =

0 1
1 0
.
Compute the character of ?2.
iii) Find an invertible matrix P ? GL2C such that for any x ? D4
one has
P ?2(x)P
-1 = ?1(x).
3) i) Let
C3 = {1, t, t2
}, t3 = 1
be the cyclic group of order three. Let V be the 2-dimensional vector
space with basis e1 and e2. Prove that there exist a unique C3-module
structure on V such that
te1 = e2, and te2 = e1 – e2
Describe the corresponding representation and decompose it as a direct
sum of irreducible representations.
4) i) Let C6 be a cyclic group of order 6 with a generator t. Describe
all 1-dimensional representations of C6.
ii) Let V be a C6-module. For each 6-th root of unity ?, we let
V? = {v ? V |tv = ?v}.
Prove that
V ~=
M
i=5
i=0
V?
i
Here ? =
1
2 +
v
3
2
i is the primitive root of unity.
iii) Let C
6 be a C6-module, where the action of t on V is given by
t(?1, ?2, ?3, ?4, ?5, ?6) = (?6, ?1, ?2, ?3, ?4, ?5).
Consider a = (1, 2, 3, 1, 2, 3). Describe the smallest submodul W ? V
such that a ? W. What is the dimension of W? Find a C6-submodule
U such that U ? W. Decompose V and W as the direct sum of irreducible
representations.
5) Let G = D8 be the dihedral group. Thus G =< x; y : x
8 = 1 =
y
2
, xy = yx-1 >. Prove that there exist a unique representation
? : G ?
for which
?(x) = 
-7 10
-5 7 
and
?(y) = 
-5 6
-4 5
find the character of ?. Is ? an irreducible representation?
6) Let C12 be the cyclic group of order 12, with a generator t and
let e1, · · · , e12 be a standard basis of C
12, which is considered as a
C10-module via the action
tei =
(
ei+1 i < 12
e1, i = 12.
Find the dimension of a minimal submodule which contains the element
e1 – e2 + e3 – 34 + e5 – e6 + e7 – e8 + e9 – e10 + e11 – e12.

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