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Fearless Symmetry by Robert Gross, Avner Ash

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GROUP REPRESENTATIONS 139
DEFINITION: A linear representation (or matrix
representation)ofagroupH is a morphism from H to a
matrix group GL(n, R), where R is some number system.
For instance, suppose that G is the absolute Galois group of Q.
Look back at our discussion in chapter 8 about how any element g in
G permutes the roots of any Z-polynomial. So if we pick a particular
polynomial, we can ask how any element of G permutes its roots. We
will explain this example in detail in chapter 14. You can also look
at our description of the n-torsion of an elliptic curve in chapter 9.
There, too, we can think of an element of G as permuting the
n-torsion. In this case, we can also produce linear representations
of G. See the end of this chapter and chapter 18 for more details.
A
4
, Symmetries of a Tetrahedron
This is a very long example. If the going gets a bit dense, don’t
worry; we will not need the details of this example in any other
part of this book. Our source group in this example is a permutation
group called A
4
. First, we will tell you about the group abstractly,
and then we will give you some other wa ys to think about it by
using representation theory.
Our group A
4
is a group of permutations of the set {1, 2, 3, 4},but
it does not contain every permutation (or else it would be
{1,2,3,4}
).
In fact, A
4
contains exactly 12 permutations, or half of all of the
permutations in
{1,2,3,4}
.
To start, A
4
contains every permutation that leaves one number
fixed, and cyclically permutes the other three numbers. For exam-
ple, A
4
contains the permutation that sends 2 to 2, and cyclically
permutes 1, 3, and 4, meaning that 1 3 4 1:
1 3
2 2
3 4
4 1.
140 CHAPTER 12
The short notation for writing this permutation is (134), where
the omission of the number 2 means that 2 is not changed by the
permutation.
There are eight such permutations.
EXERCISE: List all eight of the permutations that fix one
element of {1, 2, 3, 4} and cyclically permute the other three
numbers.
SOLUTION: It would consume too much space to list these
permutations using the arrow notation. Using the shorter
cycle notation, the eight permutations are (123), (132), (134),
(143), (124), (142), (234), and (243).
Remember that a group must also contain the neutral element
e (which leaves every element of {1, 2, 3, 4} xed), so you now know
nine elements of A
4
. The other three elements can be found because
our group must be closed under the composition law.
The result of (123) (124) (which, you remember, means first
doing the permutation (124) and then the permutation (123)) is:
1 3
2 4
3 1
4 2.
The way to write this in cycle notation is (13)(24).
There are two other permutations that result from combining
the elements of our group: (12)(34) and (14)(23). Here is the list of
all of the permutations in A
4
: e, (123), (132), (134), (143), (124), (142),
(234), (243), (12)(34), (13)(24), (14)(23).
EXERCISE: Find a permutation of {1, 2, 3, 4} that is not in A
4
.
SOLUTION: Here are all of the permutations in
{1,2,3,4}
that
are not in A
4
: (12), (13), (14), (23), (24), (34), (1234), (1324),
(1423), (1243), (1342), and (1432).
GROUP REPRESENTATIONS 141
Figure 12.1: A regular tetrahedron
This is pretty abstract stuff. Fortunately, there is another way to
think about A
4
.
DEFINITION: A regular tetrahedron is a solid figure with
four congruent equilateral triangles as its faces.
You can see a picture of a tetrahedron in F igure 12.1.
Number the vertices of the tetrahedron from 1 to 4. For the sake
of illustration, label the top vertex with the number 1, and the three
on the base 2, 3, and 4 (counterclockwise).
Now imagine rotating the tetrahedron so that when you are
done, the tetrahedron still looks the same, except that the vertices
may have moved around. For example, you may keep the vertex
numbered 1 fixed, and spin the tetrahedron 120
counterclockwise,
which will shift the other three vertices. The effect of this spinning
on the vertices is the permutation
1 1
2 3
3 4
4 2.

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