142 CHAPTER 12

In other words, this spin of the tetrahedron can be thought of as

producing an element of A

4

.

In fact, every element of A

4

comes from some way of spinning

the tetrahedron. You may have some difﬁculty ﬁnding the rotation

that corresponds to the permutation (12)(34), but it does exist. You

can ﬁnd it if you play long enough with a physical model of a

tetrahedron.

What we have done is to represent A

4

as permutations of the

vertices of a tetrahedron. Each element of A

4

gives a different

permutation. There is no way to permute the vertices of the

tetrahedron via a rotation other than by the permutations listed

in A

4

. In other words, the permutation (12), which ﬂips vertices

1 and 2 and leaves vertices 3 and 4 unchanged, is not a possible

permutation of the vertices that can be achieved by rotating the

tetrahedron, nor are any of the other 11 permutations listed above

as elements of

{1,2,3,4}

that are not in A

4

.

We have thus given our abstract group A

4

amorephysical

interpretation. The classical terminology is that A

4

is the group of

orientation-preserving symmetries of the tetrahedron. Now we can

also make a connection with ideas mentioned earlier. Look back at

the deﬁnition of the group SO(3) in chapter 2. It is the group of

rotations of a rigid sphere.

Representations of A

4

We now take our ivory sphere out of its spherical box, erasing

whatever dots and lines we have drawn on it so it is blank, and

carefully draw on it a perfect tetrahedron. This means putting four

dots on the sphere that are all equally spaced from one another. See

Figure 12.2.

Put the sphere back into its container, and on the inside of the

container put four dots touching the dots on the sphere, and number

them the same way, 1, 2, 3, 4, so that dot 1 touches dot 1, dot 2

touches dot 2, and so on. Now you are ready to play.

Rotate the sphere inside the box. You are applying a group

element from SO(3). Unless you are very careful, the four dots

GROUP REPRESENTATIONS 143

Figure 12.2: A tetrahedron with a sphere

on the sphere will no longer exactly touch the four dots on the

box. So be careful when doing this. We collect together in a set all

the elements of SO(3) that do end up with the dots on the sphere

touching the dots on the box, possibly in a new way. For example, if

you rotate the sphere around the axis that goes through dot 1 and

the center of the sphere exactly 120

◦

, the dots will line up, as we

noticed above.

Take an orange or a ball and try this out. How many different

rotations can you ﬁnd that keep the dots lined up? Remember that

one possibility is the neutral rotation e which leaves everything in

place.

You should be able to convince yourself that each element of

A

4

tells how to spin the tetrahedron around, and each of those

rotations gives us an element of SO(3). What we have found is

a way to think of elements of A

4

as determined by rotations

in SO(3).

The important thing is that this is a morphism. This morphism

is a function that we call r.Ifσ is a permutation in A

4

,thenr(σ )is

the rotation of the sphere that comes from spinning the tetrahedron

to permute the vertices the way that σ tells us to. So r is a function

from A

4

to SO(3). The morphism property means that if τ is another

permutation in A

4

,thenr(σ ◦ τ ) = r(σ ) ◦ r(τ ). In words: If you ﬁrst

do τ to {1, 2, 3, 4},thendoσ, and then see what rotation performs

that composite permutation of the numbered dots, that is the same

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