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Fearless Symmetry by Robert Gross, Avner Ash

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Before diving back into number theory, we explain the
key concept that puts together groups, permutations,
and matrices: group representations. We will give several
different examples. Except for the last example, which
uses elliptic curves, we mostly keep away from number
theory in this chapter, so the unadorned concept of group
representation can stand out more clearly.
Our central example is that of the symmetries of a
tetrahedron, which shows how geometrical symmetry can
be interpreted using group theory, and then be understood
further using representation theory. This example is a
simple version of many other, more complicated, geomet-
rical symmetry groups and their representations, which
could be the subject matter of a different book.
Morphisms of Groups
At last, we come to the heart of this book, or at least the peri-
cardium. A group representation is nothing more nor less than a
morphism from one group to another group. The reason we use the
special term “representation” is that the target group is chosen to
be one we are especially comfortable with, or one whose properties
are important for understanding the source group. The two types
of representations we will look at are permutation representations
and linear representations.
136 CHAPTER 12
What is a morphism from the group H to the group K?Itis
a function, call it f (x), for example, from H to K with just one
property:
f (x y) = f (x) f (y) (12.1)
for any two elements x and y in H.Inotherwords:f is a rule that
assigns to any element, say x,fromH, exactly one element f (x)from
K. And it does this in such a wa y that (12.1) holds. Note that in
(12.1), the little circle in x y refers to the group law in H, while
that in f (x) f (y) refers to the group law in K. W e summarize all of
this as follows: f : H K is a morphism of groups.
EXAMPLE: We start with an example that captures three of
the most basic facts about multiplication of numbers:
positive × positive = positive.
positive × negative = negative.
negative × negative = positive.
In this example, our source group H is R
×
, the set that
contains all real numbers other than 0, with the group law of
multiplication. The target group K is {+1, 1}, again with the
group law of multiplication. The morphism f is the following
rule:
f (x) =
+1ifx > 0.
1ifx < 0.
You can check that the equation f (xy) = f (x)f (y) will always
be true, if x and y are any nonzero real numbers.
In this equation, we throw away a lot of information about
the source group R
×
and preserve only the sign of the
number. Our representation aggregates the source group into
two clumps—positive numbers and negative numbers—and
ignores all of the other information about real numbers. It
captures the rules for multiplication of signed numbers at the
expense of forgetting about all of the other properties of real
numbers.

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