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Fearless Symmetry by Robert Gross, Avner Ash

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REPRESENTATIONS 11
We could alter the counting process, and stipulate that a mor-
phism be a one-to-one correspondence from the source to a subset of
the target. But that would allow us to count the three oranges on
our desk as “19, 3, 55,” for example, which is useless.
Or is it? If that is our count, then we know that there are three
oranges, because {19, 3, 55} is a set of three numbers. But how do
we know how many numbers are in the set {19, 3, 55}? We still have
to count them, so this technique has not helped us.
Suppose that we require the count to go in order of size. Then
the above example is invalid, but “3, 19, 55” is valid. As always ,
knowing the last number in the count is the point. In this case,
we would then know that the source has at most 55 objects. This
leads to the concept of less than or equal. We could now generate
the science of inequalities by using this kind of morphism.
6
Summary
If A represents B, we have three things: two objects, A and B,
which from now on will be sets, and the relation between them,
which from now on will be a morphism. When A and B have
some additional “structure”—e.g., they are finite sets, or ordered
sets—and we restrict the possible morphisms from A to B to have
something to do with that structure—e.g., morphisms must be one-
to-one correspondences, or order-preserving functions—then the
existence of a representation from A to B gives us some information
about A in terms of the standard object B—e.g., we can find out
how many elements are in A, or at most how many elements are
in A. Another example of adding structure to a set, allowing a more
profound study of that set, is given by the sets of permutations to
which we will add a group structure;seechapter3.
In this book we explore some very explicit examples of represen-
tations. The things we consider are always mathematical objects
such as sets, groups, matrices, and functions between them. We
6
For a nice discussion of counting and its extension t o infinite sets, see One, Two,
Three ...Infinity (Gamow, 1989).
12 CHAPTER 1
show you how this works in detail in one particular case that
we develop throughout the book and that gets us to our goal:
mod p linear representations of Galois groups. We explain how
these representations help to clarify the general problem of solving
systems of polynomial equations with integer coefficients, and how
they can sometimes lead to definitive results in this area.
Besides the representations discussed in this book, there are
many other kinds of representation theories used today in math-
ematics. Representation theory is often needed to formulate inter-
esting problems, as well as to solve them.

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