Characters, An Introduction
Introduction
In its most general sense, a character χ(G)
is any homomorphism that takes an abelian group G
into the units of a commutative ring R, with χ(0) = 1.
(Sometimes a particular subgroup of R* is specified as the range.)
Since characters are defined on groups and rings,
why are they placed here, below numbers?
In practice characters are often used to address questions in number theory.
For instance, characters help us extend the zeta function, which is instrumental in analytic number theory.
They are also pivotal in higher order reciprocity theorems.
This doesn't mean characters are used nowhere else;
but most of the aplications are in number theory, so characters are described here.
Finitely Generated
When G is finitely generated, it splits into a direct product of cyclic groups.
Map the generators into R, and G follows.
If a generator has order n in G, its image is an nth root of 1 in R.
If R is the complex plane, and G is finite, all of G maps into the unit circle in the plane.
Group of Characters
Given G and R, the characters from G into R define an abelian group.
If f1 and f2 are two such characters, let f3(x) = f1(x)f2(x).
This is still a group homomorphism, still a character, because R is commutative.
Mapping all of G to 1 is the trivial character,
and inverting the images in R produces the inverse character.
Thus the characters combine to form an abelian group.
Assume G is finite, and R is an integral domain.
If χ(G) = 1, then the sum of the images of x over all x in G is |G|.
Otherwise, fix y in G such that χ(y) ≠ 1, and note that y permutes the elements of G.
Thus the sum of χ(x) over all x in G is the same as the sum of χ(x+y).
This in turn is χ(y) times the sum over χ(x).
The sum of the image of G, times a unit other than one, produces the sum of the image of G.
Therefore the image of G sums to 0.
As a corollary, the quotient of two characters produces an image that sums to 0 iff the characters are different.
Next, fix an x in G, and consider the sum over all the images of x, for all the characters of G into R.
(This only makes sense when the group of characters is finite.)
If x maps to 1 across the board, as when x = 0, then the sum of the images of x equals the size of the character group.
Let x be nonzero and let f
be a character that maps x to something other than 1.
The sum over h(x), for all h(G) in the character group, is now the sum over fh(x).
This is f(x) times the sum over h(x),
and as above, the sum over h(x) has to be 0.
Multiplying Images Together
The image of G under a character becomes an abelian subgroup of the units of R.
If G is finite, the product over the image of G is 1,
unless the image contains precisely one involution t, whence the product is t.
This is a consequence of a more general theorem.
Note that t is a square root of 1, and in an integral domain, this is 1 or -1.
Mapping a Finite Group into the Complex Plane
Let c ∈ G generate a cyclic group of order pk.
This maps to any 1 of the pk roots of 1 in the unit circle.
Add two characters together and multiply the images of c, which adds angles, or exponents, around the unit circle.
This holds for all the generators of G,
hence the character group is isomorphic to G.
The characters that embed are those that map generators to primitive roots of 1, according to their cycle lengths.