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Fearless Symmetry by Robert Gross, Avner Ash

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THE GREEKS HAD A NAME FOR IT 163
Traces
There are some amazing facts about finite groups and their linear
representations. To explain them, we first need to talk about the
trace of a square matrix.
DEFINITION: The trace of a square matrix is the sum of the
diagonal elements.
The “diagonal elements” are those elements that go from the
upper left-hand corner to the lower right-hand corner of a square
matrix.
EXAMPLE: The trace of
1234
5678
9 101112
13 14 15 16
is 1 + 6 + 11 + 16 = 34.
We apply this apparently innocuous definition to matrix repre-
sentations of groups.
DEFINITION: For any matrix representation r of the finite
group H, the character of an element g in H under r is the
trace of r(g). Often the character is written χ
r
(g), where the
Greek letter χ is used because it is the initial letter of the
Greek word that means “character.
Here is a concrete example. Back on page 145, we defined
a representation V of the permutation group
{1,2,3,4}
which was
a morphism of
{1,2,3,4}
to GL(4, Z), the group of 4-by-4 integer
164 CHAPTER 15
matrices. For instance, V takes the permutation σ , described by
1 1
2 3
3 4
4 2
to the matrix
M =
1000
0001
0100
0010
.
Then χ
V
(σ ) = trace(M) = 1 + 0 + 0 + 0 = 1.
In Greek, the word “character” denotes some outstanding fea-
ture of a thing that enables us to identify that thing. Here
is the definition from Liddell and Scott’s Greek Lexicon (school
edition):
that which is cut in or marked, the impress or stamp on coins,
seals, etc. ...metaphorically the mark or token impressed on
a person or thing, a characteristic, distinctive mark, charac-
ter ... a likeness, image, exact representation.
It is the middle meaning that is relevant to us here, but notice how
at the end of the definition, the meaning is tied up with the concept
of representation!
Suppose that H is a group and r is an n-dimensional linear
representation
1
of H over k, where k is some field. This means
that r is a function from H to the group GL(n, k), and that this
function is a morphism (i.e., r(h) = r(g)r(h)). Then the character
χ
r
is a function from H to k. In symbols, if r : H GL(n, k), then
χ
r
: H k.
1
Remember that “linear representation” is synonymous with “matrix representation.”
We retain both terms for the sake of elegant variation.

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