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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