164 CHAPTER 15
matrices. For instance, V takes the permutation σ , described by
1 → 1
2 → 3
3 → 4
4 → 2
to the matrix
(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
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
Suppose that H is a group and r is an n-dimensional linear
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
is a function from H to k. In symbols, if r : H → GL(n, k), then
: H → k.
Remember that “linear representation” is synonymous with “matrix representation.”
We retain both terms for the sake of elegant variation.