Linear representation theory of cyclic group:Z3

This article discusses the linear representation theory of cyclic group:Z3, a group of order three.

Summary information
Below is summary information on irreducible representations. Note that a particular representation may make sense, and be irreducible, only for certain kinds of fields -- see the "Values not allowed for field characteristic" and "Criterion for field" columns to see the condition the field must satisfy for the representation to be irreducible there.

Trivial representation
This is a one-dimensional representation sending all the elements to $$( 1 )$$, and makes sense over any field.

One-dimensional nontrivial representations
The cyclic group of order three has two non-identity elements. Also, in a field with a primitive cube root of unity, there are two such primitive cube roots of unity.

The two one-dimensional nontrivial representations both send the identity element to $$(1)$$ and the two non-identity elements to the two primitive cube roots of unity. The representations differ in terms of which element of the group is matched with which primitive cube root of unity.

Two-dimensional representation: irreducible in the non-splitting case
If we denote the group as $$\{ e,x,x^2\}$$ where $$e$$ is the identity element, we can construct the following representation with integer matrices:

We can interpret this representation over any field $$F$$, and three cases arise as to the nature of $$F$$:

Our case of current interest is the first one, i.e., where $$F$$ does not have characteristic $$3$$ and does not contain a primitive cube root of unity.

In the case $$F$$ is the field is the field of real numbers, or more generally, any subfield of the reals containing $$\mathbb{Q}[\sqrt{3}]$$, we can provide an alternative description of this representation as rotations by multiples of $$2\pi/3$$. Note that this alternative description, though it gives an equivalent representation, does not work over fields that lack a square root of $$3$$.

Character table over a splitting field
Let $$\omega$$ be a primitive cube root of unity. The character table over a splitting field is as follows:

Note that this character table is interpreted differently depending on what the splitting field is and which of the primitive cube roots we choose to be $$\omega$$. Switching the roles of $$\omega$$ and $$\omega^2$$ in the above table simply permutes the two nontrivial one-dimensional representations and has no effect on the overall character table.

In characteristic zero, $$\omega$$ can be taken as $$e^{2\pi i/3}$$ or $$\cos(2\pi/3) + i\sin(2\pi/3)$$, which is $$(-1 + i\sqrt{3})/2$$. $$\omega^2$$ is the other primitive cube root of unity, and is given as $$e^{-2\pi i/3}$$ or $$\cos(2\pi/3) - i\sin(2\pi/3)$$ or $$(-1 - i\sqrt{3})/2$$.

Character table over a non-splitting field
For a field that is not a splitting field for the group, there are only two equivalence classes of irreducible representations. But also, the number of Galois conjugacy classes is two. The character table looks as follows:

Over a finite field, the character values are interpreted as integers modulo the field characteristic; over an infinite field, they are interpreted as rational numbers and hence field elements.

If doing character theory over the real numbers, we know that the number of irreducible representations over reals equals number of real conjugacy classes. The above is the character table both over the rationals and over the reals.