Linear representation theory of alternating group:A4

This article discusses the linear representation theory of the alternating group of degree four, a group of order four. For convenience, the underlying set is $$\{ 1,2,3,4 \}$$, and permutations are written using the cycle decomposition notation.

See alternating group:A4 and subgroup structure of alternating group:A4 for background information on the group structure.

Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.

Below are representations that are irreducible over a non-splitting field but split up over a splitting field.

Trivial representation
The trivial representation works over all fields. It is a one-dimensional representation that sends every element of the group to the $$1 \times 1$$ identity matrix $$( 1 )$$.

Two one-dimensional representations with kernel of order four
The alternating group of degree four has a unique proper nontrivial normal subgroup, namely V4 in A4. This is a subgroup of order four isomorphic to the Klein four-group, and equals the derived subgroup. It is explicitly given by:

$$\! K := \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$.

There are two one-dimensional representations with kernel $$K$$ over any field that has characteristic not equal to $$3$$ and has primitive cuberoots of unity.. These correspond to the two one-dimensional representations of the quotient group, which is cyclic of order three (see linear representation theory of cyclic group:Z3 for details).

If we denote by $$\omega$$ a primitive cube root of unity, the two representations are explicitly as follows (note that since the representations are one-dimensional, we simply write them as numbers rather than as matrices):

In the case of characteristic zero, $$\omega = e^{2\pi i/3} = \cos(2\pi/3) + i\sin(2\pi/3)$$ where $$i$$ is a square root of $$-1$$.

Three-dimensional irreducible representation
There is a three-dimensional representation that works over any field. Here is one way of describing this representation. Consider the action of the alternating group on a four-dimensional vector space, by permuting the basis vectors through its action on a set of size four. This action has an invariant subspace of codimension one: the subspace comprising vectors whose coordinates add to zero. This gives a three-dimensional vector space on which the alternating group acts, and this is an irreducible representation.

Here is a description of the values of this representation at each element:

Here is what happens with various characteristics of fields:

Two-dimensional representation: irreducible in the non-splitting case
For any field $$F$$, there is a two-dimensional representation over $$F$$ with kernel V4 in A4, given as follows:

For more on the representation of cyclic group:Z3 that gives rise to this representation, see linear representation theory of cyclic group:Z3.

Here are the three cases for the field $$F$$, and what happens in each case:

Summary
Described below for a field of characteristic not $$2$$ or $$3$$ (i.e., the non-modular case):

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 (for instance, over $$\mathbb{C}$$), $$\omega$$ 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 $$(-1 - i\sqrt{3})/2$$. Here are the characters multiplied by conjugacy class size and divided by the degree of the representation. Note that size-degree-weighted characters are algebraic integers:

Character table over a non-splitting field
For a field that is not a splitting field for the group, there are only three equivalence classes of irreducible representations. But also, the number of Galois conjugacy classes is three. Specifically, the two conjugacy classes of 3-cycles become a single conjugacy class. Here is the character table:

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 equivalence classes under real conjugacy and for the rational numbers, we know that the. The above is the character table both over the rationals and over the reals.