Linear representation theory of alternating group:A4

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A4.
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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

Item	Value
Degrees of irreducible representations over a splitting field	1,1,1,3
Maximum degree of irreducible representation over a splitting field	3
lcm of degrees of irreducible representations over a splitting field	3
Smallest ring of realization of all representations (characteristic zero)	$\mathbb{Z}[e^{2\pi i/3})]$
Smallest field of realization of all representations (characteristic zero)	$\mathbb{Q}(e^{2\pi i/3})$
Criterion for a field to be a splitting field	Any field of characteristic not 2 or 3 that contains a primitive cube root of unity (sufficient, also necessary?)
Degrees of irreducible representations over a non-splitting field	1,2,2,3
Maximum of degrees of irreducible representations over a non-splitting field	3
lcm of degrees of irreducible representations over a non-splitting field	6

Representations

The trivial representation

The trivial representation works over all fields. It is a one-dimensional representation that sends every element of the group to the identity matrix.

The three-dimensional irreducible representation

There is a unique three-dimensional irreducible representation that works over any field of chracteristic not equal to $2$. 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 hsa 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.

The 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 $K$. 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). The two representations are, explicitly as follows:

• One representation sends $(1,2,3),(3,2,4),(4,2,1),(1,3,4)$ to $e^{2\pi i/3}$ and $(1,3,2),(3,4,2),(4,1,2),(1,4,3)$ to $e^{-2\pi i/3}$.
• The other representation sends $(1,2,3),(3,2,4),(4,2,1),(1,3,4)$ to $e^{-2\pi i/3}$ and $(1,3,2),(3,4,2),(4,1,2),(1,4,3)$ to $e^{2\pi i/3}$.

The analogues of these representations work over any field that has characteristic not equal to $3$ and has primitive cuberoots of unity.

A two-dimensional irreducible representation over fields not having primitive cuberoots of unity

Suppose a field $F$ has characteristic not equal to $3$ and does not have a primitive cuberoot of unity, the two representations described in the previous section have no analogue.

Instead, there is an irreducible two-dimensional representation over $F$ with kernel the derived subgroup, corresponding to the irreducible two-dimensional representation of the cyclic group of order three over $F$. For instance, over the field of real numbers, such a representation is given by the rotation of multiples of $2\pi /3$ (note that although the description as rotations requires the use of $\sqrt{3}$, there is an alternative description that uses only integers and hence works in any field).

For more on this representation, see linear representation theory of cyclic group:Z3.

This representation is not absolutely irreducible. In fact, if we take a quadratic extension of $F$ by a primitive cube root of unity, the representation splits in this extension as a sum of the two one-dimensional irreducible representations with the same kernel mentioned earlier.

Character table

Representation/conjugacy class representative	$()$	$(1,2)(3,4)$	$(1,2,3)$	$(1,3,2)$
trivial representation	1	1	1	1
first nontrivial one-dimensional representation	1	1	$e^{2\pi i/3}$	$e^{-2\pi i/3}$
second nontrivial one-dimensional representation	1	1	$e^{-2\pi i/3}$	$e^{2\pi i/3}$
three-dimensional irreducible representation	3	-1	0	0

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:

Representation/conjugacy class representative	$()$	$(1,2)(3,4)$	$(1,2,3)$	$(1,3,2)$
trivial representation	1	3	4	4
first nontrivial one-dimensional representation	1	3	$4e^{2\pi i/3}$	$4e^{-2\pi i/3}$
second nontrivial one-dimensional representation	1	3	$4e^{-2\pi i/3}$	$4e^{2\pi i/3}$
three-dimensional irreducible representation	1	-1	0	0

Degrees of irreducible representations

Described below for a field of characteristic not $2$ or $3$:

Type of field	Condition on polynomial	Condition on $q$ for field of size $q$	Degrees of irreducible representations
Contains a primitive cuberoot of unity	$x^2+x+1$ splits	$3$ divides $q-1$ $1,1,1,3$
Does not contain a primitive cuberoot of unity	$x^2+x+1$ does not split	$3$ does not divide $q-1$	$1,2,3$

Realization information

Smallest ring of realization

Representation	Smallest ring of realization	Smallest set of elements that can be used as matrix entries for the ring
trivial representation	$\mathbb{Z}$ -- ring of integers	$\{1\}$
first nontrivial irreducible representation	$\mathbb{Z}[e^{2\pi i/3}]$	$\{1,e^{2\pi i/3},e^{-2\pi i/3}\}$
second nontrivial irreducible representation	$\mathbb{Z}[e^{2\pi i/3}]$	$\{1,e^{2\pi i/3},e^{-2\pi i/3}\}$
irreducible three-dimensional representation	$\mathbb{Z}$	$\{1,0,-1\}$
irreducible two-dimensional representation over fields not containing a primitive cuberoot of unity	$\mathbb{Z}$	$\{1,0,-1\}$