Linear representation theory of alternating group:A4

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A4.
View linear representation theory of particular groups | View other specific information about alternating group:A4

This article discusses the linear representation theory of the alternating group of degree 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.

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. This is a subgroup of order four, and equals the commutator 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 $e^{2 π i / 3}$ and $(1, 3, 2), (3, 4, 2), (4, 1, 2), (1, 4, 3)$ to $e^{- 2 π i / 3}$ .
The other representation sends $(1, 2, 3), (3, 2, 4), (4, 2, 1), (1, 3, 4)$ to $e^{- 2 π i / 3}$ and $(1, 3, 2), (3, 4, 2), (4, 1, 2), (1, 4, 3)$ to $e^{2 π i / 3}$ .

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

If the field 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, corresponding to the irreducible two-dimensional representation of the cyclic group of order three over such fields. For instance, over the field of real numbers, such a representation is given by the rotation of multiples of $2 π / 3$ .

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	$e^{2 π i / 3}$	$e^{- 2 π i / 3}$
second nontrivial one-dimensional representation	1	1	$e^{- 2 π i / 3}$	$e^{2 π i / 3}$
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	$4 e^{2 π i / 3}$	$4 e^{- 2 π i / 3}$
second nontrivial one-dimensional representation	1	3	$4 e^{- 2 π i / 3}$	$4 e^{2 π i / 3}$
three-dimensional irreducible representation	1	-1	0	0

Degrees of irreducible representations

Described below for a field of characteristic not $2$ or Failed to parse (syntax error): {\displaystyle 3<math>: {| class="wikitable" border="1" ! Type of field !! Condition on polynomial !! Condition on <math>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	$Z$ -- ring of integers	${1}$
first nontrivial irreducible representation	$Z [e^{2 π i / 3}]$	${1, e^{2 π i / 3}, e^{- 2 π i / 3}}$
second nontrivial irreducible representation	$Z [e^{2 π i / 3}]$	${1, e^{2 π i / 3}, e^{- 2 π i / 3}}$
irreducible three-dimensional representation	$Z$	${1, 0, - 1}$
irreducible two-dimensional representation over fields not containing a primitive cuberoot of unity	$Z$	${1, 0, - 1}$