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, i.e., the polynomial $x^2+x+1$ splits.
Degrees of irreducible representations over a non-splitting field	1,2,3
Maximum of degrees of irreducible representations over a non-splitting field	3
lcm of degrees of irreducible representations over a non-splitting field	6
Smallest size splitting field	Field:F7, the field with 7 elements.

Representations

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.

Name of representation type	Number of representations of this type	Values not allowed for field characteristic	Criterion for field	What happens over a splitting field?	Kernel	Degree	Schur index
trivial	1	--	any	remains the same	whole group	1	1
three-dimensional irreducible	1	--	any	remains the same	trivial subgroup, i.e., the representation is faithful	3	1
one-dimensional nontrivial	2	3	contains a primitive cube root of unity	remains the same	V4 in A4	1	1
two-dimensional, kernel of order four	1	3	does not contain a primitive cube root of unity	splits into the two one-dimensional nontrivial representations	V4 in A4	2	1

Note that although 2 divides the order of the group, and representations in characteristic two could potentially fall in the modular case, all the irreducible representations do make sense in characteristic two.

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 $1\times 1$ identity matrix $(1)$.

The three-dimensional irreducible representation

There is a unique three-dimensional irreducible representation that works over any field of characteristic 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 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:

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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 $K$ 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:

Coset of $K$ (V4 in A4)	Image of every element of this coset under the first one-dimensional representation	Image of this coset under the second one-dimensional representation
$\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$	1	1
$\{(1,2,3),(2,4,3),(1,4,2),(1,3,4)\}$	$\omega$	${\omega }^2$
$\{(1,3,2),(2,3,4),(1,2,4),(1,4,3)\}$	${\omega }^2$	$\omega$

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.

Degrees of irreducible representations

Summary

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

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

Type of field	Splitting field?	Condition on polynomial	Condition on $q$ for finite field of size $q$	Degrees of irreducible representations
Contains a primitive cube root of unity	Yes	$x^2+x+1$ splits	$3$ divides $q-1$	1,1,1,3
Does not contain a primitive cube root of unity	No	$x^2+x+1$ does not split	$3$ does not divide $q-1$	1,2,3

Character table

Character table over a splitting field

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

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

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	$\omega$	${\omega }^2$
second nontrivial one-dimensional representation	1	1	${\omega }^2$	$\omega$
three-dimensional irreducible representation	3	-1	0	0

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:

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

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:

Representation/conjugacy class representative	$()$	$(1,2)(3,4)$	$(1,2,3)$
trivial representation	1	1	1
irreducible two-dimensional representation	2	2	-1
irreducible three-dimensional representation	3	-1	0

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

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	$\mathbb{Z}$	$\{1,0,-1\}$