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. 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 $e^{2\pi i/3}$ 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

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 $2\pi /3$.

Projective representations

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