Modular representation theory of alternating group:A4 at 2

This article discusses the modular representation theory of alternating group:A4 in characteristic two, i.e., the linear representation theory in chraacteristic two, specifically, for field:F2 and its extensions.

For information on the linear representation theory in characteristic three (the other modular case) see modular representation theory of alternating group:A4 at 3.

For information on linear representation theory in other characteristics (including characteristic zero, the typical case), see linear representation theory of alternating group:A4.

Summary information
Below are the absolutely irreducible representations, i.e., representations that are irreducible in any field of characteristic two realizing them:

Below are representations that are irreducible over some fields but are not absolutely irreducible, i.e., they split in a bigger field:

Character table
Denote by $$\alpha$$ one of the elements in field:F4 not in field:F2, hence $$\alpha$$ is a primitive cube root of unity. Note that $$\alpha^2 = \alpha + 1$$ is the other primitive cube root of unity. The character table is as follows:

Brauer characters
The Brauer characters can all be realized over $$\mathbb{Q}(\zeta)$$ where $$\zeta$$ is a primitive cube root of unity. This field is isomorphic to $$\mathbb{Q}[x]/(x^2 + x + 1)$$ and can be viewed as a subfield of the complex number where $$\zeta = e^{2\pi i/3} = (-1 + i\sqrt{3})/2$$.

To establish Brauer characters, we need to choose a bijection between the primitive cube roots of unity in field:F4 and those in characteristic zero. One such bijection is $$\alpha \mapsto \zeta, \alpha^2 \mapsto \zeta^2$$. The other bijection would be $$\alpha \mapsto \zeta^2, \alpha^2 \mapsto \zeta$$. The choice of bijection simply affects the correspondence between modular characters and ordinary characters, and does not affect the set of Brauer characters.

Since all characters are one-dimensional, the Brauer characters actually are ordinary characters in the most straightforward way imaginable.