Modular representation theory of alternating group:A4 at 2

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

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

Item Value
degrees of irreducible representations over a field realizing all irreducible representations (or equivalently, degrees of irreducible Brauer characters) 1,1,1
maximum: 1, lcm: 1, number: 3
unique smallest field of realization of irreducible representations in characteristic 2 field:F4, the field of four elements
degrees of irreducible representations over the prime field field:F2 1,2

Irreducible representations

Summary information

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

Name of representation type Number of representations of this type Degree Criterion for field Kernel Liftable to ordinary representation (characteristic zero)? How does the behavior differ from the non-modular case?
trivial 1 1 field:F2 whole group Yes No difference
one-dimensional mapping to primitive cube roots of unity 2 1 field:F4 Klein four-subgroup of alternating group:A4 -- comprising the identity and the three double transpositions Yes, over $\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive cube root of unity The map surjects to $\mathbb{F}_4^\ast$

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

Name of representation type Number of representations of this type Degree Criterion for field What happens over a field where it reduces completely? Kernel Liftable to ordinary representation (characteristic zero)? How does the behavior differ from the non-modular case?
two-dimensional representation over a non-splitting field 1 field:F2 splits into the two nontrivial one-dimensionals Klein four-subgroup of alternating group:A4 2 Yes No difference

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:

Rep/conj class representative $()$ -- 2-regular $(1,2,3)$ -- 2-regular $(1,3,2)$ -- 2-regular $(1,2)(3,4)$ (not 2-regular)
trivial 1 1 1 1
one of the one-dimensional nontrivial ones 1 $\alpha$ $\alpha^2$ 1
the other one-dimensional nontrivial one 1 $\alpha^2$ $\alpha$ 1

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.

Brauer character table

Irreducible representation in characteristic two whose Brauer character we are computing Irreducible representation in characteristic zero whose character equals the Brauer character Value of Brauer character on conjugacy class of $()$ Value of Brauer character on conjugacy class of $(1,2,3)$ Value of Brauer character on conjugacy class of $(1,3,2)$
trivial trivial 1 1 1
one of the one-dimensional nontrivial ones one of the one-dimensional nontrivial ones 1 $\zeta$ $\zeta^2$
the other one-dimensional nontrivial one the other one-dimensional nontrivial one 1 $\zeta^2$ $\zeta$