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.
|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|
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 where is a primitive cube root of unity||The map surjects to|
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|
|Rep/conj class representative||-- 2-regular||-- 2-regular||-- 2-regular||(not 2-regular)|
|one of the one-dimensional nontrivial ones||1||1|
|the other one-dimensional nontrivial one||1||1|
The Brauer characters can all be realized over where is a primitive cube root of unity. This field is isomorphic to and can be viewed as a subfield of the complex number where .
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 . The other bijection would be . 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||Value of Brauer character on conjugacy class of|
|one of the one-dimensional nontrivial ones||one of the one-dimensional nontrivial ones||1|
|the other one-dimensional nontrivial one||the other one-dimensional nontrivial one||1|