Modular representation theory of alternating group:A4 at 3
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 three, specifically, for field:F3 and its extensions.
For information on the linear representation theory in characteristic two (the other modular case) see modular representation theory of alternating group:A4 at 2.
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 of realization of all irreducible representations (equivalently, degrees of irreducible Brauer characters) | 1,3 maximum: 3, lcm: 3, number: 2 |
| unique smallest field of realization of irreducible representations in characteristic three | field:F3 |
Family contexts
| Family name | Parameter values | General discussion of modular representation theory of family at prime |
|---|---|---|
| alternating group | degree | modular representation theory of alternating groups at 3 |
| projective special linear group of degree two | field:F3, i.e., the group is | modular representation theory of projective special linear group over a finite field in its defining characteristic |
Irreducible representations
Summary information
| Name of representation type | Number of representations of this type | Field of realization | What happens over a splitting field? | Kernel | Degree | Liftable to ordinary representation (characteristic zero)? | How does the behavior differ from the non-modular case? |
|---|---|---|---|---|---|---|---|
| trivial | 1 | field:F3 | remains the same | whole group | 1 | Yes | no difference |
| restriction of standard representation reduced mod 3 | 1 | field:F3 | remains the same | trivial subgroup | 3 | Yes | no difference |
Character table
| Representation/conjugacy class representative and size | -- 3-regular, size 1 | -- 3-regular, size 3 | -- not 3-regular | -- not 3-regular |
|---|---|---|---|---|
| trivial | 1 | 1 | 1 | 1 |
| restriction of standard, reduced mod 3 | 0 | 2 | 0 | 0 |
Brauer characters
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 |
|---|---|---|---|
| trivial | trivial | 1 | 1 |
| restriction of standard, reduced mod 3 | restriction of standard | 3 | -1 |