Modular representation theory of alternating group:A4 at 3

From Groupprops
Jump to: navigation, search
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 A_n degree n = 4 modular representation theory of alternating groups at 3
projective special linear group of degree two field:F3, i.e., the group is PSL(2,3) 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 (1,2)(3,4) -- 3-regular, size 3 (1,2,3) -- not 3-regular (1,3,2) -- 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 (1,2)(3,4)
trivial trivial 1 1
restriction of standard, reduced mod 3 restriction of standard 3 -1