Modular representation theory of cyclic group:Z3

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This article gives specific information, namely, modular representation theory, about a particular group, namely: cyclic group:Z3.
View modular representation theory of particular groups | View other specific information about cyclic group:Z3

This article describes the modular representation theory of cyclic group:Z3, the cyclic group of order three, i.e., the linear representation theory of the group in characteristic three, which means the linear representation theory over field:F3 and its extensions.

For the linear representation theory in other characteristics, see linear representation theory of cyclic group:Z3.

Summary

Item Value
degrees of irreducible representations (or equivalently, degrees of irreducible Brauer characters) 1
maximum: 1, lcm: 1, number: 1
smallest field of realization of irreducible representations in characteristic 3 field:F3, i.e., the field of three elements
degrees of indecomposable representations 1,2,3
maximum: 3
smallest field of realization of indecomposable representations in characteristic 3 field:F3, i.e., the field of three elements

Irreducible representations

Further information: Irreducible representation of group of prime power order in characteristic equal to underlying prime is trivial

There is a unique irreducible representation: the trivial representation, which sends all elements of the group to the matrix ( 1 ). This is a general feature common to all representations of a group of prime power order in a field of characteristic equal to the prime.

Indecomposable representations

There are (up to equivalence of linear representations) three indecomposable representations:

Representation Degree Image of identity element Image of generator of group Image of other generator of group
trivial 1 ( 1 ) (1) (1)
two-dimensional indecomposable 2 \begin{pmatrix}1 & 0 \\ 0 & 1 \\\end{pmatrix} \begin{pmatrix} 1 & 1 \\0 & 1 \\\end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}
three-dimensional indecomposable 3 \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\end{pmatrix} \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\\end{pmatrix}