Modular representation theory of cyclic group:Z3
This article gives specific information, namely, modular representation theory, about a particular group, namely: cyclic group:Z3.
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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.
|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|
|smallest field of realization of indecomposable representations in characteristic 3||field:F3, i.e., the field of three elements|
There is a unique irreducible representation: the trivial representation, which sends all elements of the group to the matrix . 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.
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|