Modular representation theory of cyclic group of prime power order

This article describes the linear representation theory of a cyclic group of prime power order in characteristic equal to the underlying prime, i.e., the linear representation theory of a cyclic group of order $$p^r$$ in characteristic $$p$$.

Irreducible representations
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) the representations of degree $$d$$ that send the generator to the upper-triangular single Jordan block with 1s on the diagonal and first superdiagonal. Here, $$1 \le d \le p^r$$, and we thus get $$p^r$$ distinct indecomposable representations.