Modular representation theory of cyclic group:Z2

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This article describes the modular representation theory of cyclic group:Z2, i.e., the linear representation theory in characteristic two (this is the only prime in which we have interesting modular behavior, because it is the only prime dividing the order of the group).

For the linear representation theory in other characteristics, see linear representation theory of cyclic group:Z2.

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 2 field:F2, i.e., the field of two elements
degrees of indecomposable representations 1,2
maximum: 2

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) two indecomposable representations:

• The trivial representation, which is also irreducible.
• A two-dimensional indecomposable representation that sends the non-identity element of the group to $\begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}$. This is equivalent to the regular representation.

All representations

Because every finite-dimensional linear representation is expressible as a direct sum of indecomposable linear representations, this means that every finite-dimensional linear representations is expressible in a form where the non-identity element goes to a block diagonal matrix with some blocks looking like the matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ and the rest looking like the identity matrix.

In particular, in degree $n$, the total number of equivalence classes of representations is $(n+2)/2$ if $n$ is even and $(n + 1)/2$ if $n$ is odd, and the representation is described by the number of size 2 indecomposable representations in it.

Degree $n$ Number of equivalence classes of representations ($(n + 2)/2$ or $(n + 1)/2$ depending on parity of $n$) List of decompositions
1 1 trivial only (integer partition: 1)
2 2 indecomposable degree two (integer partition: 2), two copies of trivial (integer partition: 1 + 1)
3 2 indecomposable degree two + trivial (integer partition: 2 + 1), three copies of trivial (integer partition: 1 + 1 + 1)
4 3 two copies of indecomposable degree two (integer partition: 2 + 2), indecomposable degree two + two copies of trivial (integer partition: 2 + 1 + 1), four copies of trivial (integer partition: 1 + 1 + 1 + 1)

Regular representation

The regular representation of cyclic group:Z2 in characteristic two is equivalent as a linear representation to the indecomposable two-dimensional representation.