Linear representation theory of Klein four-group
From Groupprops
This article gives specific information, namely, linear representation theory, about a particular group, namely: Klein four-group.
View linear representation theory of particular groups | View other specific information about Klein four-group
The Klein four-group is an example of a rational representation group in the sense that all its representations can be realized over the field of rational numbers. In fact, all its representations can be realized over the two-element set and are one-dimensional, hence the representations described for characteristic zero generalize to any situation where the characteristic is not two.
We describe the Klein four-group as a four-element group with identity element and three non-identity elements
. Recall that each of the non-identity elements has order two and the product of any two distinct ones among them is the third one.
Representations
There are four irreducible representations, all one-dimensional:
- The trivial representation: This sends all four elements to
.
- The representation with kernel
: This sends
and
to
and sends
and
to
.
- The representation with kernel
: This sends
and
to
and sends
and
to
.
- The representation with kernel
: This sends
and
to
and sends
and
to
.
Character table
Rep/element | ![]() |
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---|---|---|---|---|
trivial | 1 | 1 | 1 | 1 |
kernel ![]() |
1 | 1 | -1 | -1 |
kernel ![]() |
1 | -1 | 1 | -1 |
kernel ![]() |
1 | -1 | -1 | 1 |
Realizability information
Smallest ring of realization
Representation | Smallest ring over which it is realized | Smallest set of elements in matrix entries |
---|---|---|
trivial representation | ![]() |
![]() |
any of the nontrivial representations | ![]() |
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Smallest ring of realization as orthogonal matrices
Representation | Smallest ring over which it is realized | Smallest set of elements in matrix entries |
---|---|---|
trivial representation | ![]() |
![]() |
any of the nontrivial representations | ![]() |
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