Linear representation theory of Klein four-group

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This article gives specific information, namely, linear representation theory, about a particular group, namely: Klein four-group.
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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 \pm 1 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 e and three non-identity elements a,b,c. 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 1.
  • The representation with kernel \{ e, a \}: This sends e and a to 1 and sends b and c to -1.
  • The representation with kernel \{ e, b \}: This sends e and b to 1 and sends a and c to -1.
  • The representation with kernel \{ e, c \}: This sends e and c to 1 and sends a and b to -1.

Character table

Rep/element e a b c
trivial 1 1 1 1
kernel \{e ,a \} 1 1 -1 -1
kernel \{e,b \} 1 -1 1 -1
kernel \{e,c \} 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 \mathbb{Z} -- ring of integers \{ 1 \}
any of the nontrivial representations \mathbb{Z} -- ring of integers \{ 1, -1 \}

Smallest ring of realization as orthogonal matrices

Representation Smallest ring over which it is realized Smallest set of elements in matrix entries
trivial representation \mathbb{Z} -- ring of integers \{ 1 \}
any of the nontrivial representations \mathbb{Z} -- ring of integers \{ 1, -1 \}