Projective representation theory of Klein four-group

This article describes the projective representation theory of the Klein four-group in characteristic zero, which we call $$G$$.

First, note that the Schur multiplier $$H^2(G,\mathbb{C}^\ast)$$ is isomorphic to cyclic group:Z2. For more, see group cohomology of Klein four-group.

Summary description in terms of linear representations of a Schur covering group
We can take dihedral group:D8 as a Schur covering group for the Klein four-group (we could also alternatively take the quaternion group). Thus, all projective representations of the Klein four-group arise from ordinary representations of dihedral group:D8. Below is a complete description of the list of irreducible projective representations and the corresponding ordinary representations of dihedral group:D8.

Projective representations for trivial cohomology class
For the trivial cohomology class, there is a unique irreducible projective representation, namely the one-dimensional projective representation induced by the trivial representation. All one-dimensional representations are projectively equivalent to the trivial representation, and they all induce this projective representation.

In addition to the four irreducible representations of the Klein four-group, any one-dimensional representation of any group admitting the Klein four-group as a quotient also induces the trivial projective representation on the Klein four-group.

Projective representations for nontrivial cohomology class
There is a unique irreducible projective representation up to equivalence. This has degree two, and is described below. We denote by $$e$$, $$a$$, $$b$$, $$c$$ the four non-identity elements. Note that the values $$\lambda$$ used for each of the elements may differ from one another.

This projective representation can be realized by choosing a two-dimensional faithful irreducible linear representations of a group of the form $$K$$ with a central subgroup $$N$$ such that $$K/N \cong G$$. To see how, note that by a corollary of Schur's lemma, all elements of $$N$$ must map to scalar matrices. so the map $$K \to GL(2,\mathbb{C})$$ induces a map $$K/N \to PGL(2,\mathbb{C})$$ and hence a map $$G \to PGL(2,\mathbb{C})$$. Here are some examples of choices of $$K$$ and $$N$$ (it also turns out that for the two-dimensional faithful irreducible representation to exist, $$N$$ must be precisely the center, because if the center were any bigger, then the whole group would be abelian and hence all irreducible representations would be one-dimensional):