Linear representation theory of generalized dihedral groups

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This article gives specific information, namely, linear representation theory, about a family of groups, namely: generalized dihedral group.
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This article discusses the linear representations of the generalized dihedral group corresponding to a finite abelian group H. This group is defined as:

G := \langle H,x \mid x^2 = e, xhx = h{^-1} \ \forall \ h \in H \rangle.

The irreducible representations

For the discussion below, we let n denote the order of H. We let S denote the set of squares in H, and K = H/S is the quotient group. K is an elementary abelian 2-group, and we denote its order by 2^k.

One-dimensional representations

There are 2^{k+1} of these, described as follows.

Since K is an elementary abelian group of order 2^k, it has 2^k one-dimensional representations. Each of these gives rise to a one-dimensional representation of H, by composing with the quotient map from H to K. Further, each such representation takes values \pm 1.

For every such representation \rho of H, there are two corresponding one-dimensional representations of G whose restriction to H is \rho: one representation sends the element x to (1), and the other representation sends x to -1. Thus, we get a total of 2^{k+1} one-dimensional representations.

Two-dimensional irreducible representations

These two-dimensional representations arise from all the representations of H that do not descend to K. There are n - 2^k of these.

Start with any representation \rho of H that does not have S in its kernel. Consider the induced representation to G. Then, this induced representation is irreducible. Further, the induced representations for two representations are equal if and only if they are complex conjugates of each other -- this can readily be verified by looking at character values.

Since \rho does not descend to K, it is not equal to its complex conjugate. Thus, we obtain (n - 2^k)/2 inequivalent two-dimensional irreducible representations this way.

Orthogonality relations and numerical checks