Linear representation theory of generalized dihedral groups
This article gives specific information, namely, linear representation theory, about a family of groups, namely: generalized dihedral group.
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The irreducible representations
For the discussion below, we let denote the order of . We let denote the set of squares in , and is the quotient group. is an elementary abelian -group, and we denote its order by .
There are of these, described as follows.
Since is an elementary abelian group of order , it has one-dimensional representations. Each of these gives rise to a one-dimensional representation of , by composing with the quotient map from to . Further, each such representation takes values .
For every such representation of , there are two corresponding one-dimensional representations of whose restriction to is : one representation sends the element to , and the other representation sends to . Thus, we get a total of one-dimensional representations.
Two-dimensional irreducible representations
These two-dimensional representations arise from all the representations of that do not descend to . There are of these.
Start with any representation of that does not have in its kernel. Consider the induced representation to . 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 does not descend to , it is not equal to its complex conjugate. Thus, we obtain inequivalent two-dimensional irreducible representations this way.
Orthogonality relations and numerical checks
- The total number of irreducible representations is , which equals the number of conjugacy classes.
- The sum of squares of irreducible representations is , which is the order of . This confirms the fact that sum of squares of degrees of irreducible representations equals order of group.
- The degrees of all irreducible representations are or . This confirms the fact that degree of irreducible representation divides index of abelian normal subgroup.