For , the irreducible real representations are the trivial line, the sign line when is even, and the pairwise nonisomorphic real planes on which rotates through for .
The real regular representation of is the direct sum of the irreducible real representations of a finite cyclic group, each once. A real Fourier basis consists of the constant vector, the alternating vector when is even, and cosine-sine pairs for frequencies .
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Real representation can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Mathematics**: In mathematics, particularly in real analysis, a "real representation" often refers to expressing a mathematical object or function explicitly in terms of real numbers. For example, representing complex numbers in terms of their real and imaginary components.