Examples
All representation of the symmetric groups are real (and in fact rational), since we can build a complete set of irreducible representations using Young tableaux.
All representations of the rotation groups are real, since they all appear as subrepresentations of tensor products of copies of the fundamental representation, which is real.
Further examples of real representations are the spinor representations of the spin groups in 8k−1, 8k, and 1 + 8k dimensions for k = 1, 2, 3 ... . This periodicity modulo 8 is known in mathematics not only in the theory of Clifford algebras, but also in algebraic topology, in KO-theory; see spin representation.
Read more about this topic: Real Representation
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