representation is also a (complex) conjugate representation because the Hilbert space norm gives an antilinear bijective map from the representation to its dual representation. Self-dual complex irreducible representation correspond to either real irreducible representation of the same dimension or real irreducible representations of twice the dimension called quaternionic representations (but not both) and non-self-dual complex irreducible representation correspond to a real irreducible representation of twice the dimension. Note for the latter case, both the complex irreducible representation and its dual give rise to the same real irreducible representation. An example of a quaternionic representation would be the four-dimensional real irreducible representation of the quaternion group Q8. Definition in terms of the symmetric and alternating square If is the underlying vector space of a representation of a group , then the tensor product representation can be decomposed as the direct sum of two subrepresentations, the symmetric square, denoted (also often denoted by or ) and the alternating square, (also often denoted by , , or ). In terms of these square representations, the indicator has the following, alternate definition: where is the trivial representation. To see this, note that the term naturally arises in the characters of these representations; to wit, we haveand.Substituting either of these formulae, the Frobenius–Schur indicator takes on the structure of the natural -invariant inner product on class functions:The inner product counts the multiplicities of direct summands; the equivalence of the definitions then follows immediately. Applications Let be an irreducible complex representation of a group (or equivalently, an irreducible -module, where denotes the group ring). Then There exists a nonzero -invariant bilinear form on if and only if There exists a nonzero -invariant symmetric bilinear form on if and only if There exists a nonzero -invariant skew-symmetric bilinear form on if and only if . The above is a consequence of the universal properties of the symmetric algebra and exterior algebra, which are the underlying vector spaces of the symmetric and alternating square. Additionally, if and only if is not real-valued (these are complex representations), if and only if can be realized over (these are real representations), and if and only if is real but cannot be realized over (these are quaternionic representations). Higher Frobenius-Schur indicators Just as for any complex representation ρ, is a self-intertwiner, for any integer n, is also a self-intertwiner. By Schur's lemma, this will be a multiple of the identity for irreducible