Let act on by . Its character is zero unless and are conjugate, in which case it equals . Character column orthogonality gives the same character forwhere the run through the irreducible complex representations. Maschke's theorem therefore gives an isomorphism of the two representations.
A bilinear form on is invariant exactly when is a -homomorphism . On an irreducible representation, Schur's lemma makes a nonzero such form nondegenerate and unique up to scale.
Transposing a nonzero invariant bilinear form on an irreducible complex representation gives a scalar multiple . Transposing twice yields , so the form is symmetric or alternating.
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Dual representation refers to the ability to understand and represent the same information in different ways or formats. This concept is often discussed in various fields, including psychology, education, and cognitive science, particularly in relation to learning and comprehension. In the context of cognitive development, particularly in children, dual representation is exemplified by the ability to understand that a model or symbol (such as a map or a scale model) can represent something else in the real world.