The algebraic contragredient acts on the entire algebraic dual space. It need not be a smooth representation, even if the original representation is smooth. Taking the inverse in this formula is necessary to obtain a group action, and inverts a one-dimensional linear character.
For a finite-dimensional complex group representation, combining inverse and conjugate transpose gives another representation: both operations reverse multiplication order. This is the complex conjugate of the algebraic dual representation in matrices. If is invariant under , its orthogonal complement is invariant under , because for , . Thus irreducibility is preserved. Applying the construction twice returns , so the equivalence of irreducibility works in both directions for arbitrary groups in finite dimension.
The smooth dual consists of the smooth vectors of the algebraic contragredient representation. Their union is a vector subspace because finite intersections of open stabilizers remain open; it is invariant because translation conjugates stabilizers.
If one-dimensional smooth linear characters of a local multiplicative group have conductors tending to infinity, their direct sum can be admissible. Its algebraic dual is the product of inverse linear characters, but a smooth vector in this product has only finitely many nonzero coordinates. A common open stabilizer contains one fixed principal-unit subgroup, on which every sufficiently ramified linear character is nontrivial. Thus the smooth dual is again a direct sum.
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.