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The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
The reducible representation of the bonding of water with C 2v symmetry. An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense, n atomic orbitals combine to form n molecular orbitals, which can be numbered i = 1 to n and which may not all be the same.
The Amar Opening (also known as the Paris Opening, [1] or the Drunken Knight Opening) is a chess opening defined by the move: . 1. Nh3. Analogous to calling the Durkin Opening the "Sodium Attack," this opening could be called the Ammonia Opening, since the algebraic notation 1.Nh3 resembles the chemical formula NH 3 for ammonia.
Just like representations, a corepresentation is reducible if there is a proper subspace invariant under the operations of the corepresentation. If the corepresentation is given by matrices, it is reducible if it is equivalent to a corepresentation with each matrix in block diagonal form.
Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible.
In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible ...
For n = 3 the obvious analogue of the (n − 1)-dimensional representation is reducible – the permutation representation coincides with the regular representation, and thus breaks up into the three one-dimensional representations, as A 3 ≅ C 3 is abelian; see the discrete Fourier transform for representation theory of cyclic groups.
Spacefilling models or CPK models (also an atomic coloring scheme in representations) – the molecule is represented by overlapping spheres representing the atoms. Cartoon – a representation used for proteins where loops, beta sheets, and alpha helices are represented diagrammatically and no atoms or bonds are explicitly represented (e.g ...