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The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (x, y and z), rotations about those three coordinates (R x, R y and R z), and functions of the quadratic terms of the coordinates(x 2, y 2, z 2, xy, xz, and yz).
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.
To determine which modes are Raman active, the irreducible representation corresponding to xy, xz, yz, x 2, y 2, and z 2 are checked with the reducible representation of Γ vib. [4] A Raman mode is active if the same irreducible representation is present in both.
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.
A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the ...
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.
Isomorphic representations have the same characters. Over a field of characteristic 0, two representations are isomorphic if and only if they have the same character. [1] If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations.