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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).
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.
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 ...
The irreducible representation for the C-O stretching vibration is A 1g + E g + T 1u. Of these, only T 1u is IR active. B 2 H 6 has D 2h molecular symmetry. The terminal B-H stretching vibrations which are active in IR are B 2u and B 3u. Diborane. Fac-Mo(CO) 3 (CH 3 CH 2 CN) 3, has C 3v geometry. The irreducible representation for the C-O ...
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.
Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple ...