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  2. List of character tables for chemically important 3D point groups

    en.wikipedia.org/wiki/List_of_character_tables...

    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).

  3. Irreducible representation - Wikipedia

    en.wikipedia.org/wiki/Irreducible_representation

    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.

  4. Vibrational spectroscopy of linear molecules - Wikipedia

    en.wikipedia.org/wiki/Vibrational_spectroscopy...

    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.

  5. Character table - Wikipedia

    en.wikipedia.org/wiki/Character_table

    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).

  6. Representation theory of finite groups - Wikipedia

    en.wikipedia.org/wiki/Representation_theory_of...

    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 ...

  7. Semisimple representation - Wikipedia

    en.wikipedia.org/wiki/Semisimple_representation

    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 ...

  8. Schur's lemma - Wikipedia

    en.wikipedia.org/wiki/Schur's_lemma

    In mathematics, Schur's lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras.In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0.

  9. Schur orthogonality relations - Wikipedia

    en.wikipedia.org/wiki/Schur_orthogonality_relations

    The space of complex-valued class functions of a finite group G has a natural inner product: , := | | () ¯ where () ¯ denotes the complex conjugate of the value of on g.With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table: