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The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations. The symbol i used in the body of the table denotes the imaginary unit: i 2 = −1. Used in a column heading, it denotes the operation of inversion.
Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. 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 ...
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 representation is called an irreducible representation, if these two are the only subrepresentations. Some authors also call these representations simple, given that they are precisely the simple modules over the group algebra []. Schur's lemma puts a strong constraint on maps between irreducible representations.
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 ...
For n = 3, 4 there are two additional one-dimensional irreducible representations, corresponding to maps to the cyclic group of order 3: A 3 ≅ C 3 and A 4 → A 4 /V ≅ C 3. For n ≥ 7, there is just one irreducible representation of degree n − 1, and this is the smallest degree of a non-trivial irreducible representation.
Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure.
Schur's lemma for irreducible representations over the complex numbers states that if a matrix commutes with all matrices of the representation then it is a (complex) multiple of the identity matrix, that is, the set of commuting matrices is isomorphic to the complex numbers .