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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 ...
This is the double cover of SE(2) (see projective representation). We have two cases, one where irreps are described by an integral multiple of 1 / 2 called the helicity, and the other called the "continuous spin" representation. For the third case The only finite-dimensional unitary solution is the trivial representation called the vacuum.
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).
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.
There are three types of irreducible real representations of a finite group on a real vector space V, as Schur's lemma implies that the endomorphism ring commuting with the group action is a real associative division algebra and by the Frobenius theorem can only be isomorphic to either the real numbers, or the complex numbers, or the quaternions.
Corollary (Maschke's theorem) — Every representation of a finite group over a field with characteristic not dividing the order of is a direct sum of irreducible representations. [ 6 ] [ 7 ] The vector space of complex-valued class functions of a group G {\displaystyle G} has a natural G {\displaystyle G} -invariant inner product structure ...
If the representation V is faithful, then every irreducible representation is contained in some tensor power , and the McKay graph of V is connected. The McKay graph of a finite subgroup of SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} has no self-loops, that is, n i i = 0 {\displaystyle n_{ii}=0} for all i .