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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 ...
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. In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
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 dual of an irreducible representation is always irreducible, [9] but may or may not be isomorphic to the original representation. In the case of the group SU(3), for example, the irreducible representations are labeled by a pair ( m 1 , m 2 ) {\displaystyle (m_{1},m_{2})} of non-negative integers.
Irreducible representations are the building blocks of representation theory for many groups: if a representation is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if is finite-dimensional, then both the subrepresentation and the quotient have smaller dimension ...
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of G.
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).
In mathematics, Maschke's theorem, [1] [2] named after Heinrich Maschke, [3] is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without