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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 ...
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. In commutative algebra, a ...
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). [1] It is an example of the general mathematical notion of ...
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.
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 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.
The representation of dimension zero is considered to be neither reducible nor irreducible, [1] just as the number 1 is considered to be neither composite nor prime. Under the assumption that the characteristic of the field K does not divide the size of the group, representations of finite groups can be decomposed into a direct sum of ...
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.