In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of .
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 upper triangular unipotent matrices is indecomposable but reducible.
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In mathematics, specifically in the representation theory of groups and algebras, an irreduciblerepresentation ( ρ , V ) {\displaystyle (\rho ,V)} or...
of an irreduciblerepresentation are the scalar multiples of the identity. Irreducible representations are the building blocks of representation theory...
{\displaystyle W} be an irreduciblerepresentation of H {\displaystyle H} and let V {\displaystyle V} be an irreduciblerepresentation of G , {\displaystyle...
where Tr is the trace. A character χρ is called irreducible or simple if ρ is an irreduciblerepresentation. The degree of the character χ is the dimension...
partitions of n. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers...
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple...
the only irreducible representations. Irreducibility follows from the unitarian trick and that a representation Π of SU(2) × SU(2) is irreducible if and...
finite-dimensional representation of SO(3) decomposes as a direct sum of irreducible representations. The group SO(3) has one irreduciblerepresentation in each...
representations of T. (Note that a given irreduciblerepresentation of T may occur more than once.) Now, each irreduciblerepresentation of T is described by a linear...
field, the regular representation decomposes as a direct sum of irreducible representations, with each irreduciblerepresentation appearing in the decomposition...
picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes (see below) of irreducible unitary representations...
unknown character as a linear combination of irreducible characters, i.e. # of copies of irreduciblerepresentation Vi in V = ⟨ χ , χ i ⟩ {\displaystyle V=\left\langle...
(finite-dimensional) representation is irreducible, then the dual representation is also irreducible—but not necessarily isomorphic to the original representation. On...
finite-dimensional, irreducible projective representation of SO(3) comes from a finite-dimensional, irreducible ordinary representation of SU(2). Notable...
restricted representation is simpler to understand. Rules for decomposing the restriction of an irreduciblerepresentation into irreducible representations...
real-linear representation is irreducible if and only if the corresponding complex-linear representation is irreducible. Moreover, a complex semisimple...
different quantum states of an elementary particle give rise to an irreduciblerepresentation of the Poincaré group. Moreover, the properties of the various...
and only if every irreduciblerepresentation of G occurs as a subrepresentation of SnV (the n-th symmetric power of the representation V) for a sufficiently...
an irreduciblerepresentation of Sn may not be irreducible for Sn−1. Instead, it may be a direct sum of several representations that are irreducible for...
0 case every irreduciblerepresentation is a direct summand of the regular representation, hence is projective. Modular representation theory was developed...
while the third representation (τ) is irreducible. A set-theoretic representation (also known as a group action or permutation representation) of a group...
to ensure the fraction is actually irreducible. Every rational number has a unique representation as an irreducible fraction with a positive denominator...
from S4 to S3 also yields a 2-dimensional irreduciblerepresentation, which is an irreduciblerepresentation of a symmetric group of degree n of dimension...
spin representation of Spin(5). An example of a non-unitary quaternionic representation would be the two dimensional irreduciblerepresentation of Spin(5...
X ∈ g {\displaystyle X\in {\mathfrak {g}}} . A nonzero representation is said to be irreducible if the only invariant subspaces are V {\displaystyle V}...
corresponding to the B1 irreduciblerepresentation. Likewise, the 2pz orbital is seen to have the symmetry of the A1 irreduciblerepresentation (i.e.: none of...
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