Irreducible representation

Group elements can be represented by matrices, although the term "represented" has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to the general linear group of matrices. As notation, let a, b, c... denote elements of a group G with group product signified without any symbol, so ab is the group product of a and b and is also an element of G, and let representations be indicated by D. The representation of a is written

${\displaystyle D(a)={\begin{pmatrix}D(a)_{11}&D(a)_{12}&\cdots &D(a)_{1n}\\D(a)_{21}&D(a)_{22}&\cdots &D(a)_{2n}\\\vdots &\vdots &\ddots &\vdots \\D(a)_{n1}&D(a)_{n2}&\cdots &D(a)_{nn}\\\end{pmatrix}}}$

By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations:

${\displaystyle D(ab)=D(a)D(b)}$

If e is the identity element of the group (so that ae = ea = a, etc.), then D(e) is an identity matrix, or identically a block matrix of identity matrices, since we must have

${\displaystyle D(ea)=D(ae)=D(a)D(e)=D(e)D(a)=D(a)}$

and similarly for all other group elements. The last two statements correspond to the requirement that D is a group homomorphism.

A representation is reducible if it contains a nontrivial G-invariant subspace, that is to say, all the matrices ${\displaystyle D(a)}$ can be put in upper triangular block form by the same invertible matrix ${\displaystyle P}$. In other words, if there is a similarity transformation:

${\displaystyle D'(a)\equiv P^{-1}D(a)P,}$

which maps every matrix in the representation into the same pattern upper triangular blocks. Every ordered sequence minor block is a group subrepresentation. That is to say, if the representation is of k dimension, then we have:

${\displaystyle D'(a)=P^{-1}D(a)P={\begin{pmatrix}D^{(11)}(a)&D^{(12)}(a)\\0&D^{(22)}(a)\\\end{pmatrix}},}$

where ${\displaystyle D^{(11)}(a)}$ is a nontrivial subrepresentation. If we are able to find a good enough transformation matrix ${\displaystyle P}$ which cancel ${\displaystyle D^{(12)}(a)}$, then ${\displaystyle D(a)}$ is also decomposable.

Notice: Even if a representation is reducible, its matrix representation may still not be the upper triangular block form. In order to get the upper triangular block form matrix representation, the first few bases should be able to span the G-invariant subspace.

A representation is decomposable if all the matrices ${\displaystyle D(a)}$ can be put in block-diagonal form by the same invertible matrix ${\displaystyle P}$. In other words, if there is a similarity transformation:[1]

${\displaystyle D'(a)\equiv P^{-1}D(a)P,}$

which diagonalizes every matrix in the representation into the same pattern of diagonal blocks. Each such block is then a group subrepresentation independent from the others. The representations D(a) and D′(a) are said to be equivalent representations.[2] The representation can be decomposed into a direct sum of k > 1 matrices:

${\displaystyle D'(a)=P^{-1}D(a)P={\begin{pmatrix}D^{(1)}(a)&0&\cdots &0\\0&D^{(2)}(a)&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &D^{(k)}(a)\\\end{pmatrix}}=D^{(1)}(a)\oplus D^{(2)}(a)\oplus \cdots \oplus D^{(k)}(a),}$

so D(a) is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in D⁽ⁿ⁾(a) for n = 1, 2, ..., k, although some authors just write the numerical label without parentheses.

The dimension of D(a) is the sum of the dimensions of the blocks:

${\displaystyle \dim[D(a)]=\dim[D^{(1)}(a)]+\dim[D^{(2)}(a)]+\cdots +\dim[D^{(k)}(a)].}$

If this is not possible, i.e. k = 1, then the representation is indecomposable.[1][3]

Notice: Even if a representation is decomposable, its matrix representation may still not be the diagonal block form. In order to get the diagonal block form matrix representation, the basis should be chosen carefully.

An irreducible representation is by nature an indecomposable one. However, the converse may fail.

But under some conditions, we do have an indecomposable representation being an irreducible representation.

• When group ${\displaystyle G}$ is finite, and it has a representation over field ${\displaystyle \mathbb {C} }$, then an indecomposable representation is an irreducible representation. [4]
• When group ${\displaystyle G}$ is finite, and it has a representation over field ${\displaystyle K}$, if we have ${\displaystyle char(K)\nmid |G|}$, then an indecomposable representation is an irreducible representation.

All groups ${\displaystyle G}$ have a one-dimensional, irreducible trivial representation.

Any one-dimensional representation is irreducible by virtue since it has no proper nontrivial subspaces.

The irreducible complex representations of a finite group G can be characterized using results from character theory. In particular, all such representations decompose as a direct sum of irreps, and the number of irreps of ${\displaystyle G}$ is equal to the number of conjugacy classes of ${\displaystyle G}$.[5]

• The irreducible complex representations of ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ are exactly given by the maps ${\displaystyle 1\mapsto \gamma }$, where ${\displaystyle \gamma }$ is an ${\displaystyle n}$th root of unity.

• Let ${\displaystyle V}$ be an ${\displaystyle n}$-dimensional complex representation of ${\displaystyle S_{n}}$ with basis ${\displaystyle \{v_{i}\}_{i=1}^{n}}$. Then ${\displaystyle V}$ decomposes as a direct sum of the irreps

${\displaystyle V_{\text{triv}}=\mathbb {C} \left(\sum _{i=1}^{n}v_{i}\right)}$

and the orthogonal subspace given by

${\displaystyle V_{\text{std}}=\left\{\sum _{i=1}^{n}a_{i}v_{i}:a_{i}\in \mathbb {C} ,\sum _{i=1}^{n}a_{i}=0\right\}.}$

The former irrep is one-dimensional and isomorphic to the trivial representation of ${\displaystyle S_{n}}$. The latter is ${\displaystyle n-1}$ dimensional and is known as the standard representation of ${\displaystyle S_{n}}$.[5]

• Let ${\displaystyle G}$ be a group. The regular representation of ${\displaystyle G}$ is the free complex vector space on the basis ${\displaystyle \{e_{g}\}_{g\in G}}$ with the group action ${\displaystyle g\cdot e_{g'}=e_{gg'}}$, denoted ${\displaystyle \mathbb {C} G.}$ All irreducible representations of ${\displaystyle G}$ appear in the decomposition of ${\displaystyle \mathbb {C} G}$ as a direct sum of irreps.

• Let ${\displaystyle G}$ be a ${\displaystyle p}$ group and ${\displaystyle V=\mathbb {F} _{p}^{n}}$ be a finite dimensional irreducible representation of G over ${\displaystyle \mathbb {F} _{p}}$. By Orbit-stabilizer theorem, the orbit of every ${\displaystyle V}$ element acted by the ${\displaystyle p}$ group ${\displaystyle G}$ has size being power of ${\displaystyle p}$. Since the sizes of all these orbits sum up to the size of ${\displaystyle G}$, and ${\displaystyle 0\in V}$ is in a size 1 orbit only containing itself, there must be other orbits of size 1 for the sum to match. That is, there exists some ${\displaystyle v\in V}$ such that ${\displaystyle gv=v}$ for all ${\displaystyle g\in G}$. This forces every irreducible representation of a ${\displaystyle p}$ group over ${\displaystyle \mathbb {F} _{p}}$ to be one dimensional.

The irreps of D(K) and D(J), where J is the generator of rotations and K the generator of boosts, can be used to build to spin representations of the Lorentz group, because they are related to the spin matrices of quantum mechanics. This allows them to derive relativistic wave equations.[7]

• Simple module
• Indecomposable module
• Representation of an associative algebra

• Representation theory of Lie algebras
• Representation theory of SU(2)
• Representation theory of SL2(R)
• Representation theory of the Galilean group
• Representation theory of diffeomorphism groups
• Representation theory of the Poincaré group
• Theorem of the highest weight

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• R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
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• Bargmann, V.; Wigner, E. P. (1948). "Group theoretical discussion of relativistic wave equations". Proc. Natl. Acad. Sci. U.S.A. 34 (5): 211–23. Bibcode:1948PNAS...34..211B. doi:10.1073/pnas.34.5.211. PMC 1079095. PMID 16578292.
• E. Wigner (1937). "On Unitary Representations Of The Inhomogeneous Lorentz Group" (PDF). Annals of Mathematics. 40 (1): 149–204. Bibcode:1939AnMat..40..149W. doi:10.2307/1968551. JSTOR 1968551. MR 1503456.