Permutation representation

Last updated December 26, 2020

In mathematics, the term permutation representation of a (typically finite) group $G$ can refer to either of two closely related notions: a representation of $G$ as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two.

Abstract permutation representation

A permutation representation of a group $G$ on a set $X$ is a homomorphism from $G$ to the symmetric group of $X$ :

\rho \colon G\to \operatorname {Sym} (X).

The image $\rho (G)\subset \operatorname {Sym} (X)$ is a permutation group and the elements of $G$ are represented as permutations of $X$ .^[1] A permutation representation is equivalent to an action of $G$ on the set $X$ :

G\times X\to X.

See the article on group action for further details.

Linear permutation representation

If $G$ is a permutation group of degree $n$ , then the permutation representation of $G$ is the linear representation of $G$

\rho \colon G\to \operatorname {GL} _{n}(K)

which maps $g\in G$ to the corresponding permutation matrix (here $K$ is an arbitrary field).^[2] That is, $G$ acts on $K^{n}$ by permuting the standard basis vectors.

This notion of a permutation representation can, of course, be composed with the previous one to represent an arbitrary abstract group $G$ as a group of permutation matrices. One first represents $G$ as a permutation group and then maps each permutation to the corresponding matrix. Representing $G$ as a permutation group acting on itself by translation, one obtains the regular representation.

Character of the permutation representation

Given a group $G$ and a finite set $X$ with $G$ acting on the set $X$ then the character $\chi$ of the permutation representation is exactly the number of fixed points of $X$ under the action of $\rho (g)$ on $X$ . That is $\chi (g)=$ the number of points of $X$ fixed by $\rho (g)$ .

This follows since, if we represent the map $\rho (g)$ with a matrix with basis defined by the elements of $X$ we get a permutation matrix of $X$ . Now the character of this representation is defined as the trace of this permutation matrix. An element on the diagonal of a permutation matrix is 1 if the point in $X$ is fixed, and 0 otherwise. So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of $X$ .

For example, if $G=S_{3}$ and $X=\{1,2,3\}$ the character of the permutation representation can be computed with the formula $\chi (g)=$ the number of points of $X$ fixed by $g$ . So

\chi ((12))=\operatorname {tr} ({\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}})=1

as only 3 is fixed

\chi ((123))=\operatorname {tr} ({\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}})=0

as no elements of

X

are fixed, and

\chi (1)=\operatorname {tr} ({\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}})=3

as every element of

X

is fixed.

Related Research Articles

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix $A$ is denoted $det(A)$ , $det A$ , or $| A |$ . Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the $n$ -dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear transformation preserves or reverses the orientation of a real vector space.

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

In linear algebra, the trace of a square matrix $A$ , denoted $, is defined to be the sum of elements on the main diagonal of A .$

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset $of the quaternions under multiplication. It is given by the group presentation$

In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say $P$ , represents a permutation of $m$ elements and, when used to multiply another matrix, say $A$ , results in permuting the rows or columns of the matrix $A$ .

In mathematics, a generalized permutation matrix is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is

Representation of a Lie group Group representation

In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

In mathematics, the affine group or general affine group of any affine space over a field $K$ is the group of all invertible affine transformations from the space into itself.

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space $together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.$

In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.

In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.

In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation, master equation in Lindblad form, quantum Liouvillian, or Lindbladian is the most general type of Markovian and time-homogeneous master equation describing evolution of the density matrix $ρ$ that preserves the laws of quantum mechanics.

In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel.

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.

In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.

This is a glossary of representation theory in mathematics.

References

↑ Dixon, John D.; Mortimer, Brian (2012-12-06). Permutation Groups. Springer Science & Business Media. pp. 5–6. ISBN 9781461207313.
↑ Robinson, Derek J. S. (2012-12-06). A Course in the Theory of Groups. Springer Science & Business Media. ISBN 9781468401288.

External links

https://mathoverflow.net/questions/286393/how-do-i-know-if-an-irreducible-representation-is-a-permutation-representation

This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Dixon, John D.; Mortimer, Brian (2012-12-06). Permutation Groups. Springer Science & Business Media. pp. 5–6. ISBN 9781461207313.

[2] Robinson, Derek J. S. (2012-12-06). A Course in the Theory of Groups. Springer Science & Business Media. ISBN 9781468401288.

[1]

[2]