Regular representation

This article gives a basic definition in the following area: linear representation theory
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This article describes a notion of representation, or a group action on a certain kind of object.
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Definition

As a group action

The regular representation of a group $G$ over a field $K$ is the permutation representation arising from the regular group action, i.e, the action of the group on itself as a set by left multiplication (or right multiplication if we're using a right action convention).

The regular representation of $G$ is therefore a representation over a vector space with basis indexed by the elements of $G$. In particular, the vector space is a $|G|$-dimensional vector space. In particular, it is finite-dimensional if the group is a finite group.

As a module over the group ring

The regular representation of a group $G$ over a field $K$ corresponds to viewing $K[G]$ as a module over itself with the usual left multiplication.

Facts

• The character of the regular representation takes the value $|G|$ at the identity element and zero elsewhere. This is because the character of a permutation matrix is the number of fixed points, and multiplication by a non-identity element has no fixed points.
• Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree

The regular representation is also used in proving various things such as:

• Splitting implies characters form a basis for space of class functions
• Sum of squares of degrees of irreducible representations equals order of group
• Sum of elements in row of character table of finite group is non-negative integer

Example

Let's work out the regular representation $\rho$ of the cyclic group:Z3 over the field of complex numbers $K=\mathbb {C}$.

Say $\mathbb {Z} _{3}=\langle x:x^{3}=e\rangle$.

We thus associate each element of $\mathbb {Z} _{3}$ with a $|\mathbb {Z} _{3}|=3$-dimensional map in $\mathrm {GL} _{3}(\mathbb {C} )$. We can thus represent these linear maps with $3\times 3$ complex matrices, in the basis where each element of $\mathbb {Z} _{3}$ is a basis vector. Let's say the basis vectors $\mathbf {e_{1}}$, $\mathbf {e_{2}}$, $\mathbf {e_{3}}$ correspond to $e$, $x$, $x^{2}$ respectively.

Then:

$\rho (e)$ is represented by the matrix ${\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}$, $\rho (x)$ is represented by the matrix ${\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\\\end{pmatrix}}$, $\rho (x^{2})$ is represented by the matrix ${\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\\\end{pmatrix}}$.