Linear representations

5. Linear representations#

5.1. Matrix groups#

Let \(M_n\) be the set of all \(n\times n\) matrices \(A=(a_{ij})\), \(i,j=1,2,\ldots,n\). The identity matrix \(\mathbb{I}_n\) is the matrix with all diagonal entries equal to 1 and the remaining entries equal to 0:

\[\begin{split} \mathbb{I}_n=\begin{pmatrix}1&0&\ldots&0\\0&1&\ldots&0\\\vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&1\end{pmatrix} \end{split}\]

Multiplication of two matrices \(A=(a_{ij})\) and \(B=(b_{ij})\) in \(M_n\) produces a matrix

\[ A\cdot B=(\sum_{l=1}^n a_{il}b_{lj})\in M_n \]

The multiplication can be considered as a map

\[ M_n\times M_n\to M_n \]

It is therefore a binary operation.

Proposition

We have the following multiplication rules

  • \(A=\mathbb{I}\cdot A=A\cdot \mathbb{I}_n\)

  • \(A\cdot(B\cdot C)=(A\cdot B)\cdot C\) for any three matrices \(A,B,C\in M_n\)

A matrix \(A\in M_n\) is called invertible, or non-singular, if there exists a matrix \(B\) such that

\[ A\cdot B=B\cdot A=\mathbb{I}_n \]

Definition (Linear group)

The subset of \(M_n\) consisting of invertible matrices is denoted by \(GL_n\) and called the general linear group.

Proposition

The general linear group \(GL_n\) is a group with matrix multiplication as the binary operation

We can consider different linear groups by specyfying the domain of the matrix elements, e.g.:

  • \(GL_n(\mathbb{C})\) – invertible matrices with complex entries

  • \(GL_n(\mathbb{R})\) – invertible matrices with real entries

  • \(GL_n(\mathbb{Z})\) – invertible matrices with integer entries

Example (Linear group)

\(GL_2(\mathbb{R})=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}:a,b,c,d\in\mathbb{R}, ad-bc\neq 0\right\}\)

There are various subgroups of \(GL_n\):

  • \(SL_n=\{A\in GL_n:\det A=1\}\)

  • \(O_n=\{A\in GL_n(\mathbb{R}):A\cdot A^T=\mathbb{I}_n\}\)

  • \(SO_n=\{A\in O_n:\det A=1\}\)

  • The matrices \(D=\{\mathbb{I}_2,A,A^2,A^3,B,AB,A^2B,A^3B\}\) where \(A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) and \(B=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\) is a subgroup of \(GL_2\). We encountered this group already when we studied the subgroups of \(S_n\), namely \(D\) has the same multiplication table as the group of symmetries of a square: \(D_4\).

  • Lorentz group: let \(\eta\) be the following matrix

\[\begin{split} \eta=\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix} \end{split}\]

Then the Lorentz group is defined as:

\[ Lor=\{A\in GL_4(\mathbb{R}):A^T\eta A=\eta\}\subset GL_4(\mathbb{R}) \]

5.2. Linear representations#

Our goal is to define a mathematical structure that facilitates the action of groups on vector spaces. The first step is to observe that the set \(GL(V)\) of invertible linear maps \(V\to V\) on an \(n\)-dimensional vector space \(V\) over a field \(\mathbb{F}\) forms a group, called the general linear group of \(V\).The group multiplication is composition of maps, the identity element is the identity map and the iverse is the map inverse.

For \(V=\mathbb{F}^n\) the general linear group reduces to the previously mentioned group of invertible matrices:

\[ GL(\mathbb{F}^n)=GL_n(\mathbb{F}) \]

In a general case, the general linear group \(GL(V)\) acts linearly on the vector space \(V\). For an arbitrary group \(G\) we can find an action on a vector space by assigning to each group element in \(G\) a linear transformation in \(GL(V)\). This assignment should preserve the group structure of \(G\), which means that it should be a group homomorphism \(G\to GL(V)\). This motivates the following definition:

Definition (Representation)

A representation \(R\) of a group \(G\) is a group homomorphism \(R:G\to GL(V)\), where \(V\) is a vector space over \(\mathbb{F}\). The dimension of the representation \(R\) is defined by \(\dim(R)=\dim_\mathbb{F}(V)\).

Examples (Representation)

  • Trivial representation: let \(V\) be any vector space. The trivial representation of \(G\) on \(V\) is the group homomorphism \(G\to GL(V)\) sending every element of \(G\) to the identity transformation. That is, the elements of \(G\) all act on \(V\) trivially by doing nothing.

  • Permutation represtation of \(S_n\): Consider the \(n\)-dimensional vector space \(V\) with basis elements \(e_i\), \(i=1,2,\ldots,n\). Then the symmetric group \(S_n\) acts on \(V\) by permuting basis elements, namely for \(\sigma\in S_n\), the action is:

\[ \sigma.e_i=e_{\sigma(i)} \]

For example, for \(S_3\) this leads to the following three-dimensional representation \(S_3\to GL(\mathbb{R}^3)\):

\[\begin{split} \begin{align*} e\mapsto \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \qquad (12)\mapsto \begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}, \qquad (13)\mapsto \begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}\\ (23)\mapsto \begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}, \qquad (123)\mapsto \begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}, \qquad (132)\mapsto \begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix} \end{align*} \end{split}\]

There is an analogous representation of \(S_n\) on \(\mathbb{R}^n\) for all \(n\).

  • Alternating representation of \(S_n\): For every symmetric group \(S_n\) for any \(n\), there exists a one-dimensional representation defined by the homomorphism: for \(\sigma\in S_n\) we have \(\sigma\mapsto (1)\) if \(\sigma\) is an even permutation; and \(\sigma\mapsto (-1)\) if \(\sigma\) is an odd permutation.

  • Tautological representation of \(D_n\): The tautological representation of \(D_n\) is given by the action of \(D_n\) on regular \(n\)-gon. For example, for \(D_4\) we have a homomorphism

\[ \rho:D_4\to GL(\mathbb{R}^2) \]

that takes the rotation by \(90^\circ\) element \(A\) to \(\rho(A)=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\); and the reflection with respect to the \(x\)-axis \(B\) to \(\rho(B)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\). The tautological representation of \(D_n\) is always two-dimensional.

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