Affine group

Representing the affine group as a semidirect product of V by GL(V), then by construction of the semidirect product, the elements are pairs (M, v), where v is a vector in V and M is a linear transform in GL(V), and multiplication is given by:

${\displaystyle (M,v)\cdot (N,w)=(MN,v+Mw)\,.}$

This can be represented as the (n + 1) × (n + 1) block matrix:

${\displaystyle \left({\begin{array}{c|c}M&v\\\hline 0&1\end{array}}\right)}$

where M is an n × n matrix over K, v an n × 1 column vector, 0 is a 1 × n row of zeros, and 1 is the 1 × 1 identity block matrix.

Formally, Aff(V) is naturally isomorphic to a subgroup of GL(V ⊕ K), with V embedded as the affine plane {(v, 1) | v ∈ V}, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the n × n and 1 × 1) blocks corresponding to the direct sum decomposition V ⊕ K.

A similar representation is any (n + 1) × (n + 1) matrix in which the entries in each column sum to 1.[1] The similarity P for passing from the above kind to this kind is the (n + 1) × (n + 1) identity matrix with the bottom row replaced by a row of all ones.

Each of these two classes of matrices is closed under matrix multiplication.

The simplest paradigm may well be the case n = 1, that is, the upper triangular 2 × 2 matrices representing the affine group in one dimension. It is a two-parameter non-Abelian Lie group, so with merely two generators (Lie algebra elements), A and B, such that [A, B] = B, where

${\displaystyle A=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad B=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right)\,,}$

so that

${\displaystyle e^{aA+bB}=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)\,.}$

Aff(F_p) has order p(p − 1). Since

${\displaystyle {\begin{pmatrix}c&d\\0&1\end{pmatrix}}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\begin{pmatrix}c&d\\0&1\end{pmatrix}}^{-1}={\begin{pmatrix}a&(1-a)d+bc\\0&1\end{pmatrix}}\,,}$

we know Aff(F_p) has p conjugacy classes, namely

${\displaystyle {\begin{aligned}C_{id}&=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}}\right\}\,,\\[6pt]C_{1}&=\left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}^{*}\right\}\,,\\[6pt]{\Bigg \{}C_{a}&=\left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}\right\}{\Bigg |}a\in \mathbf {F} _{p}\setminus \{0,1\}{\Bigg \}}\,.\end{aligned}}}$

Then we know that Aff(F_p) has p irreducible representations. By above paragraph (§ Matrix representation), there exist p − 1 one-dimensional representations, decided by the homomorphism

${\displaystyle \rho _{k}:\operatorname {Aff} (\mathbf {F} _{p})\to \mathbb {C} ^{*}}$

for k = 1, 2,… p − 1, where

${\displaystyle \rho _{k}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}=\exp \left({\frac {2ikj\pi }{p-1}}\right)}$

and i² = −1, a = g^j, g is a generator of the group F^∗
_p. Then compare with the order of F_p, we have

${\displaystyle p(p-1)=p-1+\chi _{p}^{2}\,,}$

hence χ_p = p − 1 is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of Aff(F_p):

${\displaystyle {\begin{array}{c|cccccc}&{\color {Blue}C_{id}}&{\color {Blue}C_{1}}&{\color {Blue}C_{g}}&{\color {Blue}C_{g^{2}}}&{\color {Gray}\dots }&{\color {Blue}C_{g^{p-2}}}\\\hline {\color {Blue}\chi _{1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {2\pi i}{p-1}}}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {2\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{2}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Blue}e^{\frac {8\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {4\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{3}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {6\pi i}{p-1}}}&{\color {Blue}e^{\frac {12\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {6\pi (p-2)i}{p-1}}}\\{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }\\{\color {Blue}\chi _{p-1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}\dots }&{\color {Gray}1}\\{\color {Blue}\chi _{p}}&{\color {Gray}p-1}&{\color {Gray}-1}&{\color {Gray}0}&{\color {Gray}0}&{\color {Gray}\dots }&{\color {Gray}0}\end{array}}}$

According to Rafael Artzy,[2] "The linear part of each affinity [of the real affine plane] can be brought into one of the following standard forms by a coordinate transformation followed by a dilation from the origin:

${\displaystyle {\begin{aligned}{\text{1.}}&&x&\mapsto ax+by\,,&y&\mapsto -bx+ay\,,&a,b&\neq 0\,,&a^{2}+b^{2}=1\,,\\[3pt]{\text{2.}}&&x&\mapsto x+by\,,&y&\mapsto y\,,&b&\neq 0\,,&\\[3pt]{\text{3.}}&&x&\mapsto ax\,,&y&\mapsto {\tfrac {y}{a}}\,,&a&\neq 0\,,&\end{aligned}}}$

where the coefficients a, b, c, and d are real numbers."

Case 1 corresponds to similarity transformations which generate a subgroup of similarities.  Euclidean geometry corresponds to the subgroup of congruences. It is characterized by Euclidean distance or angle, which are invariant under the subgroup of rotations.

Case 2 corresponds to shear mappings. An important application is absolute time and space where Galilean transformations relate frames of reference. They generate the Galilean group.

Case 3 corresponds to squeeze mapping. These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane. The geometry associated with this group is characterized by hyperbolic angle, which is a measure that is invariant under the subgroup of squeeze mappings.

Using the above matrix representation of the affine group on the plane, the matrix M is a 2 × 2 real matrix. Accordingly, a non-singular M must have one of three forms that correspond to the trichotomy of Artzy.

• Holomorph