example of group action

Let $a,b,c$ be integers and let $[a,b,c]$ denote the mapping

$$[a,b,c]:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z},(x,y)\mapsto a{x}^2+bxy+c{y}^2.$$

Let $G$ be the group of $2\times 2$ matrices such that $detA=\pm 1\forall A\in G$. The substitution

$$\left(\begin{array}{c}x\hfill \\ y\hfill \end{array}\right)\mapsto A\left(\begin{array}{c}x\hfill \\ y\hfill \end{array}\right)$$

leads to

$$[a,b,c](a_{11}x+a_{12}y,a_{21}x+a_{22}y)=a^{{}^{\prime }}{x}^2+b^{{}^{\prime }}xy+c^{{}^{\prime }}{y}^2,$$

where

$a^{{}^{\prime }}$ $=$ $a{a}_{11}^2+b{a}_{11}{a}_{21}+c{a}_{21}^2$ (1)

$b^{{}^{\prime }}$ $=$ $2a{a}_{11}{a}_{12}+2c{a}_{21}{a}_{22}+b(a_{11}{a}_{22}+a_{12}{a}_{21})$

$c^{{}^{\prime }}$ $=$ $a{a}_{12}^2+b{a}_{12}{a}_{22}+c{a}_{22}^2$

So we define

$$[a,b,c]\ast A:=[a^{{}^{\prime }},b^{{}^{\prime }},c^{{}^{\prime }}]$$

to be the binary quadratic form with coefficients $a^{{}^{\prime }},b^{{}^{\prime }},c^{{}^{\prime }}$ of $x^2,xy,y^2$, respectively as in (1). Putting in we have $[a,b,c]\ast A=[a,b,c]$ for any binary quadratic form $[a,b,c]$. Now let $B$ be another matrix in $G$. We must show that

$$[a,b,c]\ast (AB)=([a,b,c]\ast A)\ast B.$$

Set $[a,b,c]\ast (AB):=[a^{{}^{\prime \prime }},b^{{}^{\prime \prime }},c^{{}^{\prime \prime }}]$. So we have

$a^{{}^{\prime \prime }}$ $=$ $a{\left(a_{11}{b}_{11}+a_{12}{b}_{21}\right)}^2+c{\left(a_{21}{b}_{11}+a_{22}{b}_{21}\right)}^2+b\left(a_{11}{b}_{11}+a_{12}{b}_{21}\right)(a_{21}{b}_{11}+a_{22}{b}_{21})$ (2)

$=$ $a^{{}^{\prime }}{b}_{11}^2+c^{{}^{\prime }}{b}_{21}^2+\left(2a\cdot a_{11}{a}_{12}+2c\cdot a_{21}{a}_{22}+b\left(a_{11}{a}_{22}+a_{12}{a}_{21}\right)\right)b_{11}{b}_{21}$

$c^{{}^{\prime \prime }}$ $=$ $a{\left(a_{11}{b}_{12}+a_{12}{b}_{22}\right)}^2+c\cdot {\left(a_{21}{b}_{12}+a_{22}{b}_{22}\right)}^2+b\cdot \left(a_{11}{b}_{12}+a_{12}{b}_{22}\right)\left(a_{21}{b}_{12}+a_{22}{b}_{22}\right)$ (3)

$=$ $a^{{}^{\prime }}{b}_{12}^2+c^{{}^{\prime }}{b}_{22}^2+\left(2a\cdot a_{11}{a}_{12}+2c{a}_{21}{a}_{22}+b\left(a_{11}{a}_{22}+a_{12}{a}_{21}\right)\right)b_{12}{b}_{22}$

as desired. For the coefficient $b^{{}^{\prime \prime }}$ we get

$b^{{}^{\prime \prime }}$ $=$ $2a\left(a_{11}{b}_{11}+a_{12}{b}_{21}\right)\left(a_{11}{b}_{12}+a_{12}{b}_{22}\right)$

$+$ $2c\cdot \left(a_{21}{b}_{11}+a_{22}{b}_{21}\right)\left(a_{21}{b}_{12}+a_{22}{b}_{22}\right)$

$+$ $b\left(\left(a_{11}{b}_{11}+a_{12}{b}_{21}\right)\left(a_{21}{b}_{12}+a_{22}{b}_{22}\right)+\left(a_{11}{b}_{12}+a_{12}{b}_{22}\right)\left(a_{21}{b}_{11}+a_{22}{b}_{21}\right)\right)$

and by evaluating the factors of $b_{11}{b}_{12},b_{21}{b}_{22}$, and $b_{11}{b}_{22}+b_{21}{b}_{12}$, it can be checked that

$$b^{{}^{\prime \prime }}=2a^{{}^{\prime }}{b}_{11}{b}_{12}+2c^{{}^{\prime }}{b}_{21}{b}_{22}+\left(b_{11}{b}_{22}+b_{21}{b}_{12}\right)\left(2a{a}_{11}{a}_{12}+2c\cdot a_{21}{a}_{22}+b\left(a_{11}{a}_{22}+a_{12}{a}_{21}\right)\right).$$

This shows that

$$[a^{{}^{\prime \prime }},b^{{}^{\prime \prime }},c^{{}^{\prime \prime }}]=[a^{{}^{\prime }},b^{{}^{\prime }},c^{{}^{\prime }}]\ast B$$ (4)

and therefore $[a,b,c]\ast (AB)=([a,b,c]\ast A)\ast B$. Thus, (1) defines an action of $G$ on the set of (integer) binary quadratic forms. Furthermore, the discriminant of each quadratic form in the orbit of $[a,b,c]$ under $G$ is $b^2-4ac$.

Title	example of group action
Canonical name	ExampleOfGroupAction
Date of creation	2013-03-22 13:50:00
Last modified on	2013-03-22 13:50:00
Owner	Thomas Heye (1234)
Last modified by	Thomas Heye (1234)
Numerical id	11
Author	Thomas Heye (1234)
Entry type	Example
Classification	msc 11E16
Classification	msc 16W22
Classification	msc 20M30