example of group action

example of group action

Let

a

,

b

,

c

be

integers

and let

[

a

,

b

,

c

]

denote the mapping

[

a

,

b

,

c

]

:

ℤ

×

ℤ

→

ℤ

,

(

x

,

y

)

↦

a

⁢

x

2

+

b

⁢

x

⁢

y

+

c

⁢

y

2

.

Let

G

be the group of

2

×

2

matrices such that

det

⁡

A

=

±

1

⁢

∀

A

∈

G

. The substitution

(

x

y

)

↦

A

⁢

(

x

y

)

leads to

[

a

,

b

,

c

]

⁢

(

a

11

⁢

x

+

a

12

⁢

y

,

a

21

⁢

x

+

a

22

⁢

y

)

=

a

′

⁢

x

2

+

b

′

⁢

x

⁢

y

+

c

′

⁢

y

2

,

where

a

′

=

a

⁢

a

11

2

+

b

⁢

a

11

⁢

a

21

+

c

⁢

a

21

2

(1)

b

′

=

2

⁢

a

⁢

a

11

⁢

a

12

+

2

⁢

c

⁢

a

21

⁢

a

22

+

b

⁢

(

a

11

⁢

a

22

+

a

12

⁢

a

21

)

c

′

=

a

⁢

a

12

2

+

b

⁢

a

12

⁢

a

22

+

c

⁢

a

22

2

So we define

[

a

,

b

,

c

]

∗

A

:=

[

a

′

,

b

′

,

c

′

]

to be the

binary quadratic form

with

coefficients

a

′

,

b

′

,

c

′

of

x

2

,

x

⁢

y

,

y

2

, respectively as in (

1

). Putting in

A

=

1

0

0

1

we have

[

a

,

b

,

c

]

∗

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

]

∗

(

A

⁢

B

)

=

(

[

a

,

b

,

c

]

∗

A

)

∗

B

.

Set

[

a

,

b

,

c

]

∗

(

A

⁢

B

)

:=

[

a

′′

,

b

′′

,

c

′′

]

. So we have

a

′′

=

a

⁢

(

a

11

⁢

b

11

+

a

12

⁢

b

21

)

2

+

c

⁢

(

a

21

⁢

b

11

+

a

22

⁢

b

21

)

2

+

b

⁢

(

a

11

⁢

b

11

+

a

12

⁢

b

21

)

⁢

(

a

21

⁢

b

11

+

a

22

⁢

b

21

)

(2)

=

a

′

⁢

b

11

2

+

c

′

⁢

b

21

2

+

(

2

⁢

a

⋅

a

11

⁢

a

12

+

2

⁢

c

⋅

a

21

⁢

a

22

+

b

⁢

(

a

11

⁢

a

22

+

a

12

⁢

a

21

)

)

⁢

b

11

⁢

b

21

c

′′

=

a

⁢

(

a

11

⁢

b

12

+

a

12

⁢

b

22

)

2

+

c

⋅

(

a

21

⁢

b

12

+

a

22

⁢

b

22

)

2

+

b

⋅

(

a

11

⁢

b

12

+

a

12

⁢

b

22

)

⁢

(

a

21

⁢

b

12

+

a

22

⁢

b

22

)

(3)

=

a

′

⁢

b

12

2

+

c

′

⁢

b

22

2

+

(

2

⁢

a

⋅

a

11

⁢

a

12

+

2

⁢

c

⁢

a

21

⁢

a

22

+

b

⁢

(

a

11

⁢

a

22

+

a

12

⁢

a

21

)

)

⁢

b

12

⁢

b

22

as desired.
For the coefficient

b

′′

we get

b

′′

=

2

⁢

a

⁢

(

a

11

⁢

b

11

+

a

12

⁢

b

21

)

⁢

(

a

11

⁢

b

12

+

a

12

⁢

b

22

)

+

2

⁢

c

⋅

(

a

21

⁢

b

11

+

a

22

⁢

b

21

)

⁢

(

a

21

⁢

b

12

+

a

22

⁢

b

22

)

+

b

⁢

(

(

a

11

⁢

b

11

+

a

12

⁢

b

21

)

⁢

(

a

21

⁢

b

12

+

a

22

⁢

b

22

)

+

(

a

11

⁢

b

12

+

a

12

⁢

b

22

)

⁢

(

a

21

⁢

b

11

+

a

22

⁢

b

21

)

)

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

′′

=

2

⁢

a

′

⁢

b

11

⁢

b

12

+

2

⁢

c

′

⁢

b

21

⁢

b

22

+

(

b

11

⁢

b

22

+

b

21

⁢

b

12

)

⁢

(

2

⁢

a

⁢

a

11

⁢

a

12

+

2

⁢

c

⋅

a

21

⁢

a

22

+

b

⁢

(

a

11

⁢

a

22

+

a

12

⁢

a

21

)

)

.

This shows that

[

a

′′

,

b

′′

,

c

′′

]

=

[

a

′

,

b

′

,

c

′

]

∗

B

(4)

and therefore

[

a

,

b

,

c

]

∗

(

A

⁢

B

)

=

(

[

a

,

b

,

c

]

∗

A

)

∗

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

-

4

⁢

a

⁢

c

.

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

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