Information Technology Reference
In-Depth Information
A
Algorithm
s
O
asso-
We describe an explicit algorithm that generates the oscillator system
S
ciated with the collection of split tori in
Sp
=
SL
2
(
F
p
).
A.1
Tori
Consider the standard diagonal torus
A
=
a
0
a
−
1
;
a
∈
F
p
.
0
Every split torus in
Sp
is conjugated to the torus
A
, which means that the
collection
T
of all split tori in
Sp
can be written as
gAg
−
1
;
g
T
=
{
∈
Sp
}
.
A.2
Parametrization
can be written as
gAg
−
1
A direct calculation reveals that every torus in
T
for
an element
g
of the form
g
=
1+
bc b
c
,b,c
∈
F
p
.
(3)
1
Unless
c
= 0, this presentation is not unique: In the case
c
g
=0,anelement
represents the same torus as
g
if and only if it is of the form
g
=
1+
bc b
c
0
c
−
1
−c
0
.
1
Let us choose a set of elements of the form (3) representing each torus in
T
exactly once and denote this set of representative elements by
R
.
A.3
Generators
The group
A
is a cyclic group and we can find a generator
g
A
for
A
.Thistaskis
simple from the computational perspective, since the group
A
is finite, consisting
of
p
1elements.
Now we make the following two observations. First observation is that the
oscillator basis
−
B
A
is the basis of eigenfunctions of the operator
ρ
(
g
A
).
The second observation is that, other bases in the oscillator system
s
O
can
S
be obtained from
B
A
by applying elements form the sets
R
. More specifically,
for a torus
T
of the form
T
=
gAg
−
1
,
g
∈
R,
we have
B
gAg
−
1
=
{
ρ
(
g
)
ϕ
;
ϕ
∈B
A
}
.
Concluding, we described the oscillator system
s
S
O
=
{B
gAg
−
1
;
g
∈
R
}
.
