# Orbits of U(n,n) acting on Mn: Step 3

The spreadsheets from Step 1
and Step 2 have produced the equations
that we were after, but now we would like to reduce the ``extra''
relations.
Once again, we declare an ordering, and then we read in the result
from the previous step. This is done in the file
OrbitStep3.m. One thing that needs to be
explained is the large number of user selects that are declared in
this file.

**
**

**selects={Tp[f] ** y ** Tp[y] ** f -> -1 + x ** Tp[x] + Tp[f] ** f,
Tp[F] ** Tp[x] ** x ** F -> -1 + Tp[F] ** F + Tp[y] ** y,
Tp[f]-Tp[a]-x**Tp[b],f-a-b**Tp[x],
Tp[F]-d+Tp[y]**b,F-Tp[d]+Tp[b]**y,
F ** Inv[F] -> 1,
Inv[F] ** F -> 1,
Tp[Inv[F]] ** Tp[F] -> 1,
Tp[F] ** Tp[Inv[F]] -> 1,
a ** Inv[a] -> 1,
Inv[a] ** a -> 1,
Tp[Inv[a]] ** Tp[a] -> 1,
Tp[a] ** Tp[Inv[a]] -> 1,
b ** Inv[b] -> 1,
Inv[b] ** b -> 1,
Tp[Inv[b]] ** Tp[b] -> 1,
Tp[b] ** Tp[Inv[b]] -> 1,
c ** Inv[c] -> 1,
Inv[c] ** c -> 1,
Tp[Inv[c]] ** Tp[c] -> 1,
Tp[c] ** Tp[Inv[c]] -> 1,
d ** Inv[d] -> 1,
Inv[d] ** d -> 1,
Tp[Inv[d]] ** Tp[d] -> 1,
Tp[d] ** Tp[Inv[d]] -> 1};
**

**
**
The first two equations are the two that we liked from
step 2.
The next four equations are the definitions of *f* and *F*
which were noticed in
step 1.
The next equations declare *F* to be invertible, and the
remaining equations are the rest of the inverse relations.
The reason that all of these equations are included in the user
select category has to do with our reduction algorithm. These
equations will be used to reduce the more complicated ones which we
wish to get rid of. Here are the
undigested relations
which we will try to eliminate.
If we did not select all of these inverse
relations, then they would be considered undigested, since they
contain only unkowns and they do not solve for anything. As a
result, some of the inverse relations could be eliminated, leaving
us with messier equations which are not obviously equivalent. Since
we want to get rid of as many of the complicated undigested
equations as possible,
we will force our algorithm to keep these more simple undigested equations.
After we run the file, we will have another
spreadsheet.
Most of the undigested equations have been eliminated.
Besides the inverse relations, there are
four undigested equations which remain.

If we multiply the first equation on the right by *x* we can
collect on *d* to obtain

.
We could solve for *d* if
(1 -
*x*
*x*) were invertible.
Notice that we can multiply the second equation on the left by
*b* to try to solve for *b*. This gives us the following
equation.
The third equation is merely the transpose of the first equation.
This suggests that we assume that
(1 -
*x*
*x*)
is invertible.