Back Back More More

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,
         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.

Home Back More