Let $k$ be an infinite field and $G_1=GL_n(k)$ or $Sp_n(k) (n=2n/)$ or $SO_n(k)
(n=2n/+1)$ and $G_2=GL_m(k)$ or $Sp_m (m=2m/)$ or $SO_m (m=2m/+1)$. Consider
the affine variety, $X(G_1,G_2)$, of all $n x m-$ matrices M over $k$ such
that,
$M^t J_{G_1} M=0$ και $M J_{G_2} M^t=0$,
where $J_{G_i}$ is the defining matrix of $G_i$. (For $G_i$ be a general linear
group we take $J_{G_i}=0$).
The group $G_1\times G_2$ acts on $X(G_1,G_2)$ with $(Α,Β)\cdot M=A M B^{-1}$
for $A\in G_1$, $B\in G_2$ and $M\in $X(G_1,G_2)$ . Hence the coordinate ring
of $X(G_1,G_2)$, denoted by, $A(G_1,G_2)$, is a $G_1\times G_2-$ module with
the induced action.
We fix now $k$ to be a field of characteristic $0$. Then $A(G_1,G_2)$ is a
semisimple $G_1\times G_2-$ module. Therefore it decomposes into simple
$G_1\times G_2-$ modules, $A(G_1,G_2)=\oplus A_i$ where $A_i$ are
simple$G_1\times G_2-$ modules. By [ M. Maliakas, Cauchy decompositions and
invariants, Math. Z. 235 (2000)] we know exactly which modules appear in the
above decomposition. Let $I$ be the ideal of $A(G_1,G_2)$ that generated by the
simple module $A_i$, $I=
$. This is a $G_1\times G_2-$ module and it
decomposes into simple modules. In this Ph.D. thesis we find a decomposition of
$I$ into simple $G_1\times G_2-$ modules for every group $G_1\times G_2$. So we
describe explicitly which simple $G_1\times G_2-$ modules appear in the
decomposition of $I$. The answer does not depend on the groups $G_1\times G_2$
but only on their ranks.
Let now, $A_i A_j$ be the product of two simple modules of $A(G_1,G_2)$. In
this Ph.D thesis we find a decomposition of $A_i A_j$ into simple $G_1\times
G_2-$ modules for every group $G_1\times G_2$. So we describe explicitly the
simple $G_1\times G_2-$ modules that appear in the decomposition of $A_i A_j$.
The answer does not depend on the groups but only on their ranks.
In the last section of the thesis we describe the prime and the primary
$G_1\times G_2-$ ideals of the ring $A(G_1,G_2)$ using the above Theorems.
