Dear GAP-Forum,
The problem observed by Nicola Sottocornola (creating a matrix group
with entries that are elements of a polynomial ring) fails (as the
error message, showing a call to `DefaultFieldOfMatrixGroup'
indicates) when GAP tries to construct a field containing all the
matrix elements.
In this situation the field would be a field of rational functions,
but at the moment no object for this exists in GAP.
On the other hand, as David Joyner already observed, there is in this
particular example no need to construct the group over this field, as
all matrix entries are effectively rational numbers. Thus the remedy
is obvious: Keep the matrices rational, and everything will work fine.
Nicola observed that he forced the matrices over the polynomial ring
to satisfy conditions of the singular package. This is a rather
unexpected feature of singular, which insists on matrices over a
function field instead of specifying the polynomial ring as a
separate argument.
As a workaround, I would recommend to keep the matrices as matrices
of rational numbers in GAP, and only to force them in the polynomial
ring when calling Singular. I.e. the call will become
I:=SingularInterface("invariant_basis", [10,A*One(R),B*One(R),C*One
(R)], "ideal");
All the best,
Alexander Hulpke
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