Fwd: [GAP Support] Re: [GAP Forum] Simple matrix group

[Alexander Hulpke]

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