7.3.12 liftstd (plural)

Syntax:

liftstd ( ideal_expression, matrix_name )
liftstd ( module_expression, matrix_name )
liftstd ( ideal_expression, matrix_name, module_name )
liftstd ( module_expression, matrix_name, module_name )

Type:

ideal or module

Purpose:

returns a left Groebner basis of an ideal or module and a left transformation matrix from the given ideal, resp. module, to the Groebner basis.
That is, if m is the ideal or module, sm is the left Groebner basis of m, returned by liftstd, and T is a left transformation matrix, then sm=module(transpose(transpose(T)*transpose(matrix(m)))).
If m is an ideal, sm=ideal(transpose(T)*transpose(matrix(m))).
In an optional third argument the left syzygy module will be returned.

Example:

LIB "ncalg.lib"; def A = makeUsl2(); setring A; // this algebra is U(sl_2) ideal i = e2,f; option(redSB); option(redTail); matrix T; ideal j = liftstd(i,T); // the Groebner basis in a compact form: print(matrix(j)); ==> f,2h2+2h,2eh+2e,e2 print(T); // the transformation matrix ==> 0,f2, -f,1, ==> 1,-e2f+4eh+8e,e2,0 ideal tj = ideal(transpose(T)*transpose(matrix(i))); size(ideal(matrix(j)-matrix(tj))); // test for 0 ==> 0 module S; ideal k = liftstd(i,T,S); // the third argument S = std(S); print(S); // the syzygy module ==> -ef-2h+6,-f3, ==> e3, e2f2-6efh-6ef+6h2+18h+12