Let \(F\) be a finite field and let \(A\) and \(B\) be two nilpotent associative \(F\)-algebras. Then \(B\) is called an immediate descendant of \(A\) if \(A\) is isomorphic to \(B/B^{cl(B)}\). The difference \(dim(B)-dim(A)\) is called the stepsize of the descendant. A nilpotent associative \(F\)-algebra is called capable, if it has descendants. In [EM17] an algorithm is described to calculate these immediate descendants. This algorithm and the capability check is implemented in the following functions.
‣ ImmediateDescendantsOfCanoForm ( C, s ) | ( operation ) |
Given a canonical form C this function returns the immediate descendants of C with stepsize s.
‣ ImmediateDescendantsOfCanoForm ( C, slist ) | ( operation ) |
Given a canonical form C this function returns the immediate descendants of C with stepsizes in slist.
‣ ImmediateDescendantsOfCanoForm ( C ) | ( operation ) |
Given a canonical form C this function returns the immediate descendants of C of all stepsizes.
‣ IsCapableCanoForm ( C ) | ( property ) |
This returns whether the given canonical form C is capable, i.e. whether it has descendants.
The descendant algorithm can be used for a classification of nilpotent associative \(F\)-algebras by dimension. This is used in the following functions.
‣ CanoFormsByDimAndGens ( F, d, n ) | ( operation ) |
Given a finite field F, a generator number d and a dimension n, this function calculates canonical forms for the d-generator nilpotent associative F-algebras of dimension n.
‣ CanoFormsByDim ( F, n ) | ( operation ) |
Given a finite field F and a dimension n, this function calculates canonical forms for the nilpotent associative F-algebras of dimension n.
Sometimes it will be enough to enumerate the number of immediate descendants or the number of nilpotent associative algebras of a given dimension. This is implemented in the following functions using orbit counting methods. However, note that the enumeration of \(n\)-dimensional algebras still requires the computation of the \((n-1)\)-dimensional algebras.
‣ NumberOfImmediateDescendantsOfCanoForm ( C, s ) | ( operation ) |
Given a canonical form C this returns the number of immediate descendants of C with stepsize s.
‣ NumberOfImmediateDescendantsOfCanoForm ( C, slist ) | ( operation ) |
Given a canonical form C this returns the number of immediate descendants of C with stepsizes in slist.
‣ NumberOfImmediateDescendantsOfCanoForm ( C ) | ( operation ) |
Given a canonical form C this returns the number of immediate descendants of C of all stepsizes.
‣ NumberOfNAAlgsByDimAndGens ( F, d, n ) | ( operation ) |
Given a finite field F, a generator number d and a dimension n, this function returns the number of d-generator nilpotent associative F-algebras of dimension n.
‣ NumberOfNAAlgsByDim ( F, n ) | ( operation ) |
Given a finite field F and a dimension n, this function returns the number of nilpotent associative F-algebras of dimension n.
generated by GAPDoc2HTML