Here it is briefly introduced which actions are implemented and which classes of algebras can be enumerated.
When one follows [Eic08] one gets the relation between the desired class of algebras that shall be enumerated and the action that has to be used.
In the case of class-2 nilpotent associative algebras of rank r one obtains that the general linear group mathrmGL(r,q) operates on the set of subspaces of the r^2 dimensional vector space M = F_q^r ⊗_F_q F_q^r via the Kronecker product.
To obtain the number of class-2 nilpotent algebras of rank r and dimension d over the field with q elements one hast to call the function NumberOfClassTwoAlgebras
(2.2-1) wth the second argument Kronecker.
The algorithm uses some important properties arising from the action. Further, symmetries amongst the codimensions k are used:
N_{d,r}(q) = 0 \; \text{if} \; d \notin \{r+1, \ldots, r^2\} \; \text{and} \; N_{r+r^2,r}(q)=1.
\text{For}\;\text{all} \; k \in \{1, \ldots, r^2\} \; \text{holds} \; N_{r+k,r}(q) = N_{r+r^2,r}(q)=1.
A proof can be found in [EW17].
Especially the second equation is used for the output of NumberOfClassTwoAlgebras
(2.2-1) when called with the second argument Kronecker: For a better reading, it will be returned a list of length ⌈fracr^22+1⌉, only. The latter half would be just symmetric to the printed functions.
In the future hopefully more actions will be implemented.
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