The package ClassTwoAlgLib is a small library developed together with the ClassTwoAlg which was designed to count the number of isomorphism classes of associative algebras of class 2 and small rank for all dimensions. See [EW17] for more information.
The package ClassTwoAlgLib provides the results of the enumerations of the ClassTwoAlg package. It is not designed to compute the functions, but it provides some functions that are also included in the ClassTwoAlg package.
A PORC function is in GAP designed as a multivariate polynomial over the rational numbers. The first indeterminate is displayed as q and is internally handled as indeterminate number 1000. All other indeterminates will internally have a larger number. Those indeterminates will be handled as gcd's with q. For a more detailed introduction see [EW17].
‣ ReadNumberOfClassTwoAssociativeAlgebrasByRankAndDimension ( rk, dim ) | ( function ) |
Returns a PORC function which, when evaluated, yields the number of isomorphism types of associative algebras of rank rk and dimension dim.
gap> ReadNumberOfClassTwoAssociativeAlgebrasByRankAndDimension(2,3); q-(q-0,2)+5
‣ ReadNumberOfClassTwoAssociativeAlgebrasByRank ( rk ) | ( function ) |
Returns a list of PORC functions. The i'th function belongs to the number of isomorphism classes of nilpotent associative algebras of class 2, rank rk and dimension rk+i. Hence, this function calls ReadNumberOfClassTwoAssociativeAlgebrasByRankAndDimension
(1.1-1) for every possible dimension.
gap> ReadNumberOfClassTwoAssociativeAlgebrasByRank(2); [ q-(q-0,2)+5, 3*q-(q-0,2)+6 ]
‣ ValueOfPorcPolynomial ( f, x ) | ( function ) |
Given a PORC function f then this function evaluates f at the point x.
gap> f := ReadNumberOfClassTwoAssociativeAlgebrasByRankAndDimension(3,5); q^6+q^5+3*q^4+6*q^3-2*q^2*(q-0,2)+18*q^2-7*q*(q-0,2)+q*(q-1,3)+38*q-10*(q-0,2)-1/2*(q-0,3)+89/2 gap> ValueOfPorcPolynomial(f,101); 1072348780593 gap> List([2,3,5,7,11,13], x -> ValueOfPorcPolynomial(f,x)); [ 322, 1650, 21969, 144773, 1986843, 5300147 ]
‣ InformationOnPorcPolynomial ( f ) | ( function ) |
Given a PORC function f some inforamtion of f is printed in nicely readible form. The highest degree in q can be found. Then it follows a table stating at which power of q the gcd's appear for the first time. One can stop the output when being asked "More information?" by typing 0 or false.
gap> InformationOnPorcPolynomial(f); _____________________________________________________________________________ | | SUMMARY | | Degree in 'q': 6 | | Gcd's with... 2 3 | First occurence 2 1 | (power of 'q') | | Total modulus of PORC function: 6. More information? (true/false) 0
When typing true or 1 a more detailed list on all gcd's is displayed.
gap> InformationOnPorcPolynomial(f); _____________________________________________________________________________ | | SUMMARY | | Degree in 'q': 6 | | Gcd's with... 2 3 | First occurence 2 1 | (power of 'q') | | Total modulus of PORC function: 6. More information? (true/false) 1 _____________________________________________________________________________ | | INFO on 'q' | | Degree in 'q': 6 | Polynomial is dense in 'q'. | (All powers (including q^0) are present.) _____________________________________________________________________________ | | INFO on gcds with 2 | | (q-0,2) appears for all powers of 'q' less than or equal to 2. _____________________________________________________________________________ | | INFO on gcds with 3 | | (q-0,3) appears for all powers of 'q' less than or equal to 0. | (q-1,3) first appears at power 1 of 'q'. | Then it is present for the following power(s) of 'q' only: | 1. No further gcd's are included.
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