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1 Introduction

1 Introduction

This package provides some functions for GAP [GAP17] to compute PORC functions N_d,r(q) giving the number of isomorphism classes of associative algebras of class two, rank r and dimension d.

Let F be a field and let V be a vector space over F. An algebra mathcalA over F is a vector space over F equipped with a bilinear mapping

\cdot: \mathcal{A}\times\mathcal{A}\to\mathcal{A}, (x,y) \mapsto x \cdot y =: xy

called multiplication.

An algebra is called associative if its multiplication is associative; It is called commutative if the multiplication is commutative. If there exists an element 1 ∈ mathcalA such that 1 ⋅ x = x = x ⋅ 1 holds for all elements x ∈ mathcalA the algebra is said to be an algebra with one.

An algebra mathcalA is said to be nilpotent if the power ideal series

\mathcal{A} > \mathcal{A}^2 > \ldots > \mathcal{A}^k > \ldots

terminates in the trivial algebra, hence there is a k ∈ N such that

\mathcal{A} > \mathcal{A}^2 > \ldots > \mathcal{A}^k =\{0\}.

The smallest l ∈ N with mathcalA^l ≠ {0} and mathcalA^l+1={0} is called the nilpotency class or just class of mathcalA denoted by mathrmcl(mathcalA).

The dimension of the algebra mathcalA denoted by mathrmdim_F}(mathcalA) = mathrmdim(mathcalA) is the dimension of mathcalA considered as vector space. The rank of mathcalA denoted by mathrmrk(mathcalA) is the minimal number of generators of mathcalA. It is mathrmrk(mathcalA) = mathrmdim(mathcalA/mathcalA^2).

Let q ∈ N be a prime or a power of a prime. Then N_d,r(q) denotes the number of isomorphism classes of nilpotent algebras of class two of dimension d and rank r over the finite field with q elements.

A function f on an infinite subset S of the natural numbers is called Polynomial on Residue Classes (PORC) if there exists a natural number m and associated polynomials g_0,...,g_m-1∈ Q[x] so that f(s) = g_a(s) for all s∈ S with s ≡ a mod m. This notation was introduced by Higman.

It can be shown (see [EW17]) that the number N_d,r(q), considered as a function in q, is PORC.

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