Speaker: Morten Wesche (Braunschweig)
Title: Enumeration of Nilpotent Associative Algebras of Class 2 Over Arbitrary Finite Fields
Abstract:
Higman's PORC theory implies that the number \(N_{d,r}(q)\) of isomorphism types of nilpotent associative algebras of dimension \(d\), rank \(r\) and class \(2\) over a finite field with \(q\) elements, considered as a function in \(q\), can be described by a polynomial on residue classes in \(q\). In the talk an algorithm is described that, given a rank \(r\), determines such polynomials for \(N_{d,r}(q)\) for all dimensions \(d\). Based on this the functions \(N_{d,r}(q)\) are determined for \(r \in\{1,\ldots,5\}\) and arbitrary \(d\).