# On two group functors extending Schur multipliers

## TL;DR

This paper explores the relationships between certain group functors related to Schur multipliers, providing new insights and algorithms for their computation, especially for polycyclic groups, supported by computational experiments.

## Contribution

It establishes connections between the functors K, , and , introduces efficient algorithms for , and investigates their properties through computational methods.

## Key findings

- (G) can be efficiently constructed for polycyclic groups.
- Conditions when K(G,3) is a quotient of (G) are identified.
- (G) and (G,3) are sometimes isomorphic, as shown by computations.

## Abstract

Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.