# Explicit construction of universal sampling sets for finite Abelian and symmetric groups

• Explizite Konstrution von universellen Abtast-Mengen für endliche abelsche und symmetrische Gruppen

Morotti, Lucia; Führ, Hartmut (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2014)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2014

Abstract

In this thesis I study sampling pairs and universal sampling sets for finite groups~G. Sampling pairs are pairs (Omega,Gamma), where Omega is a subset of the irreducible characters of G and Gamma is a subset of the conjugacy classes of G, such that the only linear combination of elements of Omega vanishing on Gamma is the zero function. A universal sampling set for t is a subset Gamma of the conjugacy classes of G such that, for every subset Omega of the irreducible characters of G with at most t~elements, (Omega,Gamma) is a sampling pair. I will first consider the case where G is abelian. Here t can be chosen arbitrarily and I will construct explicit universal sampling sets for t. In the special case of elementary abelian p-groups I will also give some algorithms that allow to reconstruct a linear combination of at most t irreducible characters from its restriction to Gamma. I will then study the case where G is a symmetric or an alternating group. Here I will construct explicit universal sampling sets for t small (tin{2,3} for symmetric groups and t=2 for alternating groups).