Distinguishing Groups and Algebras via Growth Functions and Representations

Project Details

Description

There is an abundance of functions that arise in everyday life and in nature, such as cooking and tasting food, washing and drying clothes and going to left or right while driving. Algebras or groups can be used to model their behaviour. Regularly a model is too complex to get full grasp on. Therefore one aims for, so-called, invariants describing it roughly.
An invariant is a number or geometric object associated with the algebra or group and that does not change under certain transformations. For example, the length of the three sides of a triangle remains unchanged after rotation. Interestingly, the value of the three sides suffices to draw the triangle (up to rotation of the sheet of paper).

In this research project the invariants under investigation have their origin in representation theory, geometric group theory and ring theory. In the last two cases the invariants encode how the elements in the group and algebra are related. These relations, so-called identities, are quantified into the notions of growth of groups and codimensions and we investigate their asymptotic behaviour and how they can be connected.

In case of representation theory, i.e. group rings RG, the invariants are extracted from linear algebra and exhibit symmetries of the object. In this project we extract information on the unit group of RG by means of its actions on geometric objects. We also investigate novel connections between previous invariants.

AcronymFWOTM927
StatusActive
Effective start/end date1/10/1830/09/21

Flemish discipline codes

  • Algebra not elsewhere classified

Keywords

  • algebra