QTalk: Jurij Volcic

The talk concerns polynomial and rational functions in noncommuting variables, and their symmetries. Given a linear action of a group, classical invariant theory aims to answer some basic questions about its polynomial invariants: are they finitely generated, what are minimal sets of generators, and what are the relations between them. On the other hand, there is also the induced action on the free associative algebra, which leads to analogous questions about noncommutative invariants. Surprisingly or not, the answers to these questions are more sparse and quite different compared to their commutative analogs.

This talk is introductory; it is explained how noncommutative polynomial invariants are both well- but unsatisfyingly understood, the benefit of using noncommutative rational expressions is outlined, results for invariants of abelian and solvable groups are given, and several problems are left open. Based on joint work with Igor Klep, James Pascoe and Gregor Podlogar.