# On some applications of group representation theory to algebraic problems related to the congruence principle for equivariant maps

Seminar
Speaker
Prof. Mikhail Muzychuk (Ben-Gurion University)
Date
23/05/2018 - 11:30 - 10:30Add to Calendar 2018-05-23 10:30:00 2018-05-23 11:30:00 On some applications of group representation theory to algebraic problems related to the congruence principle for equivariant maps Given a finite group G and two unitary G-representations V and W, possible restrictions on Brouwer degrees of equivariant maps between the representation spheres S(V) and S(W) are usually expressed in terms of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted α(V)).  Effective application of these congruences is limited by answers to the following questions:   (i) Under which conditions is α(V)>1? (ii) Does there exist an equivariant map whose degree is easy to calculate?    In my talk I'll address mainly the first question. It will be shown that α(V)>1 for every irreducible non-trivial C[G]-module if and only if G is solvable. So this result provides a new solvability criterion for finite groups.   This is a joint work with Z. Balanov and Haopin Wu. Third floor seminar room אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Third floor seminar room
Abstract

Given a finite group G and two unitary G-representations V and W, possible restrictions on Brouwer degrees of equivariant maps between the representation spheres S(V) and S(W) are usually expressed in terms of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted α(V)).  Effective application of these congruences is limited by answers to the following questions:

(i) Under which conditions is α(V)>1?

(ii) Does there exist an equivariant map whose degree is easy to calculate?

In my talk I'll address mainly the first question. It will be shown that α(V)>1 for every irreducible non-trivial C[G]-module if and only if G is solvable. So this result provides a new solvability criterion for finite groups.

This is a joint work with Z. Balanov and Haopin Wu.

Last Updated Date : 10/05/2018