On algebraic problems behind the Brouwer degree of equivariant maps

Zalman Balanov, Mikhail Muzychuk, Hao pin Wu

Research output: Contribution to journalArticlepeer-review

Abstract

Given a finite group G and two unitary G-representations V and W, possible restrictions on topological degrees of equivariant maps between representation spheres S(V) and S(W) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted by α(V)). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is α(V)>1? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that α(V)>1 for each irreducible non-trivial C[G]-module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial α-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the degree.

Original languageEnglish
Pages (from-to)45-77
Number of pages33
JournalJournal of Algebra
Volume549
DOIs
StatePublished - 1 May 2020

Keywords

  • Doubly transitive groups
  • Equivariant map
  • Norton algebras
  • Ordinary representations of finite groups
  • Solvable groups
  • Topological degree

ASJC Scopus subject areas

  • Algebra and Number Theory

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