2017/06/08 by Zalman Balanov, Balanov, Zalman, Mikhail Muzychuk +3
Mathematics · #15A69 #20B20 #20C15 #20D20 #55M25 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1706.02756
openalex publication_date 2017/06/08 · openalex created_date 2017/06/23 · openalex updated_date 2026/07/28
Given a finite group G and two unitary G-representations V and W, possible restrictions on Brouwer 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 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. 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 Brouwer degree.