# On Some Applications of Group Representation Theory to Algebraic Problems Related to the Congruence Principle for Equivariant Maps

## TL;DR

This paper explores conditions under which the congruence restrictions on equivariant map degrees are non-trivial, linking group solvability to the alpha-characteristic, and constructs explicit quadratic maps with computable degrees.

## Contribution

It characterizes when the alpha-characteristic exceeds one based on group solvability and constructs explicit equivariant quadratic maps with known Brouwer degrees.

## Key findings

- Alpha(V)>1 if and only if G is solvable.
- For non-solvable groups, complex representations with non-trivial alpha-characteristic are constructed.
- A class of Norton algebras yields explicit equivariant quadratic maps with computable degrees.

## Abstract

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 $\alpha(V)$). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is ${\alpha}(V)>1$? 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 ${\alpha}(V)>1$ 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 ${\alpha}$-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.