Representation theory of finite groups through (basic) algebraic geometry

TL;DR

This paper introduces a novel algebraic geometry-based approach to finite group representation theory that avoids character theory, providing new insights and simplified proofs for symmetric groups and hyperdeterminants.

Contribution

It develops a new geometric framework for finite group representations, linking group points to ground fields and simplifying symmetry analysis.

Findings

Identifies a finite set of points associated with any finite group for representation analysis.

Provides explicit equations for symmetries in the symmetric group $S_d$.

Offers a straightforward proof that hyperdeterminants vanish for all but two symmetry types.

Abstract

We introduce a new approach to representation theory of finite groups that uses some basic algebraic geometry and allows to do all the theory without using characters. With this approach, to any finite group $G$ we associate a finite number of points and show that any field containing the coordinates of those points works fine as the ground field for the representations of $G$. We apply this point of view to the symmetric group $S_d$, finding easy equations for the different symmetries of functions in $d$ variables. As a byproduct, we give an easy proof of a recent result by Tocino that states that the hyperdeterminant of a $d$-dimensional matrix is zero for all but two types of symmetry.