Year: 2020
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Arrondo, Enrique
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3083196679