Representation theory of finite groups through (basic) algebraic geometry

2020/09/06 by Enrique Arrondo, Arrondo, Enrique · 1 voice
Computer Science · Mathematics · #05E05 #05E10 #16P10 #20C30 #20C99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Polynomial and algebraic computation #Representation Theory (math.RT) #math.AG #math.CO #math.RT

paper · pdf · doi:10.48550/arxiv.2009.02774

openalex publication_date 2020/09/06 · arxiv published 2020/09/06 · arxiv updated 2024/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Discussions

• Reminds me of these kinds of things: arxiv.org/abs/2009.02774 arxiv.org/abs/1910.087... [bsky, 2 points, 0 comments]

Related

• Representation Theory of Finite Groups
• Representations of finite groups
• Representations and characters of finite groups
• Representation Theory of Symmetric Groups and Related Hecke Algebras
• Symmetric representation of the elements of finite groups