Abstract
Keywords Symmetric groups; Finite-dimensional algebra; First-order representation theory We revisit, clarify, and generalise classical results of Dickson and (much later) Wagner on minimal Sym(n)- and Alt(n)-modules. We present a new, natural notion of 'modules with an additive dimension' covering at once the classical, finitary case as well as modules definable in an o-minimal or finite Morley rank setting; in this context, we fully identify the faithful Sym(n)- and Alt(n)-modules of least dimension. Author Affiliation: (a) Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia (b) Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France (c) École Normale Supérieure-PSL, Département de Mathématiques et Applications, Paris, France (d) Department of Mathematics and Statistics, California State University, Sacramento, Sacramento, CA 95819, USA * Corresponding author. Article History: Received 1 June 2022; (miscellaneous) Communicated by Donna M. Testerman Byline: Luis Jaime Corredor [lcorredo@uniandes.edu.co] (a), Adrien Deloro [adrien.deloro@imj-prg.fr] (b,c), Joshua Wiscons [joshua.wiscons@csus.edu] (d,*)