2014/09/16 by Jens Hesse, Hesse, Jens
Materials Science · Mathematics · #14F40 (Primary) 15A75 #15A69 #18E10 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Quasicrystal Structures and Properties #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1409.4635
openalex publication_date 2014/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An F-zip over a field of positive characteristic is a vector space together with two filtrations whose subquotients are related in a certain way. We will define the category of F-zips and some basic constructions in it, especially exterior powers. If the ground field is algebraically closed, one can give a classification of F-Zips in terms of combinatorics. However, the way constructions and concepts in the category of F-zips manifest themselves in terms of the classifying invariant, is yet to be fully understood. The theory of F-crystals suggests that another invariant might be useful in trying to improve the understanding of F-zips. Given an F-zip, we calculate for every 1-dimensional F-zip (of which there is essentially one for every integer d) 1(d) and every r∈ℤ the dimension of the space of F-zip morphisms from 1(d) into the r-th exterior power of the given F-zip. To make sense of this however, we will have to canonically decompose these spaces of morphisms each into two subspaces that are finite-dimensional over the prime field and its prime field respectively. One result will then be a way to calculate these numbers for a given isomorphism type. Our main result however, is a negative one: The invariant does not classify F-zips.