Components of symmetric wide-matrix varieties

TL;DR

This paper proves that the number of symmetric group orbits on irreducible components of certain matrix varieties is a quasipolynomial in the matrix size, using category theory, combinatorics, and geometric methods.

Contribution

It introduces the category of affine FI^op-schemes of width one and establishes their structural properties, linking algebraic geometry with combinatorial orbit counting.

Findings

Number of Sym([n])-orbits is a quasipolynomial in n for large n.

Width-one FI^op-schemes become product forms after shifts and localizations.

Orbits correspond to lattice points in rational polyhedral cones.

Abstract

We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI^op-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of X_n, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem.