Components of symmetric wide-matrix varieties
Abstract
We show that if Xn is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps Xn into Xn-1, then the number of Sym([n])-orbits on irreducible components of Xn is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FIop-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 FIop-scheme becomes of product form, where Xn=Yn for some scheme Y in affine c-space. Furthermore, to any FIop-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 Xn, 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.