Components of symmetric wide-matrix varieties

Abstract

Abstract We show that if X n {X_{n}} is a variety of c × n {c\times n} -matrices that is stable under the group Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} of column permutations and if forgetting the last column maps X n {X_{n}} into X n - 1 {X_{n-1}} , then the number of Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} -orbits on irreducible components of X n {X_{n}} is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine 𝐅𝐈 𝐨𝐩 {\mathbf{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 {\mathbf{FI^{op}}} -scheme becomes of product form, where X n = Y n {X_{n}=Y^{n}} for some scheme Y in affine c-space. Furthermore, to any 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -scheme of width one we associate a component functor from the category 𝐅𝐈 {\mathbf{FI}} of finite sets with injections to the category 𝐏𝐅 {\mathbf{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 ] ) {\operatorname{Sym}([n])} -orbits of components of X n {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. We present applications of our methods to counting fixed-rank matrices with entries in a prescribed set and to counting linear codes over finite fields up to isomorphism.