Components of symmetric wide-matrix varieties

Abstract

We show that if Xn is a variety of c × n-matrices that is stable under the group Sym ([n]) of column permutations and if forgetting the last column maps Xn into X n−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 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 Xn = 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 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.

Symmetric ideals of the infinite polynomial ring

“…that are stable under the infinite symmetric group S satisfy the ascending chain condition; in other words, R is S-noetherian. There has been a surge of interest surrounding this theorem in recent years: it has found applications in algebraic statistics [AH, Dr, HS], algebraic geometry [DE, DK], and the theory of configuration spaces [Ra]; additionally, a number of authors have worked to understand aspects of S-ideals in more detail [DEF,GN,KLS,LNNR,LNNR2,MR,NR,NR2]. In this paper, we undertake a systematic study of S-ideals of R. We give a complete description of the equivariant spectrum of R, which yields a classification of all S-ideals up to copotency.…”

Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry

“…Hence u ∈ M n . Now using (7) one checks easily that u does not belong to the monoid generated by v 1 , . .…”

“…Examples of successful extensions include equivariant Hilbert's basis theorem [2,4,5,9,18], equivariant Hilbert-Serre theorem [11,16,17], equivariant Buchberger algorithm [8], equivariant Hochster's formula [15]. See, e.g., also [7,12,13,14,19,20,21] for related results.…”