Matrix orbit closures

Abstract

Let G be the group GLr(C) × (C × ) n . We conjecture that the finely-graded Hilbert series of a G orbit closure in the space of r-by-n matrices is wholly determined by the associated matroid. In support of this, we prove that the coefficients of this Hilbert series corresponding to certain hook-shaped Schur functions in the GLr(C) variables are determined by the matroid, and that the orbit closure has a set-theoretic system of ideal generators whose combinatorics are also so determined. We also discuss relations between these Hilbert series for related matrices, including their stabilizing behaviour as r increases.

Equivariant Chow Classes of Matrix Orbit Closures

“…The class of G(v) in R(GL r ) is determined by the matroid of v. Describing the irreducible decomposition of these representations was the motivation for studying X v and its class in K-theory. For the proofs of the corollaries, we refer the reader to [BF18].…”

“…Elements 3 and 4 are parallel, so M is a parallel extension of the uniform matroid U 2,3 , and therefore the above class can also be computed from Theorem 9.3 or Proposition 9.5 of [BF18]. The procedure described in [BF18,Theorem 9.3] is to apply a Demazure divided difference operator δ 3 to the class of the matroid U 2,3 ⊕ U 0,1 , which is (1 − t 4 /u 1 )(1 − t 4 /u 2 ) since the associated matrix orbit closure is a linear subspace of A 2×4 . Application of the divided difference gives…”

“…In this section we state the consequences of the matroid invariance of [X v ], which we studied further in [BF18]. Let v 1 , .…”

“…Let v ∈ A r×n be a matrix, and consider X • v = GL r vT n , which is the orbit of GL r × T n through v. We call the Zariski closure X v = X • v a matrix orbit closure, and it is our primary object of interest. Such varieties were studied in [BF17,BF18] and generalizations of them were studied in [LPST18,Li18].…”