Springer Science+Business Media New York (2016)
EQUIVARIANT CHOW CLASSES OF
MATRIX ORBIT CLOSURES
Department of Mathematics
Western Washington University
Bellingham, Washington, USA
School of Mathematical Sciences
Queen Mary University of London
Abstract. Let G be the product GL
(C) × (C
. We show that the G-equivariant
Chow class of a G orbit closure in the space of r-by-n matrices is determined by a matroid.
To do this, we split the natural surjective map from the G equvariant Chow ring of the
space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting
takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and
also has the property that the class of a subvariety of the Grassmannian is mapped to
the class of the closure of those matrices whose row span is in the variety.
The ﬁrst goal of this paper is to prove that the Chow class of a certain aﬃne
variety determined by a r-by-n matrix is a function of the matroid of that matrix.
Speciﬁcally, given an r-by-n matrix v with complex entries, we let X
set of those matrices that are projectively equivalent to v in the sense that they
are of the form gvt
, where g ∈ GL
(C), and t ∈ GL
(C) is a diagonal matrix.
Let G be the group consisting of pairs of matrices (g, t), which acts on the space
of r-by-n matrices via the rule (g, t)v = gvt
. A matrix orbit closure X
is the Zariski closure of X
; it is the G orbit closure of v. This variety
determines a class in the G equivariant Chow ring of A
. Theorem 4.3 states
that this class depends only on the matroid of v.
This matroid invariance is a consequence of two results. The ﬁrst result is the
matroid invariance of the class of a torus orbit closure in the torus equivariant
K-theory of the Grassmannian G(r, n). This result was shown by Speyer [Spe09]
and was used by Speyer and the second author to ﬁnd a purely algebro-geometric
interpretation of the Tutte polynomial [FS12]. The second result which our ma-
troid invariance relies on deals with the relationship between the G equivariant
Chow ring of A
and the torus equivariant Chow ring of G(r, n), which we now
The geometry of a particular subvariety Y of the Grassmannian G(r, n) (or
more generally, a partial ﬂag variety) is of interest. To study it, one constructs a
Received August 12, 2015. Accepted July 6, 2016.
Corresponding Author: A. Berget, e-mail: email@example.com.
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Vol. 22, No.
, 2017, pp.