4 research outputs found
Membership in moment cones and quiver semi-invariants for bipartite quivers
Let be a bipartite quiver with vertex set such that the number of
arrows between any two source and sink vertices is constant. Let
be a dimension vector of with positive
integer coordinates, and let be the moment cone associated
to . We show that the membership problem for can
be solved in strongly polynomial time.
As a key step in our approach, we first solve the polytopal problem for
semi-invariants of and its flag-extensions. Specifically, let
be the flag-extension of obtained by attaching a flag of
length at every vertex of , and let be
the extension of to that takes values
along the vertices of the flag for every vertex of .
For an integral weight of , let
be the dimension of the space of semi-invariants of
weight on the representation space of
-dimensional complex representations of .
We show that can be expressed as the number of
lattice points of a certain hive-type polytope. This polytopal description
together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants
allows us to use Tardos's algorithm to solve the membership problem for
in strongly polynomial time.Comment: v2: Fixed the claim about the generic quiver semi-stability problem
(see Remarks 2.8 and 5.5
Equivariant cohomology, Schubert calculus, and edge labeled tableaux
This chapter concerns edge labeled Young tableaux, introduced by H. Thomas
and the third author. It is used to model equivariant Schubert calculus of
Grassmannians. We survey results, problems, conjectures, together with their
influences from combinatorics, algebraic and symplectic geometry, linear
algebra, and computational complexity. We report on a new shifted analogue of
edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule
for the structure constants of the D. Anderson-W. Fulton ring, which is related
to the equivariant cohomology of isotropic Grassmannians.Comment: 39 page