Moment cone membership for quivers in strongly polynomial time

TL;DR

This paper demonstrates that determining membership in moment cones of acyclic quiver representations can be done efficiently in strongly polynomial time, extending previous results to a broader class of quivers using a simplified polytope construction.

Contribution

It introduces a less geometric, straightforward construction of multiplicity polytopes for all acyclic quivers, enabling strongly polynomial time decision algorithms for moment cone membership.

Findings

Membership in moment cones for acyclic quivers is decidable in strongly polynomial time.

A simplified multiplicity polytope construction applies to all acyclic quivers.

The approach leverages Tardos' algorithm and the saturation property for efficient computation.

Abstract

In this note we observe that membership in moment cones of spaces of quiver representations can be decided in strongly polynomial time, for any acyclic quiver. This generalizes a recent result by Chindris-Collins-Kline for bipartite quivers. Their approach was to construct "multiplicity polytopes" with a geometric realization similar to the Knutson-Tao polytopes for tensor product multiplicities. Here we show that a less geometric but straightforward variant of their construction leads to such a multiplicity polytope for any acyclic quiver. Tardos' strongly polynomial time algorithm for combinatorial linear programming along with the saturation property then implies that moment cone membership can be decided in strongly polynomial time. The analogous question for semi-invariants remains open.