On Some Quiver Determinantal Varieties

TL;DR

This paper introduces quiver analogues of determinantal varieties, studies their resolutions, and provides conditions for their defining equations, extending classical determinantal concepts to quiver and tensor settings.

Contribution

It develops a framework for quiver determinantal varieties, analyzes their resolutions, and generalizes results to tensor settings, including conditions for Cohen-Macaulay modules and defining equations.

Findings

Provided sufficient conditions for resolutions of quiver determinantal varieties.

Identified the set-theoretical defining equations for these varieties.

Discovered vanishing conditions for Kronecker coefficients.

Abstract

We introduce certain quiver analogue of the determinantal variety. We study the Kempf-Lascoux-Weyman's complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen-Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equations of these varieties. When the variety has codimension one, the only irreducible polynomial function is a relative tensor invariant. As a by-product, we find some vanishing condition for the Kronecker coefficients. In the end, we make a generalization from the quiver setting to the tensor setting.