Horn inequalities on a quiver with an involution

TL;DR

This work extends the Derksen–Weyman description of quiver semi-invariants by giving an inductive, Horn-type characterization of the cone $\Sigma(Q,\alpha)$ and, in the presence of an involution, a refined description for anti-invariant weights. It integrates homological data (Hom/Ext, Euler form), Schofield polynomials, generic filtrations, and Ressayre’s dominant morphisms to derive a practical set of linear inequalities defining the cone; King’s semi-stability criterion provides a key link between representations and semigroup membership. The main advance is Theorem(s) yielding a reduced, symmetric inequality system for $\Sigma(Q,\alpha)^{- au}$ when $\alpha=\tau\alpha$, highlighting a smaller set of necessary conditions in the anti-invariant case. The paper also furnishes concrete quiver examples (e.g., $\hat{D}_5$ and the Sun quiver) to illustrate the reduction in inequalities and the behavior of antisymmetric elements, with broader implications for invariant theory and Horn-type problems in quiver settings.

Abstract

Derksen and Weyman described the cone of semi-invariants associated with a quiver. We give an inductive description of this cone, followed by an example of refinement of the inequalities characterising anti-invariant weights in the case of a quiver equipped with an involution.

Figures (8)

• Figure 1: A quiver equipped with an involution.
• Figure 2: The oriented straight quiver $A_n^\rightarrow$
• Figure 3: The $n$-Kronecker quiver $\Theta_n$
• Figure 4: The $(n,m)$-Chindris quiver if $m$ is even
• Figure 5: Mutation and doubling of the oriented straight quiver $A_3^\rightarrow$

Theorems & Definitions (7)

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