Inversion Sets and Quotient Root Systems

Abstract: The main result of this paper is a recursive description of all decompositions [ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k ] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of inversion sets. Such decompositions play a central role in geometric invariant theory (GIT) in connection with studying the Littlewood-Richardson cone and related problems. This work can be considered as a continuation of the work of Dewji, Dimitrov, McCabe, Roth, Wehlau, and Wilson in which similar questions were studied for root systems of type $\mathbb{A}$. Their methods relied on properties of permutations and are not transferable to an arbitrary root system. In order to develop a type-independent approach, we go beyond root systems and consider quotient root systems (QRSs for short). We study subsets of positive roots in an arbitrary QRS $R$. We prove that every $\Phi \subseteq R^+$ can be represented in a canonical way as an inflation and develop methods to study recursively properties of such subsets. We extend the notion of an inversion to subsets of any QRS, i.e., beyond the case where a Weyl group is associated with $R$. If $\Phi \subseteq R^+$ is an inversion set, we introduce a graph $\text{G}(\Phi)$ and endow the set Comp$(\Phi)$ of connected components of $\text{G}(\Phi)$ with a partial addition. The resulting monoid-like structure (Comp$(\Phi),+)$ is a further generalization of root systems beyond QRSs. We study in detail the properties of (Comp$(\Phi),+)$ and their applications to studying the properties of $\Phi$. In particular, we investigate the relationship between $\Phi$ being primitive and $\Phi$ being irreducible. Apart from describing recursively all decompositions of $\Delta^+$ into the disjoint union of inversion sets, we provide applications to GIT and derive enumerative results which may be of independent interest.