Semi-invariants of filtered quiver representations with at most two pathways

TL;DR

This paper characterizes semi-invariant polynomials for filtered quiver representations, showing they originate from diagonal entries when the quiver has at most two pathways between vertices, and explicitly describes generators for such cases.

Contribution

It introduces a modified definition of quiver representations and provides explicit semi-invariant generators for quivers with at most two pathways, extending understanding of quiver invariants.

Findings

Semi-invariant polynomials originate from diagonal entries for quivers with at most two pathways.

Explicit generators for semi-invariants are provided for framed quivers under these conditions.

Results facilitate studies of Nakajima's quiver varieties and related geometric constructions.

Abstract

A pathway from one vertex of a quiver to another is a reduced path. We modify the classical definition of quiver representations and we prove that semi-invariant polynomials for filtered quiver representations come from diagonal entries if and only if the quiver has at most two pathways between any two vertices. Such class of quivers includes finite $ADE$-Dynkin quivers, affine $\widetilde{A}\widetilde{D}\widetilde{E}$-Dynkin quivers, star-shaped and comet-shaped quivers. Next, we explicitly write all semi-invariant generators for filtered quiver representations for framed quivers with at most two pathways between any two vertices; this result may be used to study constructions analogous to Nakajima's affine quotient and quiver varieties, which are, in special cases, $\mathfrak{M}_0^{F^{\bullet}}(n,1) := \mu_B^{-1}(0)/\!\!/B$ and $\mathfrak{M}^{F^{\bullet}}(n,1) := \mu_B^{-1}(0)^s/B$, respectively, where $\mu_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*\cong \mathfrak{gl}_n^*/\mathfrak{u}$, $B$ is the set of invertible upper triangular $n\times n$ complex matrices, $\mathfrak{b}=\lie(B)$, and $\mathfrak{u}\subseteq \mathfrak{b}$ is the biggest unipotent subalgebra.