The Le Bruyn-Procesi theorem following Lusztig

TL;DR

The paper addresses describing the invariant rings of quiver representation spaces under group actions, extending the Le Bruyn–Procesi framework to Lusztig’s setting with frozen vertices. It introduces a framing trick to give a simple proof of Lusztig’s theorem and characterizes generators for $\Bbbk[\operatorname{Rep}(A,v)]^{G_K}$ in terms of traces of cycles and contraction functions, with a precise kernel description of the restriction map $\tau_K$. The main contributions include a unified generator description for Lusztig’s invariant subalgebras across arbitrary subsets $K$, and an explicit description of the relations among these generators via an augmented quiver and elimination techniques, illustrated by a McKay-quiver example that connects to Hilbert schemes. Overall, the work provides a concrete, constructive framework to understand invariant rings of quiver representations with relations and their kernels, enabling explicit computations and geometric interpretations of the associated quotients.

Abstract

For any quiver $Q$ and dimension vector $v$, Le Bruyn-Procesi proved that the invariant ring for the action of the change of basis group on the space of representations $\text{Rep}(Q,v)$ is generated by the traces of matrix products associated to cycles in the quiver. Lusztig generalised this to allow for vertices where the group acts trivially. Here we provide a simple new proof of Lusztig's theorem and determine the relations between his algebra generators for any quiver with relations.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 2.2: Lusztig
• Remark 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• proof : Proof of Theorem \ref{['thm:Lusztig']}