[Paper Review] Normality of Marsden-Weinstein reductions for representations of quivers

William Crawley-Boevey|ArXiv.org|May 29, 2001

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper proves that Marsden-Weinstein reductions for representations of quivers—defined as symplectic quotients via moment maps—are normal algebraic varieties. Using geometric invariant theory and a deframing technique, the author establishes normality for these quotients, generalizing Kraft and Procesi's result on conjugacy class closures and extending it to arbitrary quiver representations with moment map reductions.

ABSTRACT

We prove that the Marsden-Weinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices.

Motivation & Objective

• To establish the normality of affine symplectic quotients arising from moment map reductions for quiver representations.
• To generalize Kraft and Procesi’s result on nilpotent conjugacy class closures to a broader class of algebraic varieties defined via quivers.
• To provide a geometric framework for understanding singularities with symplectic resolutions in representation theory and algebraic geometry.
• To show that the quotient $ N_Q( u, α) = \mu_\u03b1^{-1}(\nu) // \mathrm{GL}(\u03b1) $ is normal for any quiver $ Q $, dimension vector $ \u03b1 $, and regular value $ \nu $.
• To demonstrate that these quotients capture important geometric and representation-theoretic structures, including conjugacy class closures and deformations of Kleinian singularities.

Proposed method

• Define the double quiver $ \overline{Q} $ and the representation space $ \mathrm{Rep}(\overline{Q}, \u03b1) $, equipped with a natural symplectic form.
• Construct the moment map $ \mu_\u03b1: \mathrm{Rep}(\overline{Q}, \u03b1) \to \mathrm{End}(\u03b1) $, given by $ \mu_\u03b1(x)_i = \sum_{h(a)=i} x_a x_{a^*} - \sum_{t(a)=i} x_{a^*} x_a $, which is a moment map for the $ \mathrm{GL}(\u03b1) $-action.
• Form the Marsden-Weinstein reduction $ N_Q(\nu, \u03b1) = \mu_\u03b1^{-1}(\nu) // \mathrm{GL}(\u03b1) $, an affine quotient variety.
• Use the deframing technique to relate specific cases (e.g., conjugacy class closures) to such quotients via isomorphisms.
• Apply Serre’s criterion for normality and analyze the representation-theoretic structure of the quotient using root decompositions and the quadratic form $ p(\u03b1) $.
• Prove that $ \alpha \in \Sigma_\nu $, ensuring the quotient has the correct dimension and irreducibility to imply normality.

Experimental results

Research questions

• RQ1 Are the Marsden-Weinstein reductions $ N_Q(\nu, \u03b1) $ normal varieties for arbitrary quivers $ Q $, dimension vectors $ \u03b1 $, and regular values $ \nu $?
• RQ2 Can the normality of conjugacy class closures in $ \mathrm{Mat}(n,K) $ be generalized to a broader class of symplectic quotients via quiver representations?
• RQ3 Does the moment map reduction $ \mu_\u03b1^{-1}(\nu) // \mathrm{GL}(\u03b1) $ yield normal varieties even when the fiber $ \mu_\u03b1^{-1}(\nu) $ itself is not normal?
• RQ4 How do these quotients relate to known geometric objects such as Kleinian singularities, nilpotent orbits, and integrable representations of Kac-Moody algebras?
• RQ5 Can the structure of the quotient $ N_Q(\nu, \u03b1) $ be analyzed via root decompositions and the quadratic form $ p(\u03b1) $ to establish normality?

Key findings

• The Marsden-Weinstein reduction $ N_Q(\nu, \u03b1) = \mu_\u03b1^{-1}(\nu) // \mathrm{GL}(\u03b1) $ is a normal algebraic variety for any quiver $ Q $, dimension vector $ \u03b1 $, and regular value $ \nu $.
• The quotient $ N_Q(\nu, \u03b1) $ is isomorphic to a quotient of a moment map fiber, and its normality is established via the condition $ \alpha \in \Sigma_\nu $, ensuring the correct dimension and irreducibility.
• The construction generalizes Kraft and Procesi’s result: the closure of a nilpotent conjugacy class in $ \mathrm{Mat}(n,K) $ is isomorphic to $ N_Q(0, \u03b1) $ for a specific quiver $ Q $ and dimension vector $ \u03b1 $.
• The quotient $ \left\{ (M_i) \in \prod_{i=1}^k \overline{C_i} \mid \sum M_i = 0 \right\} // \mathrm{GL}(n,K) $ is normal and isomorphic to $ N_Q(\nu, \u03b1) $ for a star quiver with $ k $ arms and $ \u03b1 $ equal to $ n $ at the central vertex.
• The fiber $ \mu_\u03b1^{-1}(\nu) $ need not be normal, but the quotient $ N_Q(\nu, \u03b1) $ is still normal, showing that normality is a property of the quotient, not the fiber.
• The result applies to a wide class of singularities with symplectic resolutions, including Kleinian singularities and their deformations, via quiver constructions.