[Paper Review] Semi-invariants of Symmetric Quivers

Riccardo Aragona|arXiv (Cornell University)|Jun 22, 2010

Algebraic structures and combinatorial models29 references3 citations

TL;DR

This paper characterizes the rings of semi-invariants for orthogonal and symplectic representations of symmetric quivers of finite and tame type. It proves that these rings are generated by determinantal semi-invariants $ c^V $ and Pfaffians $ pf^V $ when the defining matrix is skew-symmetric, extending classical invariant theory to symmetric quiver representations via group actions on representation spaces.

ABSTRACT

This is my PhD thesis supervised by Professor Jerzy Weyman. A symmetric quiver $(Q,σ)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $σ$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $$ on a representation $V$ of $Q$. We shall say that $V$ is orthogonal if $$ is symmetric and symplectic if $$ is skew-symmetric. Moreover we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if $(Q,σ)$ is a symmetric quiver of finite type or of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$.

Motivation & Objective

• To describe the structure of the rings of semi-invariants for orthogonal and symplectic representations of symmetric quivers.
• To generalize classical invariant theory to symmetric quivers by defining actions of products of classical groups on representation spaces.
• To prove that semi-invariant rings are generated by $ c^V $ and $ pf^V $ for symmetric quivers of finite and tame type.
• To establish a connection between semi-invariants and combinatorial data such as weights, partitions, and reflection functors.
• To lay foundational tools for future work on cluster algebras and virtual semi-invariants in symmetric quiver settings.

Proposed method

• Define symmetric quivers $ (Q, au) $ as quivers with a contravariant involution $ au $ on vertices and arrows, enabling nondegenerate bilinear forms on representations.
• Introduce orthogonal and symplectic representations via symmetric and skew-symmetric bilinear forms induced by $ au $.
• Define group actions $ SO(Q, eta) $ and $ SSp(Q, eta) $ on spaces of orthogonal and symplectic representations, respectively.
• Use reflection functors and Coxeter functors to relate semi-invariants across different quiver types and dimension vectors.
• Apply highest weight theory and Schur modules to analyze the structure of polynomial invariants, particularly via Cauchy's formula and duality.
• Characterize the weight $ ilde{\chi} = \langle \beta, \cdot \rangle $ of semi-invariants $ c^V $, showing that non-zero components alternate between 1 and -1 at sources and sinks, and derive partition constraints via complementary column conditions.

Experimental results

Research questions

• RQ1 What is the structure of the ring of semi-invariants for orthogonal representations of symmetric quivers of finite and tame type?
• RQ2 How do semi-invariants of symmetric quivers relate to classical invariants like determinants and Pfaffians?
• RQ3 Can the generators of the semi-invariant ring be explicitly described in terms of combinatorial data such as dimension vectors and weights?
• RQ4 How do reflection functors and duality functors affect the semi-invariants of symmetric quivers?
• RQ5 What is the role of the weight $ \chi = \langle \beta, \cdot \rangle $ in determining the structure of the semi-invariant ring?

Key findings

• For symmetric quivers of finite or tame type, the ring of semi-invariants is generated by determinantal semi-invariants $ c^V $ and Pfaffians $ pf^V $ when the defining matrix is skew-symmetric.
• The weight $ \chi = \langle \beta, \cdot \rangle $ of a semi-invariant $ c^V $ has alternating 1 and -1 values at sources and sinks of the support of $ \beta $, with zero elsewhere.
• The family of partitions associated with a weight $ \chi $ is determined by complementary column conditions: $ \lambda(a_i) $ is the complement of $ \lambda(a_{i-1}) $ with respect to $ \alpha_i $, with initial and terminal columns of height $ \alpha_{m_1} $ and $ \alpha_{m_t} $.
• The dimension vector $ \beta $ of an indecomposable representation satisfies $ \alpha_{m_t} = \alpha_{m_{t-1}} - \alpha_{m_{t-2}} + \cdots \pm \alpha_{m_1} $, reflecting the alternating weight pattern.
• The semi-invariant ring $ \mathbb{K}[ORep(Q,\beta)]^{SO(Q,\beta)} $ is isomorphic to the invariant ring of Schur modules under $ SL(V) $, with non-zero components only at vertices where $ \chi(i) \neq 0 $.
• The proof relies on Cauchy’s formula and the structure of highest weight modules, showing that non-zero invariants arise only when partition sequences satisfy strict duality and complementarity conditions.