Semi-invariants of Symmetric Quivers of Finite Type

Abstract

ABSTRACT. Let (Q, σ) be a symmetric quiver, where Q = (Q0, Q1) is a finite quiver without oriented cycles and σ is a contravariant involution on Q0 ⊔ Q1. The involution allows us to define a nondegenerate bilinear form <, > on a representation V of Q. We shall call the representation orthogonal if <, > is symmetric and symplectic if <, > is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that 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 .

Semi-Invariants of Symmetric Quivers of Tame Type

“…So we can define c (x,σ (x)) := c σ (x) c x . Moreover we can prove that (c (x,σ (x)) Q, σ ) is a symmetric quiver (see Lemma 2.2 in [3]).…”

“…(1) We call E i , E ′ i and E ′′ i the simple non-homogeneous regular representations respectively of dimension e i , e ′ i and e ′′ i . (2) We call V (ϕ,ψ) , where (ϕ, ψ) ∈ P 1 (k), the indecomposable regular representation of dimension h. (3) We define E i, j to be the indecomposable regular representations with socle E i and dimension ∑ j k=i e k , where e k are vertices of the arc with clockwise orientation e i e j in ∆, without repetitions of e k . We denote E i := E i,i and similarly we define E ′ i, j and E ′′ i, j .…”

“…We adjust the technique of reflection functors to symmetric quivers. See [3] By definition of σ , x is an admissible sink (resp. a source) if and only if σ (x) is an admissible source (resp.…”

“…The underlying graph of D is a tree, so by Proposition 2.4 in [3], applying a composition of reflections at admissible sink-source pairs, from any orientation of D we can get the following orientation…”

“…It would be also interesting to generalize to symmetric quivers the results about virtual representations and virtual semi-invariants of quivers given by Igusa, Orr, Todorov and Weyman in [11]. The author, in [3], displayed a set of generators of rings of semi-invariants of symmetric quivers of finite type. In this paper, the second one extract from his PhD thesis, supervised by Professor Jerzy Weyman, we provide similar results for symmetric quivers of tame type.…”