任意维数向量的箭图的半不变量
Semi-invariants of quivers for arbitrary dimension vectors
摘要：
可通过向量空间$R(Q,α)$来参数化所述矩阵Q的维数向量$α$的表示，代数群$\Gl(α)$作用于所述向量空间，使得轨道集合与矩阵Q的表示的同构类的集合是双射的。我们用表示范畴描述了这个向量空间上的半不变多项式函数。更确切地说，我们关联到一个合适的映射之间的射影表示半不变的多项式函数，描述了当这个映射是反转的表示，我们表明，这些半不变的多项式函数形成一个生成集的所有半不变的多项式函数的特征0。如果该图没有定向圈，我们可以用考虑左垂直于维数向量$α$的某个表示的表示来代替考虑射影表示之间的逆映射。这些左垂直表示就是我们所考虑的射影表示之间的映射的余核。
英文摘要：
The representations of dimension vector $α$ of the quiver Q can be parametrised by a vector space $R(Q,α)$ on which an algebraic group $\Gl(α)$ acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi--invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi-invariant polynomial function that describes when this map is inverted on the representation and we show that these semi-invariant polynomial functions form a spanning set of all semi-invariant polynomial functions in characteristic 0. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector $α$. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider.
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