Polynomial invariants of representations of quivers
For a quiver \(Q\) with \(n\) vertices, a given dimension vector \(\alpha = (\alpha_ 1,\dots, \alpha_ n)\), and an algebraically closed field \(k\), let \(E_ 1,\dots, E_ n\) be fixed \(k\)-vector spaces with \(\dim_ k E_ i = \alpha_ i\). The space \(R(Q,\alpha)\) of \(k\)-representations of \(Q\) is the product \(\prod \text{Hom}_ k(E_ i, E_ j)\) over the arrows \(i \to j\) in \(Q\). The group \(\text{GL}(\alpha) = \text{GL}(E_ 1) \times \dots \times\text{GL}(E_ n)\) naturally acts on \(R(Q,\alpha)\) and thus on the symmetric algebra \(k[R(Q,\alpha)]\). For \(\text{char }k = 0\), \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, 585- 598 (1990; Zbl 0693.16018)] have determined a complete system of polynomial invariants of \(R(Q,\alpha)\), namely the coefficients of characteristic polynomials of products of maps along oriented cycles in \(Q\). The author obtains this result for arbitrary characteristic by use of his previous paper on ``Invariants of several matrices'' [Invent. Math. 110, 389-401 (1992)]. He points out that the result remains valid over \(\mathbb{Z}\) with stability under base change. Moreover, he proves a generalized statement with \(\text{GL}(\alpha)\) replaced by a subgroup \(\prod\text{GL}(E_ i)_{\theta_ i}\), where \(\text{GL}(E_ i)_{\theta_ i}\) denotes the centralizer of some element \(\theta_ i \in \text{End}_ k(E_ i)\).
scientific article; zbMATH DE number 590901
scientific article