Semi-invariants of quivers as determinants
Let \(Q=(Q_0,Q_1,t,h)\) be a quiver with set of vertices \(Q_0\), set of arrows \(Q_1\) and functions \(t,h\colon Q_1\to Q_0\) attaching to an arrow \(\alpha\) its tail \(t(\alpha)\) and head \(h(\alpha)\). A representation of \(Q\) of dimension vector \(\underline n=(n_i\mid i\in Q_0)\) is a collection of linear maps \(L=(L_\alpha\colon V_{t(\alpha)}\to V_{h(\alpha)}\mid\alpha\in Q_1)\), where \(V_i\) is an \(n_i\)-dimensional vector space associated with \(i\in Q_0\). Let the base field \(K\) be infinite and of any characteristic. Identifying \(V_i\) with the space \(K^{n_i}\) of column vectors, and \(L_\alpha\) with an \(n_{h(\alpha)}\times n_{t(\alpha)}\) matrix operating by left multiplication, the representation \(L\) corresponds to a point in the affine space \(R=R(Q,\underline n)=\bigoplus_{\alpha\in Q_1}K^{n_{h(\alpha)}\times n_{t(\alpha)}}\). The group \(\text{GL}(\underline n)=\prod_{i\in Q_0}\text{GL}_{n_i}(K)\) acts on \(R\) such that \(g\cdot L=(g_{h(\alpha)}L_\alpha g_{t(\alpha)}^{-1}\mid\alpha\in Q_1)\) for any \(g\in\text{GL}(\underline n)\). The purpose of the paper under review is to describe the algebra of \(\text{GL}(\underline n)\)-semi-invariants (i.e., the algebra of \(\text{SL}(\underline n)\)-invariants) of the quiver \(Q\). There is a canonical way to blow up an arbitrary quiver to obtain a bipartite quiver, i.e., a quiver for which any vertex is either a source or a sink. The crucial observation of the authors is that this construction corresponds to forming an associated fibre bundle. Algebraically this gives induction of modules over algebraic groups. Applying Frobenius reciprocity, the authors conclude that the semi-invariants of the original quiver can be obtained from those of the enlarged bipartite quiver. The authors define generating semi-invariants of bipartite quivers as partial polarizations of determinants of block matrices. The proof is very natural and is based on a translation of the first fundamental theorem of classical invariant theory. As an application the authors obtain the generators of the algebra of invariants of \(d\)-tuples of \(n\times n\) matrices under simultaneous conjugation in the case of a field of any characteristic [see \textit{S. Donkin}, Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)]. As the authors mention in the paper, similar results in the case of characteristic 0 have been independently obtained by \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)].
scientific article; zbMATH DE number 1623801
scientific article