Semi-invariants of quivers as determinants
type: Item
instance of: scholarly article
title: Semi-invariants of quivers as determinants
zbMATH Open document ID: 0984.16023
DOI: 10.1007/BF01236060
author: Matyas Domokos
author: A. N. Zubkov
published in: Transformation Groups
publication date: 2002-02-05T00:00:00Z
review text: 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 i…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 16W22
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1623801
zbMATH Keywords: matrix invariants
zbMATH Keywords: representations of quivers
zbMATH Keywords: semi-invariants of quivers
zbMATH Keywords: algebras of semi-invariants
zbMATH Keywords: fibre bundles
zbMATH Keywords: bipartite quivers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1623801