Tangent spaces of orbit closures for representations of Dynkin quivers of type \(\mathbb{D}\). (English) Zbl 1536.14037
Let \(k\) be an algebraically closed field, \(Q\) a finite quiver, and denote by \(\mathrm{rep}_d(Q)\) the affine \(k\)-scheme of representations of \(Q\) with a fixed dimension vector \(d\). Given such a representation \(M\) of \(Q\), the set of \(k\)-points in \(\mathrm{rep}_d(Q)\) isomorphic as representations to \(M\) is an orbit \(\mathcal{O}_M\) under an action of a product of general linear groups. The aim of this paper is to describe “in representation theoretic terms” the tangent space to this orbit when \(Q\) is a Dynkin quiver of type \(D\).
The description of the tangent space goes via another affine subscheme \(\mathcal{C}_M\) of \(\mathrm{rep}_d(Q)\). This subscheme is defined by suitable rank conditions associated to \(M\), which makes it easy to describe in representation-theoretic terms, and for all Dynkin and extended Dynkin quivers, \(\mathcal{O}_M\) is the reduced scheme associated to \(\mathcal{C}_M\).
The description of the tangent space goes via another affine subscheme \(\mathcal{C}_M\) of \(\mathrm{rep}_d(Q)\). This subscheme is defined by suitable rank conditions associated to \(M\), which makes it easy to describe in representation-theoretic terms, and for all Dynkin and extended Dynkin quivers, \(\mathcal{O}_M\) is the reduced scheme associated to \(\mathcal{C}_M\).
Reviewer: Michael Bate (York)
MSC:
| 14L30 | Group actions on varieties or schemes (quotients) |
| 16G20 | Representations of quivers and partially ordered sets |
| 14B05 | Singularities in algebraic geometry |
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