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Actions of parabolic subgroups in \(\text{GL}_n\) on unipotent normal subgroups and quasi-hereditary algebras. (English) Zbl 0978.16008
Let \(K\) be a field, \(n\geq 1\) an integer and \(R\) a parabolic subgroup of \(\text{GL}_n(K)\). Then \(R\) acts on its unipotent radical \(R_u\) and on any unipotent normal subgroup \(U\) by conjugation. Each parabolic subgroup \(R\) is the group of automorphisms of a finite dimensional \(K\)-algebra \(\Lambda(d)\), \(d\in\mathbb{N}^t\), which is Morita equivalent to the path algebra \(\Lambda=K\mathbb{A}_t\) of a directed Dynkin quiver \(\mathbb{A}_t\) with \(t\) vertices. The actions of \(R\) on \(U\) are described by means of the category \(\text{Mat}(B)\) of matrices in the sense of Yu. Drozd over a subbimodule \(B\) of the radical of \(\Lambda\) viewed as a \(\Lambda\)-\(\Lambda\)-bimodule. In this way, for each \(d\in\mathbb{N}^t\) there is a parabolic group \(R(d)\) of invertible elements in \(\Lambda(d)\) and a unipotent subgroup \(U(d)\) of \(R(d)\). One of the main results of the paper asserts that there is a quasi-hereditary algebra \(\mathcal A\) such that the orbits of \(R(d)\) on \(U(d)\) correspond bijectively to the isomorphism classes of \(\Delta\)-filtered \(\mathcal A\)-modules of \(\Delta\)-dimension vector \(d\). The bijection is induced by a morphism of algebraic varieties. In particular, it preserves degenerations and families. A quiver with relations of the quasi-hereditary algebra \(\mathcal A\) is determined.
Reviewer: D.Simson (Toruń)

16G20 Representations of quivers and partially ordered sets
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
14L30 Group actions on varieties or schemes (quotients)
16D90 Module categories in associative algebras
16S50 Endomorphism rings; matrix rings
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