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Multiple flag varieties of finite type. (English) Zbl 0951.14034
The authors give a complete classification of all dimension types of $$k$$-tuples $$(P_1, \dots , P_k)$$ of parabolic subgroups of $$Gl_n$$, such that the diagonal action of $$Gl_n$$ on $$G/P_1\times \cdots \times G/P_k$$ has finitely many orbits. Also the orbits are classified in each case, and explicit representations are constructed.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 20G15 Linear algebraic groups over arbitrary fields 14L30 Group actions on varieties or schemes (quotients) 14L35 Classical groups (algebro-geometric aspects)
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##### References:
 [1] Atiyah, M., On the Krull-Schmidt theorem with application to sheaves, Bull. soc. math. France, 84, 307-317, (1956) · Zbl 0072.18101 [2] Bongartz, K., On degenerations and extensions of finite dimensional modules, Adv. math., 121, 245-287, (1996) · Zbl 0862.16007 [3] Bongartz, K., Degenerations for representations of tame quivers, Ann. sci. éc. norm. sup., 28, 647-668, (1995) · Zbl 0844.16007 [4] Brion, M., Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke math. J., 58, 397-424, (1989) · Zbl 0701.14052 [5] Kac, V., Infinite root systems, representations of graphs and invariant theory, Invent. math., 56, 57-92, (1980) · Zbl 0427.17001 [6] Littelmann, P., On spherical double cones, J. algebra, 166, 142-157, (1994) · Zbl 0823.20040 [7] Riedtmann, C., Degenerations for representations of quivers with relations, Ann. sci. éc. norm. sup, 19, 275-301, (1986) · Zbl 0603.16025 [8] C. M. Ringel, Exceptional modules are tree modules, 1997 [9] Simpson, C.T., Products of matrices, Differential geometry, global analysis, and topology (Halifax, NS, 1990), CMS conf. proc., 12, (1991), Amer. Math. Soc Providence
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