Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers |
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Authors: | Tomohiro Kamiyoshi Shin-ichi Otani |
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Institution: | 1. Institute of Mathematics , University of Tsukuba , Tsukuba , Ibaraki , Japan;2. School of Engineering , Kanto-Gakuin University , Yokohama , Kanagawa , Japan |
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Abstract: | For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d(1), d(2),…, d(r)) of positive integers, we define a “torus-restricted” representation (GS, R d (Γ)) in natural way. Here we put GS = G1 × G2 × … ×Gr, where each Gi is either SL(d(i)) or GL(d(i)) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R d (Γ) has only finitely many GS-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (GS, R d (Γ)). |
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Keywords: | Dynkin quiver Root sublattice Semisimple finite prehomogeneous vector space Tilting module |
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