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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Action on Grassmannians associated with commutative semisimple algebras


Authors: Dae San Kim and Patrick Rabau
Journal: Trans. Amer. Math. Soc. 326 (1991), 157-178
MSC: Primary 05E25; Secondary 06A99, 14M15, 20F29
DOI: https://doi.org/10.1090/S0002-9947-1991-1068929-0
MathSciNet review: 1068929
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Abstract: Let $ A$ be a finite-dimensional commutative semisimple algebra over a field $ k$ and let $ V$ be a finitely generated faithful $ A$-module. We study the action of the general linear group $ {\text{GL}}_A(V)$ on the set of all $ k$-subspaces of $ V$ and show that, if the field $ k$ is infinite, there are infinitely many orbits as soon as $ A$ has dimension at least four. If $ A$ has dimension two or three, the number of orbits is finite and independent of the field; in each such case we completely classify the orbits by means of a certain number of integer parameters and determine the structure of the quotient poset obtained from the action of $ {\text{GL}}_A(V)$ on the poset of $ k$-subspaces of $ V$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1068929-0
Article copyright: © Copyright 1991 American Mathematical Society

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