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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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Article copyright: © Copyright 1991 American Mathematical Society