The natural representation of the stabilizer of four subspaces
Transactions of the American Mathematical Society
Consider the natural action of the general linear group GL(V ) on the product of four Grassmann varieties of the vector space V . In General linear group action on four Grassmannians we gave an algorithm to construct the generic stabilizer H of this action. In this paper we investigate the structure of V as an H-module, and we show the effectiveness of the methods developed there, by applying them to the explicit description of H. Then the mapping α has kernel A ⊕ B , and the induced map is an
... induced map is an isomorphism of H-modules. Similarly we obtain an isomorphism of Hmodules we see that it is a linear isomorphism. Since all the spaces involved in the definition of α are invariant under H, and all the constituent mappings of α are either projections or inclusions, it is clear that α is an H-module morphism. Remark. Proofs similar to that of Theorem 1 will also provide isomorphisms of H-modules and Corollary 2. We have isomorphisms of H-modules The second isomorphism is clear from the theorem. Corollary 3. Let (n; a, b, c, d) be such that r consecutive standard reductions are possible, leading to the flag Then we have isomorphisms of H-modules License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE STABILIZER OF FOUR SUBSPACES 5797 Corollary 4 (assumptions as in Corollary 3). We have an isomorphism of Lmodules where L is a Levi-factor of H.