On the geometry of linear involutions
Abstract
Let V be an n-dimensional left vector space over a division ring R and n ≥ 3. Denote by ## add figure 'advg.5.3.455_01.gif'##k the Grassmann space of k-dimensional subspaces of V and write ## add figure 'advg.5.3.455_02.gif'##k for the set of all pairs (S, U ) ∈ ## add figure 'advg.5.3.455_01.gif'##k ｘ ## add figure 'advg.5.3.455_01.gif'##n - k such that S + U = V. We study bijective transformations of ## add figure 'advg.5.3.455_02.gif'##k preserving the class of base subsets and show that these mappings are induced by semilinear isomorphisms of V to itself or to the dual space V* if n ≠ 2k ; for n = 2k this fails. This result can be formulated as the following: if n ≠ 2k and the characteristic of R is not equal to 2 then any commutativity preserving transformation of the set of (k, n - k )-involutions can be extended to an automorphism of the group GL(V ).
Walter de Gruyter GmbH & Co. KG
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Artikel in diesem Heft
- Semifield flocks, eggs, and ovoids of Q (4,q)
- Index of speciality and arithmetically Gorenstein subschemes
- Polar spaces embedded in projective spaces
- Equivariant periodicity for compact group actions
- The finiteness property and Łojasiewicz inequality for global semianalytic sets
- 3-dimensional loops on non-solvable reductive spaces
- A characterization of the P-geometry for M23
- A lower bound for the second sectional geometric genus of polarized manifolds
- On the geometry of linear involutions
- Removable singularities for p-harmonic maps: the subquadratic case
- Division algebras with an anti-automorphism but with no involution
