Mark Pankov

Grassmannians of classical buildings

Linear Algebra and Projective Geometry Buildings and Grassmannians Classical Grassmannians Polar and Half-Spin Grassmannians.

Geometry of polar Grassmann spaces

The Grassmann space of fc-subspaces of a polar space is defined and its geometry is examined. In particular, its cliques, subspaces and automorphisms are characterized. An analogue of Chows theorem for the Grassmann space of fc-subspaces of a polar spaces is proved.

TRANSFORMATIONS ON THE PRODUCT OF GRASSMANN SPACES

Let Gk denote the set of all k-dimensional subspaces of an n-dimensional vector space. We recall that two elements of Gk are called adjacent if their intersection has dimension k − 1. The set Gk is point set of a partial linear space, namely a Grassmann space for 1 < k < n − 1 (see Section 5) and a projective space for k ∈ {1, n − 1}. Two adjacent subspaces are—in the language of partial linear spaces—two distinct collinear points. W.L. Chow [4] determined all bijections of Gk that preserve adjacency in both directions in the year 1949. In this paper we call such a mapping, for short, an A-transformation. Disregarding the trivial cases k = 1 and k = n − 1, every Atransformation of Gk is induced by a semilinear transformation V → V or (only when k = 2n) by a semilinear transformation of V onto its dual space V . There is a wealth of related results, and we refer to [2], [6], and [9] for further references. In the present note, we aim at generalizing Chow’s result to products of Grassmann spaces. However, we consider only products of the form Gk × Gn−k, where Gk and Gn−k stem from the same vector space V . Furthermore, for a fixed k we restrict our attention to a certain subset of Gk ×Gn−k. This subset, say G, is formed by all pairs of complementary subspaces. Our definition of an adjacency on G in formula (3) is motivated by the definition of lines in a product of partial linear spaces; cf. e.g. [7]. One of our main results (Theorem 2) states that Chow’s theorem remains true, mutatis mutandis, for the A-transformations of G. However, in Theorem 1 we can show even more: Let us say that two elements (S,U) and (S, U ) of G are close to each other, if their Hamming distance is 1 or, said differently, if they coincide in precisely one of their components. Then the bijections of G onto itself which preserve this closeness relation in both directions—we call them C-transformations of G—are precisely the A-transformations of G. In this way, we obtain for 1 < k < n − 1 two characterizations of the semilinear bijections V → V and V → V ∗ via their action on the set G. Finally, we turn to the following question: What happens to our results if we replace the set G with the entire cartesian product Gk × Gn−k? Clearly, the basic notions of adjacency and closeness remain meaningful. We describe all C-transformations of Gk × Gn−k in Theorem 3. However, in sharp contrast to Theorem 1, this is a rather trivial task, and the transformations of this kind do not deserve any interest. Then, using a result of A. Naumowicz and K. Prażmowski [7], we also determine all A-transformations of Gk × Gn−k in Theorem 4. Such mappings are closely related with collineations of the underlying partial linear space, and in general they can

A characterization of geometrical mappings of Grassmann spaces

In the paper we show that mappings of Grassmann spaces sending base subsets to base subsets are induced by strong embeddings of the corresponding projective spaces.

Geometry of semilinear embeddings : relations to graphs and codes

Semilinear Mappings: Division Rings and Their Homomorphisms Vector Spaces Over Division Rings Semilinear Mappings Semilinear Embeddings Mappings of Grassmannians Induced by Semilinear Embeddings Kreuzers Example Duality Characterization of Strong Semilinear Embeddings Projective Geometry and Linear Codes: Projective Spaces Fundamental Theorem of Projective Geometry Proof of Theorem 1.2 m-independent Subsets in Projective Spaces PGL-subsets Generalized Macwilliams Theorem Linear Codes Remark on Symmetries of Linear Codes Isometric Embeddings of Grassmann Graphs: Graph Theory Elementary Properties of Grassmann Graphs Embeddings Isometric Embeddings Proof of Theorem 3.1 Equivalence of Isometric Embeddings Linearly Rigid Isometric Embeddings Remarks on Non-isometric Embeddings Some Results Related to Chows Theorem Huangs Theorem Johnson Graph in Grassmann Graph: Johnson Graph Isometric Embeddings of Johnson Graphs in Grassmann Graphs Proof of Theorem 4.2 Classification Problem and Relations to Codes Characterizations of Apartments in Building Grassmannians Characterization of Isometric Embeddings: Result, Corollaries and Remarks Special Subsets Connectedness of the Apartment Graph Intersections of J(n, k)-subsets of Different Types Proof of Theorem 5.1 Semilinear Mappings of Exterior Powers: Exterior Powers Grassmannians Grassmann Codes

Chow's Theorem and Projective Polarities

We give a generalization of the well-known Chow Theorem (on adjacency preserving transformations of Grassmannians) and apply it to polarities.

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ? k ( V ) be the Grassmann graph formed by k-dimensional subspaces of V, 1 < k < n - 1 . Denote by ? ( n , k ) q the restriction of ? k ( V ) to the set of all non-degenerate linear n , k q codes. We show that for any two codes the distance in ? ( n , k ) q coincides with the distance in ? k ( V ) only in the case when n < ( q + 1 ) 2 + k - 2 , i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ? k ( V ) and ? ( n , k ) q are distinct. We describe one class of such pairs.

Transformations preserving adjacency and base subsets of spine spaces

Transformations of spine spaces which preserve base subsets preserve also adjacency. They either preserve the two sorts of projective adjacency or interchange them. Lines of a spine space can be defined in terms of adjacency, except one case where projective lines have no proper extensions to projective maximal strong subspaces, and thus adjacency preserving transformations are collineations.

Base subsets of Grassmannians: infinite-dimensional case

Let V be an infinite-dimensional vector space. We define Grassmannians of V as orbits of the action of the group GL(V) on the set of proper subspaces of V and study transformations of Grassmannians preserving so-called base subsets.

Zigzag Structure of Thin Chamber Complexes

Zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all zigzags in the simplex