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Mark Pankov

Publications and source records attributed to Mark Pankov.

68 records · Page 4Linked to original sources
arXiv2010

Isometric embeddings of Johnson graphs in Grassmann graphs

Let $V$ be an $n$-dimensional vector space ($4\le n <\infty$) and let ${\mathcal G}_{k}(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $Γ_{k}(V)$. We describe all isometric embeddings of Johnson graphs $J(l,m)$, $1<m<l-1$ in $Γ_{k}(V)$, $1<k<n-1$ (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of $J(n,k)$ in $Γ_{k}(V)$ is an apartment of ${\mathcal G}_{k}(V)$ if and only if $n=2k$. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in $Γ_{k}(V)$, $1<k<n-1$.

math.CO↗
arXiv2010

Metric characterization of apartments in dual polar spaces

Let $Π$ be a polar space of rank $n$ and let ${\mathcal G}_{k}(Π)$, $k\in \{0,\dots,n-1\}$ be the polar Grassmannian formed by $k$-dimensional singular subspaces of $Π$. The corresponding Grassmann graph will be denoted by $Γ_{k}(Π)$. We consider the polar Grassmannian ${\mathcal G}_{n-1}(Π)$ formed by maximal singular subspaces of $Π$ and show that the image of every isometric embedding of the $n$-dimensional hypercube graph $H_{n}$ in $Γ_{n-1}(Π)$ is an apartment of ${\mathcal G}_{n-1}(Π)$. This follows from a more general result (Theorem 2) concerning isometric embeddings of $H_{m}$, $m\le n$ in $Γ_{n-1}(Π)$. As an application, we classify all isometric embeddings of $Γ_{n-1}(Π)$ in $Γ_{n'-1}(Π')$, where $Π'$ is a polar space of rank $n'\ge n$ (Theorem 3).

math.CO↗
arXiv2010

Grassmannians of classical buildings

This book is dedicated to Grassmannians associated with buildings of classical types: usual, polar, and half-spin Grassmannians. Grassmannians of vector spaces and Grassmannians consisting of totally isotropic subspaces of non-degenerate alternating, Hermitian, and symmetric forms are special cases of these "building" Grassmannians.

math.CO↗
arXiv2007

Order preserving transformations of the Hilbert grassmannian: complex case

Let $H$ be a separable complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every continuous order preserving bijective transformation of ${\mathcal G}_{\infty}(H)$ is induced by an invertible bounded semi-linear operator.

math.FA↗
arXiv2006

Base subsets of polar Grassmannians

Let $Δ$ be a thick building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$. Let also ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of the associated polar space $Π$ (of rank $n$). We write ${\mathfrak G}_k$ for the corresponding shadow space of type $\textsf{X}_{n,k}$. Every bijective transformation of ${\mathcal G}_k$ sending base subsets to base subsets (the shadows of apartments) is a collineation of ${\mathfrak G}_k$, and it is induced by a collineation of $Π$ if $n\ne 4$ or $k\ne 1$.

math.CO↗
arXiv2006

Order preserving transformations of the Hilbert grassmannian

Let $H$ be a separable real Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of ${\mathcal G}_{\infty}(H)$ can be extended to an automorphism of the lattice of closed subspaces of $H$. It follows from Mackey's result \cite{Mackey} that automorphisms of this lattice are induced by invertible bounded linear operators.

math.FA↗
arXiv2006

Base subsets of symplectic Grassmannians

Let $V$ and $V'$ be $2n$-dimensional vector spaces over fields $F$ and $F'$. Let also $Ω: V\times V\to F$ and $Ω': V'\times V'\to F'$ be non-degenerate symplectic forms. Denote by $Π$ and $Π'$ the associated $(2n-1)$-dimensional projective spaces. The sets of $k$-dimensional totally isotropic subspaces of $Π$ and $Π'$ will denoted by ${\mathcal G}_{k}$ and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect ${\mathcal G}_{k}$ and ${\mathcal G}'_{k}$ by so-called base subsets. We show that every mapping of ${\mathcal G}_{k}$ to ${\mathcal G}'_{k}$ sending base subsets to base subsets is induced by a symplectic embedding of $Π$ to $Π'$.

math.CO↗
arXiv2005

On geometry of linear involutions

Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal G}_{k}$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak G}_{k}$ for the set of all pairs $(S,U)\in {\mathcal G}_{k}\times {\mathcal G}_{n-k}$ such that $S+U=V$. We study bijective transformations of ${\mathfrak G}_{k}$ 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 $n\ne 2k$; for $n=2k$ this fails. This result can be formulated as the following: if $n\ne 2k$ and the characteristic of $R$ is not equal to 2 then any commutativity preserving transformation of the set of $(k,n-k)$-involutions is extended to an automorphism of the group {\rm GL}(V).

math.GR↗
arXiv2005

On geometry of symplectic involutions

Let $V$ be a $2n$-dimensional vector space over a field $F$ and $Ω$ be a non-degenerate symplectic form on $V$. Denote by ${\mathfrak H}_{k}(Ω)$ the set of all $2k$-dimensional subspaces $U\subset V$ such that the restriction $Ω|_{U}$ is non-degenerate. Our main result (Theorem 1) says that if $n\ne 2k$ and $\max(k,n-k)\ge 5$ then any bijective transformation of ${\mathfrak H}_{k}(Ω)$ preserving the class of base subsets is induced by a semi-simplectic automorphism of $V$. For the case when $n\ne 2k$ this fails, but we have a weak version of this result (Theorem 2). If the characteristic of $F$ is not equal to 2 then there is a one-to-one correspondence between elements of ${\mathfrak H}_{k}(Ω)$ and symplectic $(2k,2n-2k)$-involutions and Theorem 1 can be formulated as follows: for the case when $n\ne 2k$ and $\max(k,n-k)\ge 5$ any commutativity preserving bijective transformation of the set of symplectic $(2k,2n-2k)$-involutions can be extended to an automorphism of the symplectic group.

math.GR↗
arXiv2004

Transformations of Grassman Spaces

This is a version of a part of the book ``Transformations of Grassman Spaces'' (in progress). We study transformations of Grassman spaces preserving certain geometrical constructions related to buildings. The next part will be devoted to Grassman spaces associated with polar spaces.

math.GM↗
arXiv2003

On Tits Buildings of Type $A_n$

Let $P$ and $P'$ be projective spaces having the same dimension $n$ assumed to be finite. Let $Δ$ and $Δ'$ be buildings of type $A_n$. We consider mappings of chambers of $Δ$ into chambers of $Δ'$ which send appartments to appartments. It is shown that shuch mappings are induced by strong embeddings of $P$ into $P'$ or into the dual space $P^{'*}$.

math.GR↗
arXiv2001

Transformations of Grassmannians and automorphisms of linear groups

We consider transformations preserving certain linear structure in Grassmannians and give some generalization of the Fundamental Theorem of Projective Geometry and the Chow Theorem [Ch]. It will be exploited to study linear $(k,n-k)$-involutions, $1<k<n-1$. The analogy of the well-known J. Dieudonné and C. E. Rickart result will be obtained.

math.RA↗