arXiv subjects
Publications and source records attributed to Mark Pankov.
We consider the Grassmann graphs and dual polar graphs over the same finite field and show that, up to graph automorphism, for every dual polar graph there is the unique isometric embedding in the corresponding Grassmann graph.
Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements and let $Γ_{k}(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of $V$, $1<k<n-1$. Denote by $Γ(n,k)_{q}$ the restriction of $Γ_{k}(V)$ to the set of all non-degenerate linear $[n,k]_{q}$ codes. We show that for any two codes the distance in $Γ(n,k)_{q}$ coincides with the distance in $Γ_{k}(V)$ 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 $Γ_{k}(V)$ and $Γ(n,k)_{q}$ are distinct. We describe one class of such pairs.
We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider $3$-embeddings, i.e. embeddings preserving all distances not greater than $3$, of dual polar graphs whose diameter is not less than $3$ in polar Grassmann graphs formed by non-maximal singular subspaces. Using the same arguments we show that every such an embedding can be reduced to an embedding in a Grassmann graph.
We describe isometric embeddings of polar Grassmann graphs formed by non-maximal singular subspaces. In almost all cases, they are induced by collinearity preserving injections of polar spaces. As a simple consequence of this result, we get a metric characterization of apartments in polar Grassmannians.
We consider the Cayley graph ${\rm C}(W,S)$ of a Coxeter system $(W,S)$ and describe all maximal $2$-cliques in this graph, i.e. maximal subsets in the vertex set such that the distance between any two distinct elements is equal to $2$. As an application, we show that every automorphism of the half of Cayley graph is uniquely extendable to an automorphism of the Cayley graph if $|S|\ge 5$.
Let $Π$ be a polar space of type $\textsf{D}_{n}$. Denote by ${\mathcal G}_δ(Π)$, $δ\in \{+,-\}$ the associated half-spin Grassmannians and write $Γ_δ(Π)$ for the corresponding half-spin Grassmann graphs. In the case when $n\ge 4$ is even, the apartments of ${\mathcal G}_δ(Π)$ will be characterized as the images of isometric embeddings of the half-cube graph $\frac{1}{2}H_n$ in $Γ_δ(Π)$. As an application, we describe all isometric embeddings of $Γ_δ(Π)$ in the half-spin Grassmann graphs associated to a polar space of type $\textsf{D}_{n'}$ under the assumption that $n\ge 6$ is even.
Let $V$ and $V'$ be vector spaces over division rings (possible infinite-dimensional) and let ${\mathcal P}(V)$ and ${\mathcal P}(V')$ be the associated projective spaces. We say that $f:{\mathcal P}(V)\to {\mathcal P}(V')$ is a PGL-{\it mapping} if for every $h\in {\rm PGL}(V)$ there exists $h'\in {\rm PGL}(V')$ such that $fh=h'f$. We show that for every PGL-bijection the inverse mapping is a semicollineation. Also, we obtain an analogue of this result for the projective spaces associated to normed spaces.
The set of all subspaces of a given dimension in a finite classical polar space has a structure of a symmetric association scheme. If the dimension is zero, this is the scheme of the collinearity graph of the space; If the dimension is maximum, it is the dual polar scheme. In this note, we determine the full automorphism group of this scheme.
Let $Π$ be a polar space of rank $n\ge 3$. Denote by ${\mathcal G}_{k}(Π)$ the polar Grassmannian formed by singular subspaces of $Π$ whose projective dimension is equal to $k$. Suppose that $k$ is an integer not greater than $n-2$ and consider the relation ${\mathfrak R}_{i,j}$, $0\le i\le j\le k+1$ formed by all pairs $(X,Y)\in {\mathcal G}_{k}(Π)\times {\mathcal G}_{k}(Π)$ such that $\dim_{p}(X^{\perp}\cap Y)=k-i$ and $\dim_{p} (X\cap Y)=k-j$ ($X^{\perp}$ consists of all points of $Π$ collinear to every point of $X$). We show that every bijective transformation of ${\mathcal G}_{k}(Π)$ preserving ${\mathfrak R}_{1,1}$ is induced by an automorphism of $Π$ and the same holds for the relation ${\mathfrak R}_{0,t}$ if $n\ge 2t\ge 4$ and $k=n-t-1$. In the case when $Π$ is a finite classical polar space, we establish that the valencies of ${\mathfrak R}_{i,j}$ and ${\mathfrak R}_{i',j'}$ are distinct if $(i,j)\ne (i',j')$.
We describe all adjacency preserving bijections of certain products of Grassmann spaces.
Let $V$ and $V'$ be vector spaces over division rings. Suppose $\dim V$ is finite and not less than 3. Consider a mapping $l:V\to V$ with the following property: for every $u\in {\rm GL}(V)$ there is $u'\in {\rm GL}(V')$ such that $lu=u'l$. Our first result states that $l$ is a strong semilinear embedding if $l|_{V\setminus{0}}$ is non-constant and the dimension of the subspace of $V'$ spanned by $l(V)$ is not greater than $n$. We present examples showing that these conditions can not be omitted. In some special cases, this statement can be obtained from Dicks and Hartley (1991) and Zha (1996). Denote by ${\mathcal P}(V)$ the projective space associated with $V$ and consider the mapping $f:{\mathcal P}(V)\to {\mathcal P}(V')$ with the following property: for every $h\in {\rm PGL}(V)$ there is $h'\in {\rm PGL}(V')$ such that $fh=h'f$. By the second result, $f$ is induced by a strong semilinear embedding of $V$ in $V'$ if $f$ is non-constant and its image is contained in a subspace of $V'$ whose dimension is not greater than $n$, we also require that $R'$ is a field.
Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\mathcal X}\subset {\mathcal P}(V)$ such that every permutation on ${\mathcal X}$ can be extended to an element of ${\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups.
Let $V$ be an $n$-dimensional left vector space over a division ring $R$. We write ${\mathcal G}_{k}(V)$ for the Grassmannian formed by $k$-dimensional subspaces of $V$ and denote by $Γ_{k}(V)$ the associated Grassmann graph. Let also $V'$ be an $n'$-dimensional left vector space over a division ring $R'$. Isometric embeddings of $Γ_{k}(V)$ in $Γ_{k'}(V')$ are classified in \cite{Pankov2}. A classification of $J(n,k)$-subsets in ${\mathcal G}_{k'}(V')$, i.e. the images of isometric embeddings of the Johnson graph $J(n,k)$ in $Γ_{k'}(V')$, is presented in \cite{Pankov1}. We characterize isometric embeddings of $Γ_{k}(V)$ in $Γ_{k'}(V')$ as mappings which transfer apartments of ${\mathcal G}_{k}(V)$ to $J(n,k)$-subsets of ${\mathcal G}_{k'}(V')$. This is a generalization of the earlier result concerning apartments preserving mappings \cite[Theorem 3.10]{Pankov-book}.
Let $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $Γ_{k}(V)$ in the Grassmann graph $Γ_{k'}(V')$.
Buildings of types $\textsf{C}_n$ and $\textsf{D}_n$ are defined by rank $n$ polar spaces. The associated building Grassmannians are polar and half-spin Grassmannians. Apartments in dual polar spaces and half-spin Grassmannians were characterized in \cite{CKS}. We characterize apartments in all polar Grassmannians consisting of non-maximal singular subspaces. This characterization is a partial case of more general results concerning embeddings of polar Johnson graphs in polar Grassmann graphs.
We consider the infinite-dimensional hypercube graph. This graph is not connected and has isomorphic connected components. We describe the restrictions of its automorphisms to the connected components and the automorphism group of connected component.
We consider the {\it infinite Johnson graph} $J_{\infty}$ whose vertex set consists of all subsets $X\subset {\mathbb N}$ satisfying $|X|=|{\mathbb N}\setminus X|=\infty$ and whose edges are pairs of such subsets $X,Y$ satisfying $|X\setminus Y|=|Y\setminus X|=1$. An automorphism of $J_{\infty}$ is said to be {\it regular} if it is induced by a permutation on $\mathbb{N}$ or it is the composition of the automorphism induced by a permutation on $\mathbb{N}$ and the automorphism $X\to {\mathbb N}\setminus X$. The graph $J_{\infty}$ admits non-regular automorphisms. Our first result states that the restriction of every automorphism of $J_{\infty}$ to any connected component ($J_{\infty}$ is not connected) coincides with the restriction of a regular automorphism. The second result is a characterization of regular automorphisms of $J_{\infty}$ as order preserving and order reversing bijective transformations of the vertex set of $J_{\infty}$ (the vertex set is partially ordered by the inclusion relation). As an application, we describe automorphisms of the associated {\it infinite Kneser graph}.
Given the algebra $T$ of ternions (upper triangular $2\times 2$ matrices) over a commutative field $F$ we consider as set of points of a projective line over $T$ the set of all free cyclic submodules of $T^2$. This set of points can be represented as a set of planes in the projective space over $F^6$. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that $T$ admits an $F$-linear antiautomorphism, the plane model of our projective line does not admit any duality.