Searcharxiv⌕ Search

arXiv subjects

Mark Pankov

Publications and source records attributed to Mark Pankov.

At least 19 recordsLinked to original sources
arXiv2026

Four relations on the set of point-hyperplane anti-flags

There are precisely four arrangements of two point-hyperplane anti-flags. We consider the corresponding relations on the set of such anti-flags and show that each of them can be recovered from any other except in one special case. If the field consists of two elements, then one of the relations cannot be used to recover each of the remaining three. This is related to a bijection between anti-flags and exterior points of the hyperbolic polar space which exists in this case.

math.CO↗
arXiv2026

Recovering of the Grassmann graph from the subgraph of non-degenerate subspaces

Let ${\mathbb F}$ be a (not necessarily finite) field. A subspace of the vector space ${\mathbb F}^n$ is called {\it non-degenerate} if it is not contained in a coordinate hyperplane. We show that the Grassmann graph of $k$-dimensional subspaces of ${\mathbb F}^n$, $1 n-k$. In the case when ${\mathbb F}={\mathbb F}_q$ is the field of $q$ elements, this subgraph is known as the graph of non-degenerate linear $[n,k]_q$ codes.

math.CO↗
arXiv2025

The full automorphism groups of the five symmetric $(15,8,4)$-designs

It is clear that the full automorphism group of the $(15,8,4)$-design of points and hyperplane complements of ${\rm PG}(3,2)$ is ${\rm GL}(4,2)$. Using methods of point-line geometries, we determine the full automorphism groups of the remaining four symmetric $(15,8,4)$-designs and describe their actions on the sets of points and blocks.

math.CO↗
arXiv2024

Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$

Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge 4$ the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric $(n,2m,m)$-designs. The main results concern the case $k=4$ and give a geometric interpretation of the five well-known symmetric $(15,8,4)$-designs.

math.CO↗
arXiv2024

A non-surjective Wigner-type theorem in terms of equivalent pairs of subspaces

Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of ${\mathcal G}_{\infty}(H)$ sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let $f$ be such a transformation. We show that there is a unique up to a scalar multiple linear or conjugate linear isometry $L:H\to H$ such that for every $X\in {\mathcal G}_{\infty}(H)$ the image $f(X)$ is the sum of $L(X)$ and a certain closed subspace $O(X)$ orthogonal to the range of $L$. In the case when $H$ is separable, we give the following sufficient condition to assert that $f$ is induced by a linear or conjugate linear isometry: if $O(X)=0$ for a certain $X\in {\mathcal G}_{\infty}(H)$, then the same holds for all $X\in {\mathcal G}_{\infty}(H)$.

math-ph↗
arXiv2024

Maximal cliques in the graph of $5$-ary simplex codes of dimension two

We consider the induced subgraph of the corresponding Grassmann graph formed by $q$-ary simplex codes of dimension $2$, $q\ge 5$. This graph contains precisely two types of maximal cliques. If $q=5$, then for any two maximal cliques of the same type there is a monomial linear automorphism transferring one of them to the other. Examples concerning the cases $q=7,11$ finish the note.

math.CO↗
arXiv2023

On maximal cliques in the graph of simplex codes

The induced subgraph of the corresponding Grassmann graph formed by simplex codes is considered. We show that this graph, as the Grassmann graph, contains two types of maximal cliques. For any two cliques of the first type there is a monomial linear automorphism transferring one of them to the other. Cliques of the second type are more complicated and can contain different numbers of elements.

math.CO↗
arXiv2023

Point-line geometries related to binary equidistant codes

We investigate point-line geometries whose singular subspaces correspond to binary equidistant codes. The main result is a description of automorphisms of these geometries. In some important cases, automorphisms induced by non-monomial linear automorphisms surprisingly arise.

math.CO↗
arXiv2023

Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

Let $H$ be an infinite-dimensional complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian formed by closed subspaces of $H$ whose dimension and codimension both are infinite. We say that $X,Y\in {\mathcal G}_{\infty}(H)$ are {\it ortho-adjacent} if they are compatible and $X\cap Y$ is a hyperplane in both $X,Y$. A subset ${\mathcal C}\subset {\mathcal G}_{\infty}(H)$ is called an $A$-{\it component} if for any $X,Y\in {\mathcal C}$ the intersection $X\cap Y$ is of the same finite codimension in both $X,Y$ and ${\mathcal C}$ is maximal with respect to this property. Let $f$ be a bijective transformation of ${\mathcal G}_{\infty}(H)$ preserving the ortho-adjacency relation in both directions. We show that the restriction of $f$ to every $A$-component of ${\mathcal G}_{\infty}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of $f$ to distinct components can be related to different operators.

math-ph↗
arXiv2023

On the graph of non-degenerate linear $[n,2]_2$ codes

Consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field, $1<k<n-1$. Every automorphism of this graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space; the second possibility is realized only for $n=2k$. Let $Γ(n,k)_q$ be the subgraph of the Grassman graph formed by all non-degenerate linear $[n,k]_q$ codes. If $q\ge 3$ or $k\ge 3$, then every isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph can be uniquely extended to an automorphism of the Grassmann graph. For $q=k=2$ there is an isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph which does not have this property. In this paper, we show that such exceptional isomorphism is unique up to an automorphism of the Grassmann graph.

math.CO↗
arXiv2022

The graphs of non-degenerate linear codes

We consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field and its subgraph $Γ(n,k)_q$ formed by non-degenerate linear $[n,k]_q$ codes. We assume that $1<k<n-1$. It is well-known that every automorphism of the Grassmann graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space; the second possibility is realized only if $n=2k$. Our results are the following: if $q\ge 3$ or $k\ne 2$, then every isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph can be uniquely extended to an automorphism of the Grassmann graph; in the case when $q=k=2$, there are subgraphs of the Grassmann graph isomorphic to $Γ(n,k)_{q}$ and such that isomorphisms between these subgraphs and $Γ(n,k)_{q}$ cannot be extended to automorphisms of the Grassmann graph.

math.CO↗
arXiv2021

Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

Let $H$ be a complex Hilbert space of dimension not less than $3$ and let ${\mathcal C}$ be a conjugacy class of compact self-adjoint operators on $H$. Suppose that the dimension of the kernels of operators from ${\mathcal C}$ not less than the dimension of their ranges. In the case when ${\mathcal C}$ is formed by operators of finite rank $k$ and $\dim H=2k$, we assume that $k\ge 4$. We show that every bijective transformation of C preserving the commutativity in both directions is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces of the same dimension.

math.FA↗
arXiv2021

Automorphisms of graphs corresponding to conjugacy classes of finite-rank self-adjoint operators

We consider the graph whose vertex set is a conjugacy class ${\mathcal C}$ consisting of finite-rank self-adjoint operators on a complex Hilbert space $H$. The dimension of $H$ is assumed to be not less than $3$. In the case when operators from ${\mathcal C}$ have two eigenvalues only, we obtain the Grassmann graph formed by $k$-dimensional subspaces of $H$, where $k$ is the smallest dimension of eigenspaces. Classical Chow's theorem describes automorphisms of this graph for $k>1$. Under the assumption that operators from ${\mathcal C}$ have more than two eigenvalues we show that every automorphism of the graph is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces with the same dimensions. In contrast to this result, Chow's theorem states that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality if ${\mathcal C}$ is formed by operators with precisely two eigenvalues.

math.CO↗
arXiv2021

Automorphisms and some geodesic properties of ortho-Grassmann graphs

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $Γ^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $Γ^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $Γ^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.

math.CO↗
arXiv2020

A geometric approach to Wigner-type theorems

Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then every orthogonality preserving transformation of ${\mathcal P}(H)$ is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of ${\mathcal P}(H)$ to itself is induced by a linear or conjugate-linear isometry ($H$ is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

math-ph↗
arXiv2020

Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators

Two distinct projections of finite rank $m$ are adjacent if their difference is an operator of rank two or, equivalently, the intersection of their images is $(m-1)$-dimensional. We extend this adjacency relation on other conjugacy classes of finite-rank self-adjoint operators which leads to a natural generalization of Grassmann graphs. Let ${\mathcal C}$ be a conjugacy class formed by finite-rank self-adjoint operators with eigenspaces of dimension greater than $1$. Under the assumption that operators from ${\mathcal C}$ have at least three eigenvalues we prove that every automorphism of the corresponding generalized Grassmann graph is the composition of an automorphism induced by a unitary or anti-unitary operator and the automorphism obtained from a permutation of eigenspaces with the same dimensions. The case when the operators from ${\mathcal C}$ have two eigenvalues only is covered by classical Chow's theorem which says that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality.

math.CO↗
arXiv2020

Connectedness of projective codes in the Grassmann graph

Using the concept of projective systems for linear codes and elementary linear algebra, we show that projective $[n,k]_q$ codes form a connected subgraph in the Grassmann graph consisting of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field.

math.CO↗