arXiv subjects
Publications and source records attributed to Mark Pankov.
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.
Let $H$ be a complex Hilbert space and let ${\mathcal G}_{k}(H)$ be the Grassmannian formed by $k$-dimensional subspaces of $H$. Suppose that $\dim H>2k$ and $f$ is an orthogonality preserving injective transformation of ${\mathcal G}_{k}(H)$, i.e. for any orthogonal $X,Y\in {\mathcal G}_{k}(H)$ the images $f(X),f(Y)$ are orthogonal. If $\dim H=n$ is finite, then $n=mk+i$ for some integers $m\ge 2$ and $i\in \{0,1,\dots,k-1\}$ (for $i=0$ we have $m\ge 3$). We show that $f$ is a bijection induced by a unitary or anti-unitary operator if $i\in \{0,1,2,3\}$ or $m\ge i+1\ge 5$; in particular, the statement holds for $k\in \{1,2,3,4\}$ and, if $k\ge 5$, then there are precisely $(k-4)(k-3)/2$ values of $n$ such that the above condition is not satisfied. As an application, we obtain a result concerning the case when $H$ is infinite-dimensional.
We give a complete description of the distance relation on the graph of $4$-ary simplex codes of dimension $2$. This is a connected graph of diameter $3$. For every vertex we determine the sets of all vertices at distance $i\in\{1,2,3\}$ and describe their symmetries.
Let $H$ be a complex Hilbert space and let ${\mathcal C}$ be a conjugacy class of finite rank self-adjoint operators on $H$ with respect to the action of unitary operators. We suppose that ${\mathcal C}$ is formed by operators of rank $k$ and for every $A\in {\mathcal C}$ the dimensions of distinct maximal eigenspaces are distinct. Under the assumption that $\dim H\ge 4k$ we establish that every bijective transformation $f$ of ${\mathcal C}$ preserving the commutativity in both directions is induced by a unitary or anti-unitary operator, i.e. there is a unitary or anti-unitary operator $U$ such that $f(A)=UAU^{*}$ for every $A\in {\mathcal C}$. A simple example shows that the condition concerning the dimensions of maximal eigenspaces cannot be omitted.
A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a common vertex, 2) for any three consecutive edges the first and the third edges are disjoint and the face containing the first and the second edges is distinct from the face which contains the second and the third. A map is $z$-knotted if it contains a single zigzag. Such maps are closely connected to Gauss code problem and have nice homological properties. We show that every triangulation of a connected closed $2$-dimensional surface admits a $z$-knotted shredding.
We investigate zigzags in triangulations of connected closed $2$-dimensional surfaces and show that there is a one-to-one correspondence between triangulations with homogeneous zigzags and closed $2$-cell embeddings of directed Eulerian graphs in surfaces. A triangulation is called $z$-knotted if it has a single zigzag. We construct a family of tree structured $z$-knotted spherical triangulations whose zigzags are homogeneous.
Let $Γ$ be a triangulation of a connected closed $2$-dimensional (not necessarily orientable) surface. Using zigzags (closed left-right paths), for every face of $Γ$ we define the $z$-monodromy which acts on the oriented edges of this face. There are precisely $7$ types of $z$-monodromies. We consider the following two cases: (M1) the $z$-monodromy is identity, (M2) the $z$-monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph $Γ^{*}$ formed by edges whose $z$-monodromies are of types (M1) and (M2), respectively, both are forests. We apply this statement to the connected sum of $z$-knotted triangulations.
Let $H$ be a complex Hilbert space whose dimension is not less than $3$ and let ${\mathcal F}_{s}(H)$ be the real vector space formed by all self-adjoint operators of finite rank on $H$. For every non-zero natural $k<\dim H$ we denote by ${\mathcal P}_{k}(H)$ the set of all rank $k$ projections. Let $H'$ be other complex Hilbert space of dimension not less than $3$ and let $L:{\mathcal F}_{s}(H)\to {\mathcal F}_{s}(H')$ be a linear operator such that $L({\mathcal P}_{k}(H))\subset {\mathcal P}_{m}(H')$ for some natural $k,m$ and the restriction of $L$ to ${\mathcal P}_{k}(H)$ is injective. If $H=H'$ and $k=m$, then $L$ is induced by a linear or conjugate-linear isometry of $H$ to itself, except the case $\dim H=2k$ when there is another one possibility (we get a classical Wigner's theorem if $k=m=1$). If $\dim H\ge 2k$, then $k\le m$. The main result describes all linear operators $L$ satisfying the above conditions under the assumptions that $H$ is infinite-dimensional and for any $P,Q\in {\mathcal P}_{k}(H)$ the dimension of the intersection of the images of $L(P)$ and $L(Q)$ is not less than $m-k$.
Consider the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the field of $q$ elements ($1<k<n-1$) and denote by $Π(n,k)_q$ the restriction of this graph to the set of projective $[n,k]_q$ codes. In the case when $q\ge \binom{n}{2}$, we show that the graph $Π(n,k)_q$ is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension $3$ are precisely maximal singular subspaces of a non-degenerate quadratic form.
We show that the transformations of Grassmannians (of complex Hilbert spaces) induced by linear or conjugate-linear isometries can be characterized as transformations preserving some of principal angles (corresponding to the orthogonality, adjacency and ortho-adjacency relations).
An embedded graph is called $z$-knotted if it contains the unique zigzag (up to reversing). We consider $z$-knotted triangulations, i.e. $z$-knotted embedded graphs whose faces are triangles, and describe all cases when the connected sum of two $z$-knotted triangulations is $z$-knotted.
We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.
Zigzags and generalized 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 generalized zigzags in the simplex $α_{n}$, the cross-polytope $β_{n}$, the $24$-cell, the icosahedron and the $600$-cell are equal to the Coxeter numbers of $A_{n}$, $B_{n}=C_{n}$, $F_{4}$ and $H_{i}$, $i=3,4$, respectively. Also, we discuss the following problem: in which cases two faces in a thin chamber complex can be connected by a zigzag?
We define the Grassmannians of an infinite-dimensional vector space $V$ as the orbits of the action of the general linear group ${\rm GL}(V)$ on the set of all subspaces. Let ${\mathcal G}$ be one of these Grassmannians. An apartment in ${\mathcal G}$ is the set of all elements of ${\mathcal G}$ spanned by subsets of a certain basis of $V$. We show that every bijective transformation $f$ of ${\mathcal G}$ such that $f$ and $f^{-1}$ send apartments to apartments is induced by a semilinear automorphism of $V$. In the case when ${\mathcal G}$ consists of subspaces whose dimension and codimension both are infinite, a such kind result will be proved also for the connected components of the associated Grassmann graph.
We consider zigzags in thin complexes. The main result states that the sum of the lengths of all zigzags in an $n$-complexe is equal to the sum of the lengths of all zigzags in all $(n-1)$-faces of this complex, and this sum also is the twice of the sum of the lengths of all zigzags in all $(n-2)$-faces. For simplicial and cubical $n$-complexes, the sum depends on the rank $n$ and the number of $(n-1)$-faces only. We also describe the sum of the lengths of all generalized zigzags, it depends on the rank and the number of flags. As an application, we find the number of zigzags in Coxeter complexes.
Let $Γ_{k}(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field ${\mathbb F}_{q}$ consisting of $q$ elements and $1<k<n-1$. Denote by $Γ(n,k)_q$ the restriction of the Grassmann graph to the set of all non-degenerate linear $[n,k]_q$ codes. We describe maximal cliques of the graph $Γ(n,k)_q$ and show that every automorphism of this graph is induced by a monomial semilinear automorphism of $V$.
Let $H$ be a complex Hilbert space of finite dimension $n\ge 3$. Denote by ${\mathcal G}_{k}(H)$ the Grassmannian consisting of $k$-dimensional subspaces of $H$. Every orthogonal apartment of ${\mathcal G}_{k}(H)$ is defined by a certain orthogonal base of $H$ and consists of all $k$-dimensional subspaces spanned by subsets of this base. For $n\ne 2k$ (except the case when $n=6$ and $k$ is equal to $2$ or $4$) we show that every bijective transformation of ${\mathcal G}_{k}(H)$ sending orthogonal apartments to orthogonal apartments is induced by an unitary or conjugate-unitary operator on $H$. The second result is the following: if $n=2k\ge 8$ and $f$ is a bijective transformation of ${\mathcal G}_{k}(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments then there is an unitary or conjugate-unitary operator $U$ such that for every $X\in {\mathcal G}_{k}(H)$ we have $f(X)=U(X)$ or $f(X)$ coincides with the orthogonal complement of $U(X)$.
Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal L}(H)$ be the logic formed by all closed subspaces of $H$. For every natural $k$ we denote by ${\mathcal G}_{k}(H)$ the Grassmannian consisting of $k$-dimensional subspaces. An orthogonal apartment of ${\mathcal G}_{k}(H)$ is the set consisting of all $k$-dimensional subspaces spanned by subsets of a certain orthogonal base of $H$. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of ${\mathcal G}_{k}(H)$. We show that every bijective transformation $f$ of ${\mathcal G}_{k}(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments (in other words, $f$ preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of ${\mathcal L}(H)$.