On modular homology in projective space

AbstractFor a vector space V over GF(q) let Lk be the collection of subspaces of dimension k. When R is a field let Mk be the vector space over it with basis Lk. The inclusion map ∂:Mk→Mk−1 then is the linear map defined on this basis via ∂(X)≔∑Y where the sum runs over all subspaces of co-dimension 1 in X. This gives rise to a sequenceM:0←M0←M1←⋯←Mk−1←Mk←⋯which has interesting homological properties if R has characteristic p>0 not dividing q. Following on from earlier papers we introduce the notion of π-homological, π-exact and almost π-exact sequences where π=π(p,q) is some elementary function of the two characteristics. We show that M and many other sequences derived from it are almost π-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from M. For infinite-dimensional spaces we give a general construction which yields π-exact sequences for finitary ideals in the group ring RPΓL(V)

Saturated simplicial complexes

AbstractAmong shellable complexes a certain class has maximal modular homology, and these are the so-called saturated complexes. We extend the notion of saturation to arbitrary pure complexes and give a survey of their properties. It is shown that saturated complexes can be characterized via the p-rank of incidence matrices and via the structure of links. We show that rank-selected subcomplexes of saturated complexes are also saturated, and that order complexes of geometric lattices are saturated

The Modular Homology of Inclusion Maps and Group Actions

Communictated by the Managing Editors Let 0 be a finite set of n elements, R a ring of characteristic p>0 and denote by Mk the R-module with k-element subsets of 0 as basis. The set inclusion map: Mk Mk&1 is the homomorphism which associates to a k-element subset 2 the sum (2)=11+12+}}}+1kof all its (k&1)-element subsets 1i. In this paper we study the chain 0 M 0 M 1 M 2}}}M k M k+1 M k+2}}} (*) arising from. We introduce the notion of p-exactness for a sequence and show that any interval of (*) not including Mn 2 or Mn+1 2 respectively, is p-exact for any prime p>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on 0 of order not divisible by p. If an interval of (*) , or an equivalent sequence arising from a permutation group on 0, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application are p-rank formulae for orbit inclusion matrices. 1996 Academic Press, Inc. 1

On modular homology of simplicial complexes: shellability

AbstractFor a simplicial complex Δ and coefficient domain F let FΔ be the F-module with basis Δ. We investigate the inclusion map given by ∂:τ↦σ1+σ2+σ3+…+σk which assigns to every face τ the sum of the co-dimension 1 faces contained in τ. When the coefficient domain is a field of characteristic p>0 this map gives rise to homological sequences. We investigate this modular homology for shellable complexes

On Modular Homology of Simplicial Complexes: Saturation

... homology, and these are the so-called saturated complexes. We show that certain conditions on the links of the complex imply saturation. We prove that Coxeter complexes and buildings are saturated

An algebraic formulation of the graph reconstruction conjecture

The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $\psi(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $\psi(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $\psi(n) = d(n)$. We give a framework based on Kocay's lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) \geq \mathsf{rank}_\mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $\psi(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = \mathsf{rank}_\mathbb{R}(M)$.Comment: 12 pages, 2 figure

On eigenvectors of the discrete Fourier transform over finite Gaussian fields

Основная статьяThe problem of furnishing orthogonal systems of eigenvectors for the discrete Fourier transform (DFT) is fundamental to image processing with applications in image compression and digital watermarking. This paper studies some properties of such systems for DFT over finite fields that may be considered as ”finite complex planes”. Some applications for multiuser communication schemes are also considered