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

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 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

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

On Stanley’s inequalities for character multiplicities

Let G be a group of automorphisms of a ranked poset Q and let N k denote the number of orbits on the elements of rank k in Q . What can be said about the N k for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group