Computation Of Eigenvalues Of Laplace Operators For A Class Of Graded Posets

As we all know, homology algebra is an important mathematics branch. We can find its figure in the field of group theory, commutative algebra, algebraic topology,algebraic geometry, differential geometry,differential topology, algebraic number theory and partial differential equation. The study of homology theory and its applied problems also gets attention from many scholars all the time.Laplace method is one of the important tools in studying homological problems.For half a century, a lot of papers have studied the eigenvalues, eigenvalue estimating and the maximum or minimum eigenvalue of the Laplace operators of homological problems comparatively systematically. Among them, the one that everyone concerns most is the computation of eigenvalues.This paper first introduces the background of using Laplace method to study cohomological problems as well as the internal and foreign research status. Then,we introduce the definitions of poset, Hasse diagram,incidence algebra and homology, which is necessary prepared knowledge. Based on this, we consider a class of partial order on the set P={（i,j）|i=1,2,3,j=1,2,...,n}We define three linear spaces R（V）,R（E）,R（F） on real number field on the base of Hasse diagrams that correspond to the graded posets. Next, we present two linear mappings and by using the view of cohomology respectively. Using Z0Y=o we can get the following complex Furthermore, we define three corresponding Laplace operators as follows:L₀：Y·Y^YL₁=Z·Z^Y+Y^Y·YL₂=Z^Y·ZAt last, we compute eigenvalues of these three operators completely.