On a class of partially ordered sets and their linear invariants (Q1184501)

Let \(({\mathcal L},<)\) be a finite partially ordered set with a unique minimal element and with a rank function. Let \({\mathcal L}_ k\) denote the set of elements of rank \(k\). \(\mathcal L\) can be described by a set of incidence matrices \(\{S_ k\}\), where the rows of \(S_ k\) are indexed by \({\mathcal L}_ k\) and the columns by \({\mathcal L}_{k+1}\). The \((x,y)\)- entry of \(S_ k\) is one if \(x< y\), zero otherwise. For the special case where \(\mathcal L\) is what the author calls join-meet regular, one has matrix equations of the form \(S_ k(S^ T_ k)-c_ k(S_{k-1})^ T S_{k- 1}=(d^ +_ k-c_ k d^ -_ k)\text{Id}\), where the coefficients \(c_ k\), \(d^ +_ k\), and \(d^ -_ k\) can be described combinatorially. From these equations, the eigenvalues (with multiplicities) of the matrices \(S_ k S^ T_ k\) are deduced. Similar arguments determine the spectra of some matrices associated with tactical decompositions.