Ela Incidence Matrix and Cover Matrix of Nested Interval Orders

YAOKUN WU AND SHIZHEN ZHAO Abstract. For any poset P , its incidence matrix n and its cover matrix C are the P × P (0, 1) matrices such that n(x, y) = 1 if any only if x is less than y in P and C(x, y) = 1 if any only if x is covered by y in P . It is shown that n and C are conjugate to each other in the incidence algebra of P over a field of characteristic 0 provided P is the nested interval order. In particular, when P is the Bruhat order of a dihedral group, which consists of a special family of nested intervals, n and C turn out to be conjugate in the incidence algebra over every field. Moreover, n and C are proved to be conjugate in the incidence algebra over every field when P is the weak order of a dihedral group. Many relevant problems and observations are also presented in this note.