Research problem: combinatorial and multilinear aspects of sign-balanced posets

Abstract

We present basic results and open problems related to the study of sets L(P) of linear extensions of posets (P, ⪯). In our study, these linear extensions are regarded as permutations σ with the alternating character (-1)inv(σ). On the combinatorial side (Section 2), we present a number of open problems related to the fundamental study of sign-balanced posets whose Hasse diagrams are specified by Ferrers diagrams of partitions. On the multilinear-algebraic side (Section 3), we propose the general study of operators associated with P L(P), and the alternating character. We derive an antisymmetric decomposition of such operators as shuffles of anti-symmetric operators associated with symmetric groups. In general, we show that the set L(P)=G is a group if and only if P is an ordinal sum of antichains and G is a direct product of symmetric groups. As a consequence, the antisymmetric decomposition becomes a tensor product if and only if L(P) is a group.