Extending the ab-index
Elena Hoster, Christian Stump, Lorenzo Vecchi
TL;DR
This work proves that for finite, graded, bounded posets of rank $n$, the Poincaré-extended $ab$-index exPsi$(y,a,b)$ is obtained from the ordinary $ab$-index $\Psi_P(a,b)$ via the $omega$-transformation, and likewise exPsi_tilde$(y,a,b) = (1+y) \omega(exPsi_tilde_P(a,b))$. It provides a unified, conceptual mechanism linking the $ab$-index to the Poincaré-extended index and, through a flag-$h$ vector expansion, yields nonnegativity results beyond $R$-labeled posets. The results recover and unify several prior developments, including Chow polynomial decompositions, gamma-positivity, and numerical decompositions in the Chow ring, by expressing extended indices and polynomials as explicit omega-transforms of classical invariants. The proofs rely on recursive decompositions, the incidence-algebra identity, and evaluations of the omega-transformation, enabling streamlined derivations and broad generalizations to nonnegative flag $h$-vectors and related polynomials.
Abstract
We prove for finite, graded, bounded posets, that the Poincaré-extended ab-index is obtained from the ab-index via the omega-transformation. This proves a conjecture by Dorpalen-Barry, Maglione, and the second author, and provides a more conceptual approach to ab-indices and Chow polynomials beyond R-labeled posets.