Bruhat and balanced graphs

TL;DR

This paper introduces balanced digraphs as a generalization of Eulerian posets, establishing their properties, dualities, and invariants like the ${\bf cd}$-index, with applications to Bruhat graphs and quasisymmetric functions.

Contribution

It generalizes chain enumeration to balanced digraphs, proves the existence of the ${\bf cd}$-index for these graphs, and links them to Hopf algebra structures and Bruhat graphs.

Findings

Balanced digraphs include classical Eulerian posets.

Existence of the non-homogeneous ${\bf cd}$-index is established.

Elementary proofs for the ${\bf cd}$-index properties in Bruhat graphs.

Abstract

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an $R$-labeling, imply the existence of the (non-homogeneous) ${\bf cd}$-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete ${\bf cd}$-index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture nonnegativity of the ${\bf cd}$-index for acyclic digraphs having a balanced linear edge labeling.