The cd-index of the Boolean lattice

TL;DR

This paper explores properties of the {f cd}-index of the Boolean lattice, establishing algebraic structures and identities that parallel known properties of the {f ab}-index, and extends some results to the cubical lattice.

Contribution

It introduces a new algebra structure on polynomial algebra for the {f cd}-index and proves identities linking the {f cd}-index to Dehn-Sommerville relations.

Findings

New algebra structure on polynomial algebra for {f cd}-index

Equivalence of Dehn-Sommerville relations to algebraic identities

Extension of results to the cubical lattice

Abstract

We study some properties of the {\bf cd}-index of the Boolean lattice. They are extremely similar to the properties of the {\ab}-index, or equivalently, the flag $h$-vector of the Boolean lattice and hence may be viewed as their {\bf cd}-analogues. We define a different algebra structure on the polynomial algebra $k < \cv, \dv>$ and give a derivation on this algebra. It is of significance for the Boolean lattice and forms our main tool. Using similar methods, we also prove some results for the {\bf cd}-index of the cubical lattice. We show that the Dehn-Sommerville relations for the flag $f$-vector of an Eulerian poset are equivalent to certain simple identities that exist in our algebra.