Poset subdivisions and the mixed cd-index

TL;DR

This paper introduces the mixed cd-index, an invariant for subdivisions of Eulerian posets, extending the classical cd-index and linking it to the mixed h-polynomial, with theoretical development and example computations.

Contribution

It defines the mixed cd-index for strong formal subdivisions of posets, extending the cd-index framework and proving a related decomposition theorem.

Findings

The mixed cd-index determines the mixed h-polynomial.

The decomposition theorem is extended to strong formal subdivisions.

Examples illustrate the computation of the mixed cd-index.

Abstract

The cd-index is an invariant of Eulerian posets expressed as a polynomial in noncommuting variables c and d. It determines another invariant, the h-polynomial. In this paper, we study the relative setting, that of subdivisions of posets. We introduce the mixed cd-index, an invariant of strong formal subdivisions of posets, which determines the mixed h-polynomial introduced by the second author with Stapledon. The mixed cd-index is a polynomial in noncommuting variables c',d',c,d, and e and is defined in terms of the local cd-index of Karu. Here, use is made of the decomposition theorem for the cd-index. We extend the proof of the decomposition theorem, originally due to Ehrenborg-Karu, to the class of strong formal subdivisions. We also compute the mixed cd-index in a number of examples.