M-vector analogue for the cd-index

TL;DR

This paper proves a conjecture relating the coefficients of the cd-index of simplicial spheres to the dimensions of graded algebra components, extending known face number conditions to more complex combinatorial invariants.

Contribution

It establishes the conjecture for simplicial spheres and provides explicit algebraic constructions, advancing the understanding of cd-index coefficients.

Findings

Proved the conjecture for simplicial spheres.

Constructed explicit multi-graded algebra using lattice paths.

Provided numerical evidence for shellable spheres.

Abstract

A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector of dimensions of graded pieces of a standard graded algebra A. Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the coefficients of the cd-index of a poset P. The coefficients of the cd-index are conjectured to be the dimensions of graded pieces in a standard multigraded algebra A. We prove the conjecture for simplicial spheres and we give numerical evidence for general shellable spheres. In the simplicial case we construct the multi-graded algebra A explicitly using lattice paths.