Joint ICE-TCS and Combinatorics seminar - Martina Kubitzke - The Lefschetz property for barycentric subdivisions of simplicial complexes

Martina Kubitzke (Philipps-Universität Marburg), who will be joining the Combinatorics  group as a postdoc in the fall, will give a talk on The Lefschetz property for barycentric subdivisions of simplicial complexes.

_________________________

Abstract

A classical problem in the field of face enumeration of simplicial complexes is the classification of special classes of simplicial complexes in terms of their h- and f-vectors, respectively. McMullen conjectured that the g-vector of the boundary complex of a simplicial polytope is an M-sequence and that, in particular, it is unimodal.

In 1971 this was shown to be true by Stanley. In his proof he used the Hard Lefschetz  theorem for toric varieties. In the same year Billera and Lee succeeded in constructing a simplicial polytope whose boundary has as g-vector a given M-sequence. The result of Stanley and Billera/Lee is known as the g-theorem. McMullen's original conjecture
was later generalized by himself and Björner/Swartz to simplicial spheres and homology spheres, respectively. This conjecture is known as the g-conjecture.

Brenti and Welker showed that the h-vector of the barycentric subdivision of a Cohen-Macaulay complex is unimodal which led to the conjecture that the g-vector of such a complex is an M-sequence.  We verify this conjecture in the affirmative by showing that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable simplicial complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the h-vector is unimodal, attains in maximum value in its middle degree (one of them if the dimension of the complex is even), and that its g-vector is an M-sequence. It is remarkable that those numerical results hold in the greater generality of Cohen-Macaulay complexes even though the algebraic result does only hold for the smaller class of shellable simplicial complexes.

Our result in particular shows the g-conjecture for barycentric subdivisions of homology spheres.  Brenti and Welker further showed that the h-vector of the barycentric subdivision of a simplicial complex can be expressed as a positive linear combination of the h-vector of the original simplicial complex. The coefficients occurring in this representation are a efinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of 1.

Using the algebraic result, i.e., the almost strong Lefschetz property of the barycentric subdivision of a shellable simplicial complex, we derive new inequalities for those statistics on permutations. Algebraic and combinatorial notion and background, such as the notion of Lefschetz elements and M-sequences will be provided during the talk.

This is joint work with Eran Nevo, Cornell University.