# Alexander Lazar: Partition and Cohen-Macaulay Extenders

**Time: **
Wed 2020-09-09 10.15 - 11.15

**Location: **
KTH, 3418

**Participating: **
Alexander Lazar

If a pure simplicial complex is partitionable or Cohen-Macaulay, the entries of its h-vector have well-known combinatorial interpretations.

In this talk, based on joint work with Joseph Doolittle (Freie Universit\"{a}t Berlin) and Bennet Goeckner (University of Washington), we consider complexes that do not have such properties. We show that, given any complex $\Delta$, there exists a complex $\Gamma \supseteq \Delta$ such that $\Gamma$ and the relative complex $(\Gamma,\Delta)$ are both partitionable. This allows us to view the h-vector of $\Delta$ as the difference of h-vectors of partitionable complexes, and yields a combinatorial interpretation of the entries of the h-vector of $\Delta$.

By contrast, there is a large class of complexes $\Delta$ for which there is no complex $\Gamma \supseteq \Delta$ such that $\Gamma$ and $(\Gamma,\Delta)$ are both Cohen-Macaulay. We give a complete characterization of when such a Cohen-Macaulay $\Gamma$ exists, and give a straightforward construction of all such $\Gamma$.

Finally, we discuss the possibility of a similar construction for shellable complexes, and a connection to Simon’s conjecture on extendable shellability.

This talk will not assume much background knowledge of the combinatorics of simplicial complexes.

The seminar will take place in 3418 at KTH and will be streamed via Zoom (meeting ID: 657 2277 4049).

https://kth-se.zoom.us/j/65722774049?pwd=dFlsRmdjQW1EWi9VQWcrWDBZL2kzZz09

Please contact Petter R at petterre@kth.se for the password.