On chain polynomials of geometric lattices

Athanasiadis and Kalampogia-Evangelinou recently conjectured that the chain polynomial of any geometric lattice has only real zeros. We verify this conjecture for families of geometric lattices including perfect matroid designs, Dowling lattices, and for a class of geometric lattices that contains all lattices of flats of paving matroids. We also investigate how the conjecture behaves with respect to certain operations such as direct products, ordinal sums and single-element extensions.

💡 Research Summary

This paper addresses a conjecture by Athanasiadis and Kalampogia-Evangelinou which posits that the chain polynomial of any geometric lattice is real-rooted, meaning all its zeros are real numbers. The authors verify this conjecture for several significant families of geometric lattices and investigate its behavior under standard lattice operations.

The paper begins by establishing the necessary background on real-rooted polynomials, interlacing sequences, and introduces the key concept of a TN-poset. A TN-poset is a quasi-rank uniform poset whose associated matrix of rank-counts is totally nonnegative. A fundamental result from prior work is that posets constructed from TN-posets, or those that are TN-posets themselves, have real-rooted chain polynomials.

The first major result concerns triangular semimodular lattices. A poset is triangular if the number of maximal chains in any interval depends only on the ranks of the interval’s endpoints. The authors prove that any triangular semimodular lattice is a TN-poset (Theorem 3.7). Consequently, its chain polynomial is real-rooted with all zeros in