ALTOP - ALgebraic and TOPological combinatorics

Funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022

In the last two decades, both algebraic and topological combinatorics witnessed exciting breakthroughs, coming from natural open problems about polynomials that arise in the field. We give three prominent examples:

• the Rota-Heron-Welsh conjecture (1970) of log-concavity of the coefficients of the characteristic polynomials of matroids, solved in 2018
• the Kazhdan-Lusztig conjecture (1979) of positivity of the polynomials named after them, solved in 2014
• the Macdonald conjecture (1988) of positivity of the q,t-Kostka coefficients,  solved in 2001.

Each of these breakthroughs has been both the culmination of an intense research activity, and the catalyst of exciting new developments in algebraic and topological combinatorics. The aim of this project is to pursue new results in these three lines of research, promoting a fruitful cooperation among our research units and the international community. Hence, the project is divided into three parts.

The first part regards the combinatorics and topology arising from hyperplane arrangements and subspace arrangements, like matroids and polymatroids. We plan to study how far the methods in the pioneering work [AHK] can be extended to the polymatroid case, following a line started in [PP]. In particular, can we prove log-concavity statements for certain classes of polymatroids? Can we define Kazhdan-Lusztig (KL) polynomials of polymatroids, and give them both a combinatorial and a geometric description?

The second part concerns the combinatorics and topology related to KL polynomials of Coxeter groups and Kazhdan-Lusztig-Vogan (KLV) polynomials, like special matchings [BCM] and local intersection cohomology of Schubert varieties [KL]. The goals will be to make progress on the combinatorial invariance conjecture: can we prove combinatorially the new breakthrough formula in [D+]? What about an analogous statement for the KLV polynomials, which are to symmetric varieties what the KL polynomials are to Schubert varieties? Can we relate KLV polynomials to singularities of symmetric quiver representation varieties?

The third part involves the combinatorics and topology related to the symmetric functions, like labelled Dyck paths (occurring in the famous shuffle conjecture [HHLRU], proved in [CM1], and in the Delta conjectures [HRL], of which one has been proved in [DM2]), affine Springer fibers (occurring already in [CO] and [CM2] in connection to diagonal coinvariants), and graph colorings (occurring in the Stanley’s symmetric function [St2] and their categorification [SY]). Can we make progress on the valley Delta conjecture and on a related conjecture from [HS] on the Hilbert series of super diagonal coinvariants? Can we find a categorification for the Tutte symmetric function introduced in [St3]?