In their preface, the editors describe algebraic combinatorics as the area of combinatorics concerned with exact, as opposed to approximate, results and which puts emphasis on interaction with other areas of mathematics, such as algebra, topology, geometry, and physics. It is a vibrant area, which saw several major developments in recent years. The goal of the 2022 conference Open Problems in Algebraic Combinatorics 2022 was to provide a forum for exchanging promising new directions and ideas. The current volume includes contributions coming from the talks at the conference, as well as a few other contributions written specifically for this volume.
The articles cover the majority of topics in algebraic combinatorics with the aim of presenting recent important research results and also important open problems and conjectures encountered in this research. The editors hope that this book will facilitate the exchange of ideas in algebraic combinatorics.
Articles
Francois Bergeron, $(\operatorname {GL}_k \times \ \mathbb {S}_n)$-modules of
multivariate diagonal harmonics
Francesco Brenti, Some open problems on Coxeter groups and unimodality
R. Kenyon, Some combinatorial problems arising in the dimer model
Per Alexandersson and Valentin Feray, A positivity conjecture on the
structure constants of shifted Jack functions
Greta Panova, Complexity and asymptotics of structure constants
Jonathan Novak and Brendon Rhoades, Increasing subsequences and Kronecker
coefficients
Nathan Williams, Combinatorics and braid varieties
Isabella Novik and Hailun Zheng, Neighborly spheres and transversal numbers
Hugh Thomas, Realizing simplicial complexes as the boundary of the totally
non-negative part of a variety
Sam Hopkins, Order polynomial product formulas and poset dynamics
Thomas Lam, An invitation to positive geometries
Satoshi Murai, An upper bound problem for triangulations of manifolds
Igor Pak, What is a combinatorial interpretation?
James Propp, Trimer covers in the triangular grid: Twenty mostly open
problems
Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling and Mike
Zabrocki, The mystery of plethysm coefficients
Tri Lai, Problems in the enumeration of tilings
Zhao Gao, Claudiu Raicu and Keller VandeBogert, Some questions arising from
the study of cohomology on flag varieties
Alexander Yong, Castelnuovo-Mumford regularity and Schubert geometry
Ralf Schiffler, Perfect matching problems in cluster algebras and number
theory
Christine Berkesch, University of Minnesota, Twin Cities, Minneapolis, MN, Benjamin Brubaker, University of Minnesota, Twin Cities, Minneapolis, MN, Gregg Musiker, University of Minnesota, Twin Cities, Minneapolis, MN, Pavlo Pylyavskyy, University of Minnesota, Twin Cities, Minneapolis, MN, and Victor Reiner, University of Minnesota, Twin Cities, Minneapolis, MN.
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