Coxeter-Catalan Combinatorics

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Catalan combinatorics is a highly active area of research within algebraic and geometric combinatorics, with multiple connections to other fields like invariant theory, algebraic geometry, and representation theory. A key ingredient is to uniformly describe phenomena which can be found in the various fields, and thereby exhibiting relations between a priori unrelated concepts. Those fields of research have in common that they can be classified according to Cartan-Killing types, and for which natural objects counted by generalizations of the famous Catalan numbers are of fundamental interest. In this project, connections between the following avenues of research are studied: (1) root systems, (2) reflection groups, (3) cluster algebras, (4) subword complexes. The objective of the project is to further establish their combinatorial properties and connections. The tools are algebraic techniques from commutative and noncommutative algebra, geometric techniques from root systems, hyperplane arrangements and simplicial complexes, representation theoretical techniques from character theory, and combinatorial techniques from symmetric function theory, and from enumerative and bijective combinatorics.