Browsing by Subject "Catalan"
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Item Applications of geometric techniques in Coxeter-Catalan combinatorics(2017-09) Douvropoulos, TheodosiosIn the seminal work [Bes 15], Bessis gave a geometric interpretation of the noncrossing lattice NC(W) associated to a well-generated complex reflection group W. He used it as a combinatorial recipe to construct the universal covering space of the arrangement complement V⧵ ⋃ H, and to show that it is contractible, hence proving the K(π,1) conjecture. Bessis' work however relies on a few properties of NC(W) that are only known via case by case verification. In particular, it depends on the numerological coincidence between the number of chains in NC(W) and the degree of a finite morphism, the LL map. We propose a (partially conjectural) approach that refines Bessis' work and transforms the numerological coincidence into a corollary. Furthermore, we consider a variant of the LL map and apply it to the study of finer enumerative properties of NC(W). In particular, we extend work of Bessis and Ripoll and enumerate the so-called ``primitive factorizations" of the Coxeter element c. That is, length additive factorizations of the form c=w⋅ t1⋯ tk, where w belongs to a prescribed conjugacy class and the ti's are reflections.Item Cyclic Actions in Combinatorial Invariant Theory(2021-07) Stucky, EricThe major original contributions of this thesis are as follows: Theorem 3.3.1 and Proposition 3.3.3 together show that a natural q-analogue of the rational Schr\"oder polynomial is (separately) unimodal in both its even and odd coefficient sequences. Theorem 4.1.2 which, for certain parameters, defines an elementary (WxC)-action on the classical parking space for a Weyl group. When this action is defined, it agrees with the more technical algebraic construction of Armstrong, Reiner, and Rhoades. Theorem 5.1.3 is a general cyclic sieving result which in particular recovers the q=-1 phenomenon for Catalan necklaces, as well as higher-order sieving for a more general family of necklaces.