Browsing by Subject "Combinatorics"

Now showing 1 - 4 of 4
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    Applications of geometric techniques in Coxeter-Catalan combinatorics
    (2017-09) Douvropoulos, Theodosios
    In 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.
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    Cataland
    (2013-08) Williams, Nathan Ferd
    We study two combinatorial miracles relating purely poset-theoretic objects with purely Coxeter-theoretic objects. The first miracle is that there are the same number of linear extensions of the root poset as reduced words of the longest element (in certain types), while the second is that there are the same number of order ideals in the root poset as certain group elements (in more types). We place these miracles on remarkably similar footing in Theorem 3.1.9 and Conjecture 4.4.9. Strangely, the first miracle is less general than the second. We motivate it in Chapter 3 by showing that behind each of its instances lurks a trivial result. This philosophy gives us a unified framework that collects several known theorems while allowing us to obtain new results. We set up the language for the second miracle in Chapter 4, in which we review the many equivalent ways to define W-Catalan objects. Of note, we show how to interpret both the Kreweras complement and Cambrian rotation as certain walks on N. Reading's Cambrian lattice, which may be conjecturally mirrored on the nonnesting partitions. In Chapter 5, we test the limits of the second miracle by turning inward to parabolic quotients. In doing so, we introduce a new generalization of W-Catalan combinatorics.
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    Cones of Hyperplane Arrangements
    (2021-07) Dorpalen-Barry, Galen
    Hyperplane arrangements dissect $\R^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers calledWhitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This thesis concerns cones of hyperplane arrangement in two ways. First we consider cones within the braid arrangement, consisting of the reflecting hyperplanes $x_i=x_j$ inside $\R^n$ for the symmetric group, thought of as the type $A_{n-1}$ reflection group. Here,\begin{itemize} \item cones correspond to posets, \item chambers within the cone correspond to linear extensions of the poset, \item the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. \end{itemize} We interpret this refinement explicitly for two families of posets: width two posets, and disjoint unions of chains. In the latter case, this gives a geometric re-interpretation to Foata's theory of cycle decomposition for multiset permutations, and leads to a simple generating function compiling these Whitney numbers. Secondly, we give an interpretation of the coefficients the Poincaré polynomial of a cone of an arbitrary arrangement via the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel'fand gave a simple presentation for this ring for an arbitrary arrangement, along with a filtration and associated graded ring whose Hilbert series is the Poincaré polynomial.We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul.
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    Parametrizations of Irreducible Rational Representations of Coxeter Groups
    (2024-04) Corsi, Craig
    We develop two closely related methods for parametrizing the rational irreducible characters of an arbitrary finite Coxeter group W. The goal is to provide a uniform approach to such a parametrization, independent of Coxeter type. The two methods generalize two approaches to describing the irreducible representations of the symmetric groups, which coincide in type A but do not coincide generally. Our methods associate characters to pairs of reflection subgroups, in one case by considering common constituents of permutation and signed permutation modules, and in the other case by a generalization of the Specht modules. We ask whether, using either method, we can identify a set of subgroup pairs for which the matrix of multiplicities of rational irreducibles in the characters associated to these subgroup pairs is unitriangular. Such a unitriangular matrix provides a parametrization of the irreducible rational characters. For the noncrystallographic types H and I, we give a positive answer to this question. In type H, we show computationally that we can parametrize the irreducible rational characters of H4 using generalized Specht modules, and we can parametrize the irreducible characters of H3 using both methods. Moreover, we give an explicit decomposition of the generalized common constituents for the dihedral groups I2(n) for all n, and we prove that we can always exhibit a unitriangular multiplicity matrix using generalized common constituents. In type A our theory coincides with the classical theory of Specht modules and does not give any new information. In type B the approach we take is closely related to an existing parametrization of the irreducible characters, but it appears to have some novel elements.