Noncrossing Partitions Research Articles

On all-loop integrands of scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.

New interpretations for noncrossing partitions of classical types

We interpret noncrossing partitions of type B and type D in terms of noncrossing partitions of type A. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type B, type C and type D which are different from those in the recent work of Fink and Giraldo. We also define Catalan tableaux of type B and type D, and find bijections between them and noncrossing partitions of type B and type D respectively.

Submaximal factorizations of a Coxeter element in complex reflection groups

When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. Lorsque $W$ est un groupe de réflexion fini, le treillis $NC(W)$ des partitions non-croisées de type $W$ est un objet combinatoire très riche, qui généralise la notion de partitions non-croisées d'un $n$-gone. Une formule (seulement prouvée au cas par cas à l'heure actuelle) exprime le nombre de chaînes de longueur donnée dans $NC(W)$ sous la forme d'un nombre de Fuß-Catalan généralisé, qui dépend des degrés invariants de $W$. Nous décrivons une stratégie visant à comprendre certaines spécifications de cette formule de manière uniforme, en utilisant une interprétation des chaînes de $NC(W)$ comme fibres d'un "revêtement de Lyashko-Looijenga''. Ce revêtement est construit à partir de la géométrie de l'hypersurface du discriminant de $W$. Nous en déduisons de nouvelles formules de comptage pour certaines factorisations d'un élément de Coxeter de $W$.

Double homotopy Cohen-Macaulayness for the poset of injective words and the classical NC-partition lattice

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homotopy Cohen-Macaulay posets, which can be considered as an extension of the classical poset fiber theorem for homotopy Cohen-Macaulay posets. Dans cet article, nous étudions certaines propriétés topologiques du poset des mots injectifs et du treillis des partitions non-croisées classiques. Plus précisément, nous montrons qu'après suppression des plus petit et plus grand élément (s'ils existent), ces posets sont doublement Cohen-Macaulay. C'est une extension du fait bien connu que ces deux posets sont épluchables ("shellable''). Ces deux résultats reposent sur un nouveau théorème poset-fibre pour les posets doublement homotopiquement Cohen-Macaulay, que l'on peut voir comme extension du théorème poset-fibre classique pour les posets homotopiquement Cohen-Macaulay.

Front Representation of Set Partitions

Let $\pi$ be a set partition of $[n]=\{1,2,\ldots,n\}$. The standard representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block which does not contain any integer between i and j. The front representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block whose smallest integer is i. Using the front representation, we find a recurrence relation for the number of $12\cdots k12$-avoiding partitions for $k\geq2$. Similarly, we find a recurrence relation for the number of k-distant noncrossing partitions for $k=2,3$. We also prove that the front representation has several joint symmetric distributions for crossings and nestings as the standard representation does.

Chain enumeration of k-divisible noncrossing partitions of classical types

We give combinatorial proofs of the formulas for the number of multichains in the k-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and Müller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of k-divisible noncrossing partitions of type A invariant under a 180° rotation in the cyclic representation.

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ ≀ S n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B .

Noncrossing partitions and the shard intersection order

We define a new lattice structure $(W,\preceq)$ on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC (W) as a sublattice. The new construction of NC (W) yields a new proof that NC (W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of $(W,\preceq)$ , like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC (W). There is a natural dimension-preserving bijection between simplices in the order complex of $(W,\preceq)$ (i.e. chains in $(W,\preceq)$ ) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC (W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The lattice $(W,\preceq)$ is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.

Sortable elements in infinite Coxeter groups

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Zeons, lattices of partitions, and free probability

Central to the theory of free probability is the notion of summing multiplicative functionals on the lattice of non-crossing partitions. In this paper, a graph-theoretic perspective of partitions is investigated in which independent sets in graphs correspond to non-crossing partitions. By associating particular graphs with elements of “zeon” algebras (commutative subalgebras of fermion algebras), multiplicative functions can be summed over segments of lattices of partitions by employing methods of “zeon-Berezin” operator calculus. In particular, properties of the algebra are used to “sieve out” the appropriate segments and sub-lattices. The work concludes with an application to joint moments of quantum random variables.

Crossings and Nestings in Set Partitions of Classical Types

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.

Bijections between noncrossing and nonnesting partitions for classical reflection groups

We present an elementary type-preserving bijection between noncrossing and nonnesting partitions for all classical reflection groups, which answers a question of Athanasiadis.

A note on 2-distant noncrossing partitions and weighted Motzkin paths

We prove a conjecture of Drake and Kim: the number of 2 -distant noncrossing partitions of { 1 , 2 , … , n } is equal to the sum of weights of Motzkin paths of length n , where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.

A noncrossing basis for noncommutative invariants of [formula omitted

Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL ( 2 , C ) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory of free stochastic measures this provides a combinatorial proof of the Molien–Weyl formula in this setting.

Explicit formulae for Kerov polynomials

We prove two formulae expressing the Kerov polynomial Σ k as a weighted sum over the set of noncrossing partitions of the set {1,?,k+1}. We also give a combinatorial description of a family of symmetric functions specializing in the coefficients of Σ k .

DUAL PRESENTATION AND LINEAR BASIS OF THE TEMPERLEY-LIEB ALGEBRAS

The braid group Bn maps homomorphically into the Temper- ley-Lieb algebra TLn. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group Bn form a basis for the vector space underlying the Temperley-Lieb algebra TLn. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno's theorem, using the interpretation of simple elements as non-crossing partitions.

From permutahedron to associahedron

AbstractFor each finite real reflection group W, we identify a copy of the type-W simplicial generalized associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalized associahedron and the elements of the type-W non-crossing partition lattice that is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for W.

Hopf algebras and the logarithm of the $S$-transform in free probability

Let $k$ be a positive integer and let $\mathcal {G}_k$ denote the set of all joint distributions of $k$-tuples $(a_1,\ldots ,a_k)$ in a noncommutative probability space $(\mathcal {A},\varphi )$ such that $\varphi (a_1)=\cdots =\varphi (a_k) = 1$. $\mathcal {G}_k$ is a group under the operation of the free multiplicative convolution $\boxtimes$. We identify $\bigl ( \mathcal {G}_k, \boxtimes \bigr )$ as the group of characters of a certain Hopf algebra $\mathcal {Y}^{(k)}$. Then, by using the log map from characters to infinitesimal characters of $\mathcal {Y}^{(k)}$, we introduce a transform $LS_{\mu }$ for distributions $\mu \in \mathcal {G}_k$. $LS_{\mu }$ is a power series in $k$ noncommuting indeterminates $z_1, \ldots , z_k$; its coefficients can be computed from the coefficients of the $R$-transform of $\mu$ by using summations over chains in the lattices $NC(n)$ of noncrossing partitions. The $LS$-transform has the “linearizing” property that \[ LS_{\mu \boxtimes \nu } =LS_{\mu } +LS_{\nu }, \ \ \forall \mu , \nu \in \mathcal {G}_k \mbox { such that } \mu \boxtimes \nu = \nu \boxtimes \mu . \] In the particular case $k=1$ one has that ${\mathcal Y}^{(1)}$ is naturally isomorphic to the Hopf algebra $\mbox {Sym}$ of symmetric functions and that the $LS$-transform is very closely related to the logarithm of the $S$-transform of Voiculescu by the formula \[ LS_{\mu } (z) = -z \log S_{\mu } (z), \ \ \forall \mu \in \mathcal {G}_1. \] In this case the group $(\mathcal G_1, \boxtimes )$ can be identified as the group of characters of $\mbox {Sym}$, in such a way that the $S$-transform, its reciprocal $1/S$ and its logarithm $\log S$ relate in a natural sense to the sequences of complete, elementary and, respectively, power sum symmetric functions.

Orbites d'Hurwitz des factorisations primitives d'un élément de Coxeter

We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton's formula for the number of chains of given length in the non-crossing partitions lattice ncp W . Our proof uses the properties of the Lyashko–Looijenga covering and the geometry of the discriminant of W.

Crossings and nestings in set partitions of classical types

In this extended abstract, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely. Dans ce résumé, nous étudions des bijections entre diverses classes de partitions d'ensemble de types classiques qui préservent les "openers'' et les "closers''. D'une part, nous présentons des bijections pour les types $B$ et $C$ qui échangent croisées et emboôtées, qui généralisent une construction de Kasraoui et Zeng pour le type $A$. D'autre part, nous généralisons une bijection pour le type $B$ et $C$ qui échange la cardinalité d'un croisement maximal avec la cardinalité d'un emboîtement maximal comme il a été fait par Chen, Deng, Du, Stanley et Yan pour le type $A$. Pour le type $D$, nous avons seulement construit une bijection entre les partitions non croisées et non emboîtées. Pour tout les types classiques, nous montrons que l'ensemble des "openers'' et l'ensemble des "closers'' déterminent une partition non croisées ou non emboîtées essentiellement de façon unique.