We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our bijections come from triangular base changes between the diagram basis and the various bases indexed by noncrossing partitions, by explicitly describing the orders giving triangularity. These orders were introduced in a joint paper with Williams and provide exotic lattice structures on noncrossing partitions. Several combinatorial objects are introduced along the way, including an involution on the set of noncrossing partitions.
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra
LISN (UMR CNRS 9015), Université Paris Saclay, INRIA, CNRS（巴黎萨克雷大学）；Universitat de Barcelona & Centre de Recerca Matemàtica（巴塞罗那大学）；LIGM, Université Gustave Eiffel（巴黎东马恩拉瓦莱大学）
An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular in the finite case. They index naturally a basis of the generalized Temperley--Lieb algebra. In this work we deal with any finite or affine Coxeter group $W$, and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrres…
We solve two open problems in Coxeter-Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type-uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter-Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.
We study two families of polynomials that play the same role, in the generalized Temperley Lieb algebra of a Coxeter group, as the Kazhdan Lusztig and R polynomials in the Hecke algebra of the group. Our results include recursions, closed formulas, and other combinatorial properties for these polynomials. We focus mainly on non branching Coxeter graphs.
We describe an inner product on the diagrams on which the Temperley-Lieb algebra can be represented. We exhibit several constructions which are in natural combinatorial bijection with these diagrams, which are generalizations of various constructions counted by the Catalan numbers. We use a method similar to the existing ones for orthogonalizing the Temperley-Lieb algebra to construct an orthogonal basis for the vector space over these diagrams.
We investigate nil-Temperley-Lieb algebras of type A. We give a general description of the structure of monomials formed by the generators. We also show that the dimensions of these algebras are the famous Catalan numbers by providing a bijection between the monomials and Dyck paths. We show that the distribution of these monomials by degree is the same as the distribution of Dyck paths by the sum of the heights of the peaks minus the number of peaks.
In a Coxeter group $W$, an element is fully commutative if any two of its reduced expressions can be linked by a series of commutation of adjacent letters. These elements have particularly nice combinatorial properties, and also index a basis of the generalized Temperley--Lieb algebra attached to $W$. We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that it always satisfies a linear recurrence with constant coefficients, by showing that reduced expressions of fully commutative elements form a regular language. Then we classify …
In this thesis, I present an associative diagram algebra that is a faithful representation of a particular Temperley--Lieb algebra of type affine $C$, which has a basis indexed by the fully commutative elements of the Coxeter group of the same type. The Coxeter group of type affine $C$ contains an infinite number of fully commutative elements, and so the corresponding Temperley--Lieb algebra is of infinite rank. With the exception of type affine $A$, all other generalized Temperley--Lieb algebras with known diagrammatic representations are of finite rank. In the finite rank case, counting argu…
We consider two families of polynomials that play the same role in the Temperley Lieb algebra of a Coxeter group as the Kazhdan Lusztig and R polynomials play in the Hecke algebra of the group. We study these polynomials from a combinatorial point of view. More precisely we obtain recursions, non recursive formulas, symmetry properties, and expressions for the constant terms, of these polynomials.
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we…
In [9], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan and Shi arrangements in [11]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type $A_2$.
An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine t…
The star operation, originally introduced by Kazhdan and Lusztig, was later specialized by Ernst to the so-called weak star reduction on the set of fully commutative elements of a Coxeter group. In this paper, we classify the star and weak star irreducible fully commutative elements in Coxeter groups of affine types $\widetilde{B}_{n+1}$ and $\widetilde{D}_{n+2}$. Focusing then on the case of type $\widetilde{D}_{n+2}$, we use the classification of star irreducible elements to provide a new proof of the faithfulness of a diagrammatic representation of the corresponding generalized Temperley-Li…
We prove that the restriction of Bruhat order to noncrossing partitions in type $A_n$ for the Coxeter element $c=s_1s_2 ...s_n$ forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the change-of-basis from the graphical basis of the Temperley-Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we give new bijections---fixing the set of reflections---between noncrossing partitions associated to distinct Coxet…
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subsequences of the entries of the alternating Jacobsthal triangle and show that they arise in a variety of combinatorial constructions. The generating functions of these sequences enable us to define their…
We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan--Lusztig bases of these groups have a nice projection property to the Temperley--Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\Bbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Co…
This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to relate…
An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n)- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the …
Let $(W,S)$ be a Coxeter system and suppose that $w \in W$ is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with $s \in S$. If there exists $t\in S$ such that $s$ and $t$ do not commute and $tw$ (respectively, $wt$) is no longer fully commutative, we say that $w$ is left (respectively, right) weak star reducible by $s$ with respect to $t$. In this paper, we classify the fully commutative elements in Coxeter groups of types $B$ and affine $C$ that are irreducible under weak star reductions. In a sequel to this paper, the classificat…
We continue the study of the nonkissing complex that was introduced by Petersen, Pylyavskyy, and Speyer and was studied lattice-theoretically by the second author. We introduce a theory of Grid-Catalan combinatorics, given the initial data of a nonkissing complex, and show how this theory parallels the well-known Coxeter-Catalan combinatorics. In particular, we present analogues of Chapoton's F-triangle, H-triangle, and M-triangle and give combinatorial, lattice-theoretic, and geometric interpretations of the objects defining these. In our Grid-Catalan setting, we prove that Chapoton's F-trian…
When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a 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 NCP_W as a generalised 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 NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the …
We define a tower of injections of $\tilde{C}$-type Coxeter groups $W(\tilde C_{n})$ for $n\geq 1$. We define a tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. Let $W^c(\tilde C_{n})$ be the set of fully commutative elements in $W(\tilde C_{n})$, we classify the elements of $W^c(\tilde C_{n})$ and give a normal form for them. We use this normal form to define two injections from $W^c(\tilde C_{n-1})$ into $W^c(\tilde C_{n})$. We then define the tower of affine Temperley-Lieb algebras of type $\tilde{C }$ and use th…
Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal. Motivated by classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture [Crelle 1998], and continuing beyond work of the sec…
We introduce an associative algebra $\M_k(x)$ whose dimension is the $2k$-th Motzkin number. The algebra $\M_k(x)$ has a basis of "Motzkin diagrams," which are analogous to Brauer and Temperley-Lieb diagrams, and it contains the Temperley-Lieb algebra $\TL_k(x)$ as a subalgebra. We prove that for a particular value of $x$, the algebra $\M_k(x)$ is the centralizer algebra of $\uqsl$ acting on the $k$-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible $\uqsl$-modules. We show that $\M_k(x)$ is generated by special diagrams $\ell_i, t_i, r_i \ (1 \le i < k)$ and $p_j …
Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking functions}, and we call the permutation representation of $W$ on the set of $W$-parking functions the (standard) $W$-{\sf parking space}. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new $W$-parking spaces, called the {\sf noncrossing parking space} and the {\sf algebraic parking space}, with the following features: 1) They are defined more g…
We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical types different from $D_n$ the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type $D_n$).This property implies positivity properties of the polynom…
