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Sadek Al Harbat

Publications and source records attributed to Sadek Al Harbat.

10 recordsLinked to original sources
arXiv2025

Canonical reduced expression in affine Coxeter groups of type $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{D}_n$

We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by a simple reflection, a study of the right descent set, and a proof that the affine length is preserved along the tower of affine Coxeter groups of type $\tilde A$, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower regardless of the ground ring. We give a similar canonical reduced expression for the elements of $W(\tilde{B}_n)$ and $W(\tilde{D}_n)$.

math.RT↗
arXiv2024

Catalan numbers: from FC elements to classical diagram algebras

Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$ and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of $W^c(A_n)$. We determine the unique bijection between $W^c(A_n)$ and the set of non-crossing diagrams of $n+1$ strings that respects the diagrammatic multiplication by concatenation in the $A_n$-type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse.

math.CO↗
arXiv2022

Canonical reduced expression for elements of affine Coxeter groups Part I -- Type $\tilde{A}_n$

We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by a simple reflection, a study of the right descent set, and a proof that the affine length is preserved along the tower of affine Coxeter groups of type $\tilde A$, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower.

math.RT↗
arXiv2019

Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatorics

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.

math.CO↗
arXiv2016

On the fully commutative elements of type $\tilde C$ and faithfulness of related towers

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 the injections above to prove the faithfulness of this tower.

math.GR↗
arXiv2016

Tower of fully commutative elements of type $\tilde A$ and applications

Let $W^c(\tilde A_{n})$ be the set of fully commutative elements in the affine Coxeter group $W(\tilde A_{n})$ of type $\tilde{A}$. We classify the elements of $W^c(\tilde A_{n})$ and give a normal form for its elements. We give a first application of this normal form to fully commutative affine braids. We then use this normal form to define two injections from $W^c(\tilde A_{n-1})$ into $W^c(\tilde A_{n})$ and examine their properties. We then consider the tower of affine Temperley-Lieb algebras of type $\tilde{A }$ and use the injections above to prove the injectivity of this tower.

math.GR↗
arXiv2015

A note on affine links

We view the $\tilde{A}$-type affine braid group as a subgroup of the $B$-type braid group. We show that the $\tilde{A}$-type affine braid group surjects onto the $A$-type braid group and we detect the kernel of this surjection using Schreier's Theorem. We then describe an injection of the $B$-type braid group into the $A$-type braid group which allows us finally to give a definition of affine links, as closures of affine braids viewed as A-type braids after composing the above injections, and we prove that the two conditions of Markov are necessary and sufficient to get the same affine closure of any two affine braids.

math.GR↗
arXiv2015

Markov elements in affine Temperley-Lieb algebras

We define a tower of affine Temperley-Lieb algebras of type $\tilde{A_{n}}$ and we define Markov elements in those algebras. We prove that any trace over an affine Temperley-Lieb algebras of type $\tilde{A_{2}}$ is uniquely defined by its values on the Markov elements.

math.GR↗
arXiv2013

A classification of affine fully commutative elements

We classify fully commutative elements in the affine Coxeter group of type $\tilde{A_{n}}$. We give a normal form for such elements, then we propose an application of this normal form: we lift these fully commutative elements to the affine braid group of type $\tilde{A_{n}}$ and we get a form for "fully commutative braids".

math.GR↗