Sadek Salem AL HARBAT

                                                      Liste de publications



                                                      dans des revues à comité de lecture


  1. S. AL HARBAT, Markov trace on a tower of affine Temperley-Lieb algebras of type $\tilde{A}$, arXiv:1311.7092v2,
    J. Knot Theory Ramifications 24 (2015) 1--28.

    We define a tower of affine Temperley-Lieb algebras of type $\tilde{A}$. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov-Ocneanu-Jones trace on the tower of Temperley-Lieb algebras of type $A$. We define an invariant of special kind of links as an application of this trace.

  2. S. AL HARBAT, Tower of fully commutative elements of type $\tilde A$ and applications, arXiv:1507.01521,
    Journal of Algebra 465 (2016) 111---136.

    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 them. 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 finally consider the tower of affine Temperley-Lieb algebras of type $\tilde{A }$ and use the injections above to prove the injectivity of this tower.

  3. S. AL HARBAT, On the fully commutative elements of type $\tilde C$ and the faithfulness of related towers, arXiv:1509.01033,
    Journal of Algebraic Combinatorics 45(3) (2017) 803---824.

    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 faithfullness of this tower.

  4. S. AL HARBAT, On fully commutative elements of type $\tilde B$ and $\tilde D$, arXiv:1803.04945,
    Journal of Algebra 530 (2019) 1---33.

    We define a tower of injections of $\tilde{B}$-type (resp. $\tilde{D}$-type) Coxeter groups $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$). Let $W^c(\tilde B_{n})$ (resp. $W^c(\tilde D_{n})$) be the set of fully commutative elements in $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$), we classify the elements of this set by giving a normal form for them. We define a $\tilde{B}$-type tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. We use this normal form to define two injections from $W^c(\tilde B_{n-1})$ into $W^c(\tilde B_{n})$. We then define the tower of affine Temperley-Lieb algebras of type $\tilde{B }$ and use the injections above to prove the faithfulness of this tower. We follow the same track for $\tilde{D}$-type objects.

  5. S. AL HARBAT, C. GONZALEZ, D. PLAZA, Type $\tilde C$ Temperley-Lieb algebra quotients and Catalan combinatorics, arXiv:1904.08351,
    Journal of Combinatorial Theory, Series A 180 (2021) 1--40.

    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.


                                                      soumises ou en cours de révision


  1. S. AL HARBAT, C. BLONDEL, Poincaré polynomial for fully commutative elements in the symmetric group, 2020, arXiv:2010.03417, 1--21.

    Let $W^c(A_n)$ be the set of fully commutative elements of the Coxeter group $W(A_n)$. Let $a_n(q)= \sum_{w \in W^c(A_n)} q^{l(w)}$. We compute $a_n(q)$.

  2. S. AL HARBAT, Canonical reduced expression for elements of affine Coxeter groups Part I - Type Ã_n, 2021, arXiv:2105.07417, 1--27.

    We classify the elements of W(Ã_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 Ã, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower.

  3. S. AL HARBAT, Catalan numbers: from FC elements to classical diagram algebras, 2023, arXiv:2308.10100, 1--27.

    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 an algorithm that allows to express directly an element of the monomial basis as a non-crossing diagram and an algorithm which determines the fully commutative element corresponding to a given non-crossing diagram.

  4. S. AL HARBAT, Canonical reduced expression in affine Coxeter groups of type Ã_n, ˜B_n, ˜D_n, 2025, arXiv:2511.01702, 1--36.

    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 $W(\tilde{D}_n)$.

                                                      autres textes


  1. S. AL HARBAT, Groupe de tresses affine, algèbre de Temperley-Lieb affine et trace de Markov. Thèse de doctorat, Université Paris-Diderot-Paris 7, Décembre 2013.

    2. Le résumé est ici.

  2. S. AL HARBAT, Markov elements in affine Temperley-Lieb algebras, arXiv:1501.06756, 2015.

    We define a tower of affine Temperley-Lieb algebras of type $\tilde{A}$ 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. This is the case $n=2$ of Theorem 4.6 in [1] (2015), that contains the proof for $n\ge 3$ while refering to this manuscript for $n=2$.

  3. S. AL HARBAT, A note on affine links, arXiv:1502.00273, 2015.

    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.