Algebraic Combinatorics of the Symmetric Groups and Coxeter Groups
5-9 July 2021

How to get there


This workshop aims at developing further the collaborations and existing scientific partnerships between France and Italy, especially but not exclusively the Universities of Aix-Marseille, l'Aquila, Côte d'Azur, Sapienza Università di Roma and Università Tor Vergata. Université du Quéec à Montréal was associated originally in the project but its members will be present only through distant talk sessions due to the pandemic. Its scientific aim is to expose the state of the art of the Algebraic combinatorics of the symmetric groups and their generalizations to Coxeter groups. The topics covered include: symmetric groups and other Coxeter groups combinatorics, geometric approaches and representation theory, invariant theory, structure species and combinatorial Hopf algebras, symmetric functions and their generalizations.
Organizers E-mail
Christophe Hohlweg
Claudia Malvenuto
Frédéric Patras


  • Clemens Berger (Université Côte d'Azur)
  • Francois Bergeron (UQAM online)
  • Francesco Brenti (Università di Roma 2 Tor Vergata)
  • Patrick Cassam-Chenaï (Université Côte d'Azur)
  • Adrian Celestino (NTNU)
  • Alessandro D'Andrea (Sapienza Università di Roma)
  • Sabino Di Trani (Sapienza Università di Roma)
  • Justine Falque (Université de Marne-La-Vallée)
  • Christophe Hohlweg (UQAM online)
  • Evelyne Hubert (INRIA Méditerranée, Nice)
  • Alessandro Iraci (UQAM)
  • Martina Lanini (Università di Roma 2 Tor Vergata)
  • Claudia Malvenuto (Sapienza Università di Roma)
  • Dominique Manchon (Université Clermont-Ferrand)
  • Tobias Metzlaff (INRIA Sophia-Antipolis)
  • Jean-Christophe Novelli (Université de Marne-La-Vallée)
  • Paolo Papi (Sapienza Università di Roma)
  • Frédéric Patras (Université Côte d'Azur)
  • Valentina Pepe (Sapienza Università di Roma)
  • Claudio Procesi (Sapienza Università di Roma)
  • Christophe Reutenauer (UQAM online)
  • Luigi Santocanale (Université d'Aix-Marseille)
  • Viola Siconolfi (Bielefeld University)

  • Location

    The conference will take place in Grand Hotel San Michele, Cetraro (CS, Italy). The accomodation will be a full board treatment.
    Please let us know (email to if you are accompanied or not and which type of room you prefer:
    * Double room (double occupancy)
    * Double room (simple occupancy)

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    How to get there

    From Rome the simplest way to reach Cetraro is by train: check the schedule on The hotel has a pick-up free service from Cetraro train station.

    From abroad, the two nearest airports are Naples and Lamezia Terme Airport. Cetraro can then be reached by public transportation: trains from Naples and Lamezia centrale (a bus connects Lamezia airport to Lamezia centrale). Schedules and connections can be found on As due to covid there might be fewer flights than usually, an alternative solution is to fly to Rome and take a train.

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    Please find here the SCHEDULE AND ABSTRACTS of the Workshop.

    The conference will start on Monday morning, July 5 and end on Friday 9 at 12. Participants are expected to arrive on Sunday July 4 and to leave on Saturday July 10.

    The talks will take place in the Hotel main Conference room.

    All participants are invited to submit talks proposals. The selection will made shortly before the conference, "Oberwolfach style". As the goal of the conference is to enhance existing interactions and create new ones, time will be left for group discussions and group work.

    Talks (so far)

    Clemens Berger
    On a certain filtration of the universal bundle of a finite Coxeter group
    The universal bundles of the symmetric groups assemble into an E-infinity operad which plays an important role in the theory of infinite loop spaces. This E-infinity operad admits a filtration by E_n-suboperads thereby producing combinatorial models for point configurations in n-dimensional Euclidean space. We shall define an analogous filtration for the universal bundle of any finite Coxeter group and study the homotopy types of the resulting filtration stages.

    François Bergeron (online)
    Irrational Catalan Combinatorics
    Following the recent work of Blasiak-Haiman-Morse-Pun-Seelinger: A Shuffle Theorem for Paths under any Line, we extend Catalan Combinatorics (Dyck paths, parking function, Macdonald eigenoperators, elliptic Hall algebra, etc.) to the context of any "triangular" partitions. These are partitions whose Ferrers diagram is the set of cells that sit below a line joining pints (0,s) and (r,0) for any positive real numbers s and r; with the classical case corresponding to r=s=n an integer. We first prove general enumerative results, and then discuss properties of associated (GL_k x S_n) character formulas (if time allows), as well as ties with many other subjects.

    Francesco Brenti
    Graphs, stable permutations, and Cuntz algebra automorphisms
    Stable permutations are a class of permutations that arises in the study of the automorphism group of the Cuntz algebra. In this talk I will present a characterization of stable permutations in terms of certain associated graphs. As a consequence of this characterization we prove a conjecture in [Advances in Math., 381(2021) 107590], namely that almost all permutations are not stable, and we characterize explicitly stable 4 and 5-cycles.
    (Joint work with Roberto Conti and Gleb Nenashev)

    Patrick Cassam-Chenaï
    Invariants and covariants in Quantum Chemistry: applications and open problems
    We will review the use of invariants and covariants in Quantum Chemistry from the introduction of these algebraic techniques by A. Schmelzer, J. Murrell, [Int. J. Quantum Chem. 28, 287- 195 (1985)] to their recent combination with artificial intelligence tools [K. Shao, J. Chen, Z. Zhao, and D. H. Zhang, J. Chem. Phys. 145, 071101 (2016)]. A special emphasis will be put on open problems arising from infinite symmetry groups such as SO(3).

    Adrian De Jesus Celestino Rodriguez
    Pre-Lie Magnus expansion and relations between cumulants.
    In non-commutative probability, a notion of independence can be understood as a rule to compute mixed moments. For each notion of independence, the respective cumulants can be defined. Relations between different brands of cumulants have been studied recently. In this talk, we will focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants - those formulas were missing in the literature. The approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion.

    Sabino Di Trani
    Small Representations in the Exterior Algebra and Generalized Exponents
    Let g be a simple Lie algebra over C, and consider the exterior algebra \Lambda g as g-representations. In 1997 Mark Reeder conjectured that it is possible to determine the graded multiplicities in \Lambda g of certain irreducible representations reducing the computations to a combinatorial problem involving suitable Weyl group representations. We will give an idea of the strategy we used to prove the conjecture in the classical cases. Moreover, we will expose how our formulae can be rearranged involving the Generalized Exponents, obtaining a generalization of some classical results for graded multiplicities in the exterior algebra for adjoint and little adjoint representations.

    Christophe Hohlweg
    Shi arrangements and Garside shadows in Coxeter groups
    Given a Coxeter group W, the Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangement; this arrangement was introduced by Shi to study the Kazhdan-Lusztig cells in the case of an affine Weyl group. Shi showed that each region of a Shi arrangement contains exactly one element of minimal length. Garside shadows were introduced in relation to the word problem in Artin-Tits (braid) group (joint work with Dehornoy and Dyer). In this talk I will discuss the following conjecture: the set of minimal length elements of the regions in a Shi arrangement is a Garside Shadow. In particular, I will outline a proof in the case of affine Weyl groups
    (Joint work with Nathan Chapelier)

    Evelyne Hubert
    Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials
    Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.
    (Joint work with M. Singer)

    Alessandro Iraci
    Delta Conjectures and Theta Operators
    Macdonald polynomials are a family of symmetric functions whose interesting combinatorial properties generated many results in algebraic combinatorics, such as the famous shuffle theorem, which gives a combinatorial interpretation for the bigraded Frobenius characteristic of the module of diagonal harmonics. In this talk, we will discuss the state of the art about the multiple generalisations of the shuffle theorem, give a sketch of its proof (focusing on the combinatorics), and state the currently open problems and some new conjectures related to the Delta and Theta operators.

    Martina Lanini
    To Be Announced

    Claudia Malvenuto
    Primitive elements in the Poirier-Reutenauer Hopf Algebra of tableaux
    In 1995 Poirier and Reutenauer introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by Malvenuto and Reutenauer in 1995. In 2006 Aguiar and Sottile introduced a new basis of this algebra, by Moebius inversion in the poset of weak Bruhat order, to describe the primitive elements of the Hopf algebra of permutations. Using this method, we determine the primitive elements of the Poirier-Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.
    (Joint work with Christophe Reutenauer)

    Dominique Manchon
    Families of algebraic structures
    Algebraic structures may come into families, where each operation at hand is replaced by a family of operations indexed by some parameter set, which often bears a semigroup structure. I will introduce Rota-Baxter families, then address other family structures (dendriform, duplicial, pre-Lie,...), and finally give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters.
    (Based on recent joint works with Loïic Foissy, Xing Gao and Yuanyuan Zhang)

    Jean-Christophe Novelli
    Lagrange inversion, noncrossing partitions and a quasi-symmetric analogue of the Farahat-Higman algebra
    We study the structure constants of the noncommutative analog of the Lagrange basis of symmetric functions. This gives rise to analogs of classical constructions and surprising combinatorial results connected to noncrossing partitions.

    Paolo Papi
    The role of affine Weyl groups in some problems of combinatorics and representation theory
    I will discuss some instances of the emergence of affine Weyl groups as a basic tool to solve problems in representation theory and combinatorics. I will consider on some detail the study of B-orbits on abelian ideals (joint works with Gandini, Maffei and Mosender Frajria) and an analog of Panyushev`s rootlet theory for infinitesimal symmetric spaces.
    (Developed in the very recent PhD thesis of my student Federico Stara).

    Frédéric Patras
    Wick polynomials in non-commutative probability
    Wick polynomials and Wick products have classically a rich combinatorics closely related to the one of set partitions. They are studied here in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular combinatorial Hopf algebra.
    (Joint work with Kurusch Ebrahimi-Fard, Nikolas Tapia, Luca Zambotti)

    Valentina Pepe
    Widened derangements and generalized Laguerre polynomials
    Let D_h, E_k and F_a be sets of size h, k, a respectively, with k smaller or equal to h. We define a strongly widened derangement to be a permutation of D_h \cup E_k \cup F_a such that the elements of D_h are not fixed and the elements of E_k cannot occupy a site originally occupied by an object of the same type or by an object of F_a. We will show a connection between strongly widened derangements and generalized Laguerre polynomials that provides a generalization, for integer values of a, of Even and Gillis (Math Proc Camb Philos Soc, 1976) different from the one presented in Foata and Zeilberger (SIAM J Discrete Math, 1988).
    (Joint work with S. Capparelli and A. Del Fra)

    Claudio Procesi
    First and second fundamental theorems of invariant theory, old and new
    These names were given by Hermann Weyl in his book 'Classical groups' and refer to the description by generators and relations for invariants of the direct sum of copies of the fundamental representation and its dual of a classical group. In his book these are the basis for the representation theory of classical groups given by tensor symmetry. After reviewing these ideas we shall give a new form of these theorems which is suggested by quantum information theory.

    Luigi Santocanale
    Linear orders on involutive quantales
    An involutitve quantale Q is a sort of (possibly non commutative) generalized Boolean algebra with a conjunction \otimes and a negation *.
    We consider functions \chi : X^2 - Delta_X --> Q such that \chi(x,y) \otimes \chi(y,z) \leq \chi(x,z) and \chi(y,x) = \chi(x,y)^*.
    If X is totally ordered and the unit of the quantale is such that 1^* \leq 1, then these functions can be ordered yielding a lattice. If X = [n] and Q = 2, then these functions bijectively correspond to transitive tournaments, that is, linear orders, and the ordering is the weak ordering on permutations. If $Q$ is the Sugihara monoid on the three element chain, then these functions yields pseudo-permutations and the facial weak ordering.
    I'll present in this talk general results of these functions, that I call linear orders or skew metrics on an involutitve quantale, present some more examples, and expose then the case where Q is a Sugihara monoid on a finite chain of odd length. I'll give a combinatorial model for the poset of linear orders on these quantales, whose elements turns out to be in bijective correspondence with maximal elements of the intersection poset of affine braid arrangements.

    Christophe Reutenauer
    The stylic monoid
    The stylic monoid Styl(A) is a finite quotient of the plactic monoid. It is obtained by the natural action (left Schensted insertion) of the free monoid A^* on columns over A . It turn out that the elements of Styl(A) are faithfully represented by certain semi-standard tableaux on A, that we call N-tableaux; the bijection is obtained by an insertion algorithm, which is a variant of Schensted`s one. Consequently there is a bijection from Styl(A) onto the set of (set-theoretic) partitions of subsets of A. The cardinality of Styl(A) is therefore B_{n+1}, the Bell number, n=|A|. A presentation of the monoid Styl(A) is obtained by adding to the plactic (Knuth) relations the idempotent relations a^2=a, a in A. The natural involution of A^*, which reverses words and exchanges the order on A, induces an anti-automorphism of Styl(A); it is computed by a variant of evacuation (Schützenberger involution) which works on standard immaculate tableaux (they are naturally in bijection with N-tableaux). The monoid Styl(A) is J-trivial, and the J-order is graded. The monoid Styl(A) is the syntactic monoid of the function mapping each word onto the length of its longest strictly decreasing sequence.
    (Joint work with Antoine Abram)

    Viola Siconolfi
    Ricci curvature, graphs and Coxeter groups
    I will talk about a notion of curvature for graphs introduced by Schmuckenschläger which is defined as an analogue of Ricci curvature. This quantity can be computed explicitly for various graphs and allows to find bounds on the spectral gap of the graph and isoperimetric-type inequalities. I will present some general results on the computation of the discrete Ricci curvature of any locally finite graph. I will then focus on graphs associated with Coxeter groups: Bruhat graphs, weak order graphs and Hasse diagrams of the Bruhat order.
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