



Organizers  
Christophe Hohlweg  hohlweg.christophe@uqam.ca 
Claudia Malvenuto  claudia@mat.uniroma1.it 
Frédéric Patras  frederic.patras@unice.fr 
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
claudia@mat.uniroma1.it) if you are accompanied or not and
which type of room you prefer:
* Double room (double occupancy)
* Double room (simple occupancy)
From Rome the simplest way to reach Cetraro is by train: check the schedule on www.trenitalia.com The hotel has a pickup 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 www.trenitalia.com As due to covid there might be fewer flights than usually, an alternative solution is to fly to Rome and take a train.
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
Einfinity operad which plays an important role in the theory of
infinite loop spaces. This Einfinity operad admits a filtration by
E_nsuboperads thereby producing combinatorial models for point
configurations in ndimensional 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 BlasiakHaimanMorsePunSeelinger: 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 5cycles.
(Joint work with Roberto Conti and Gleb Nenashev)
Patrick CassamChenaï
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
PreLie Magnus expansion and relations between cumulants.
In noncommutative 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 preLie algebra
structure on cumulant functionals. Relations among cumulants are described in terms of
the preLie 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 grepresentations. 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
KazhdanLusztig 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 ArtinTits (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 PoirierReutenauer 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 PoirierReutenauer 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 RotaBaxter families, then address other family structures (dendriform, duplicial, preLie,...), 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)
JeanChristophe Novelli
Lagrange inversion, noncrossing partitions and a quasisymmetric
analogue of the FarahatHigman 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 Borbits 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 noncommutative 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 noncommutative 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 EbrahimiFard, 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 pseudopermutations 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 semistandard tableaux on A, that we call Ntableaux; 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
(settheoretic) 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 antiautomorphism 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 Ntableaux).
The monoid Styl(A) is Jtrivial, and the Jorder 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
SchmuckenschlÃ¤ger 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
isoperimetrictype 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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