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* Laboratoire d'Algèbre, de Combinatoire et d'Informatique Mathématique (LACIM, UQAM)
* "Fondi di Ateneo 2020" Sapienza Università di Roma
| Organizers | |
| Christophe Hohlweg | hohlweg.christophe@uqam.ca |
| Claudia Malvenuto | claudia@mat.uniroma1.it |
Participants (Confirmed so far)
Wearing a mask in public transportation (bus, taxi, train, ships) is mandatory in Italy. Due to the present contagious situation, the use of the mask in all indoor environments and in all cases of crowded outdoor events is highly recommended. For more information, see https://www.italia.it/en/covid-19-italy-travel-guidelines
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 or Naples
From Rome or Naples, the simplest way to reach Cetraro is by train:
check the schedule on
https://www.trenitalia.com.
On the website of Trenitalia, use "Roma Termini" for trains departing from Rome,
and "Napoli centrale" for trains departing from Naples.
If you arrive in Rome Fiumicino Airport, there is a direct train going from
"Fiumicino aeroporto" to "Roma Termini" (central station), it runs about every
15 minutes, lasts 32 minutes and costs 14,00 eu, you can purchase the Fiumicino
Aeroporto to Roma Termini train ticket from the same website
https://www.trenitalia.com.
If you arrive in Aeroporto Internazionale di Napoli, there is a bus service
going to Napoli Centrale train station, called Alibus; the buses run every 15 minutes,
lasts 15 minutes and should cost 5,00 eu.
The hotel has a pick-up service from both Cetraro and Paola train station:
(a) From Cetraro train station: 5 euros per passenger.
(b) From Paola train station: service by Car (3 persons max., 50 euros each way)
or by Minibus (8 persons max., 77 euros each way).
Both prices are for the vehicle
and eventually can be shared by the passengers if more than one: in that case individual
bills will be made by the Hotel.
From Lamezia Terme
The nearest airport is Lamezia Terme Airport. From there are then two ways to arrive
to the hotel:
(a) going to Cetraro by train from Lamezia Terme Centrale (a bus connects
Lamezia airport to Lamezia centrale), then the pick-up service
in Cetraro see above;
again you can check the schedule on
https://www.trenitalia.com.
(b) The Hotel pick-up service from Lamezia Terme Airport, by Car
(3 persons max., 95 euros each way) or by Minibus (8 persons max., 127 euros each way).
Both prices are for the vehicle and so can be shared by the passengers. Individual
bills will be made by the Hotel.
Please note that for night shuttle services (between 1.00 am and 5.00 am) there is a
30,00 euro supplement.
In order for us to arrange and group the pick-up service, please let us know as
soon as possible (if you did not do so already) of your arrival/departure times at
Cetraro (or Paola) train station, or Lamezia airport.
Please feel free to write to us for any questions you may have.
The final SCHEDULE AND ABSTRACTS of the Workshop will be available later.
The conference will start on Monday morning, July 25 and end on Friday 29 at 12. Participants are expected to arrive on Sunday July 24 and to leave on Saturday July 30.
The talks will take place in the Hotel main Conference room.
There will be three conferences in the morning and one by the end of the afternoon. 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: In between the conferences (in the afternoon), the conference room will be available for us on the spot for working group talks.
Talks (so far)
Riccardo Biagioli
Diagrammatic representations of affine Temperley-Lieb algebras
The Temperley-Lieb (TL) algebra is a well studied finite dimensional
associative algebra: it can be realized as a diagram algebra and it has a
basis indexed by the fully commutative elements in the Coxeter group of type
A. Generalized TL algebras exist for finite and affine Coxeter types.
A few years ago, D. Ernst introduced a diagrammatic representation of the
generalized TL algebra of type affine C. The proof that such representation
is faithful is quite involved and the same author wonders if an easier proof
exists.
In this talk, we present a new combinatorial framework to describe Ernst's
algebra homomorphism, from which the injectivity follow more easily. Our
results are based on a new classification of the fully commutative elements
of type affine C in terms of heaps of pieces, and on certain operations that
we define on such heaps. We then show new faithful diagrammatic
representations of the generalized TL algebras of affine types B and D.
(This talk is based on joint works with Gabriele Calussi, Giuliana Fatabbi and
Elisa Sasso)
Srecko Brlek
Une identité remarquable sur les mots lie diverses mesures
de complexité : en facteurs, palindromique, pseudo-
palindromique, facteurs carrés.
Lors de l'exposé à "TBA" dédié à
Christophe Reutenauer, l'identité de BR palindromique a
été étendue aux pseudo-palindromes, et plus tard (2017)
aux carrés.
Cette identité suggère une approche pour démontrer
la conjecture de Fraenkel et Simpson, à savoir que le nombre
de facteurs carrés non vides de tout mot fini de longueur n est
au plus n. A la aide de la construction des graphes d'un mot
(Rauzy - De Bruijn - Flye Sainte-Marie) la solution est remarquablement
simple : à chaque facteur carré est associé
un unique circuit dans un unique graphe et donc le nombre cyclomatique
est une borne (large) et naturelle.
Cette ouvre de nouvelles perspectives pour l'amélioration de la
borne supérieure et également pour l'estimation de la
complexité d'autres motifs.
Nathan Chapelier
The Shi variety of an affine Weyl group
In this talk I will introduce for each affine Weyl W group an affine variety,
called the Shi variety of W, that has the property of having its set of
integral points in bijection with W. Subsequently we will see some
combinatorial consequences in type A. I will also discuss how we can express
the corresponding extended affine Weyl group via this variety.
Alessandro D'Andrea
An invitation to Hecke-Kiselman monoids
The q=0 version of Hecke algebras may be read as the monoid algebra of
semigroups known as Hecke monoids. They admit special quotients which were
first introduced by Kiselman in a convexity theory setting. I will give a
brief introduction to Kiselman quotients of Hecke monoids, discussing their
finiteness and describing some natural monoid action realizing them in terms
of dynamical systems on graphs.
Bérénice Delcroix-Ogier
A spoonful of dendrology: from hypertrees to Cayley trees
Hypertrees are generalizations of trees introduced by Berge in the 1980s.
McCammond, Meier and their coauthors introduced and studied a poset structure
on hypertrees in a series of papers in the 2000s to study automorphisms of
free groups and free products. In particular, they computed the Euler
characteristic of the hypertree poset on n vertices, which is precisely
the number of Cayley trees on n-1 vertices. Chapoton conjectured in
2007 than the action of the symmetric group on the homology of
the hypertree poset is exactly the same as the anticyclic one on Cayley trees.
This conjecture was proven in 2012 using species theory. In a work in
progress with Clement Dupont (IMAG), we study more deeply the link between
hypertree posets and
operads in terms of Lie brackets of planar trees.
We will first introduce in the talk the notions of hypertrees, species,
operads and
posets before fully explaining the link between hypertrees and the other
mentioned
trees.
Loïc Foissy
Pairs of cointeracting bialgebras and combinatorial applications
Pairs of cointeracting bialgebras appear recently in the literature
of combinatorial Hopf algebras, with examples based on formal series,
on trees (Calaque, Ebrahimi-Fard, Manchon and Bruned, Hairer, Zambotti),
graphs (Manchon), posets...
These objects have one product (a way to combine two elements in a single one)
and two coproducts (the first one reflecting a way to decompose a single
element into two parts, maybe into several ways,
the second one reflecting a way to contract parts of an element in order to
obtain a new one).
All these structures are related by convenient compatibilities.
We will give several results obtained on pairs of cointeracting
bialgebras:
actions on the group of characters, antipode, morphisms to quasi-symmetric
functions...
and we will give applications to Hopf algebras of graphs, posets, multigraphs
or mixed graphs.
Thomas Gobet
Toric reflection groups
We introduce and study a three-parameter family of (in general infinite) reflection-like groups that includes, among other, finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter's truncated braid groups on three strands. We give a classification of these groups, and show that they can naturally be associated with a torus knot group that behaves like their "braid group". We also show that they have a cyclic center, and that the quotient by their center is an alternating subgroup of a Coxeter group of rank three. This gives a new explanation of a phenomenon which was previously observed on a case-by-case basis for some finite complex reflection groups of rank two.
Jéremie Guilhot
Recognizing Kazhdan-Lusztig cells using representations
The aim of this talk is to show on various examples that one can recognize
Kazhdan-Lusztig cells using representations of Hecke algebras.
This idea is especially interesting in the case of affine Weyl groups where
one can
use representations that have a very nice combinatorial descriptions in
terms of
alcove paths.
(Joint work with James Parkinson)
Evelyne Hubert
Polynomial description for the T-Orbit Spaces of
Multiplicative Actions of Weyl groups Polynomials
A finite group with an integer representation has a multiplicative action on the ring of
Laurent polynomials, which is induced by a nonlinear action on the complex torus,
leaving the compact torus T invariant. The associated T-orbit space is well described
by the image of the compact torus by the fundamental invariants. For the Weyl groups of
types A, B, C and D, this image is a basic semi-algebraic set and we present the
defining polynomial inequalities explicitly. We show how orbits correspond to solutions
in the complex torus of symmetric polynomial systems and give a characterization of the
orbit space as the positivity-locus of a symmetric real matrix polynomial.
The resulting domain is the region of orthogonality for a family of generalized
Chebyshev polynomials, which have connections to topics such as Fourier a nalysis and
representations of Lie algebras. We shall present how our results can be used to
approach the spectral bound on the chromatic number of some infinite graphs by
polynomial optimization.
(Joint work with Tobias Metzlaf and Cordian Riener)
Cedric Lecouvay
Positively multiplicative graphs and random walks on alcoves
Positively multiplicative graphs are graphs whose adjacency matrix can be
embedded in a matrix algebra admitting
a distinguished basis labelled by its vertices with nonnegative structure
constants (or polynomials with nonnegative
coefficients as structure constants). It is easy to get such graphs from the
group algebra of the character algebra of a finite
group. Other simple examples are obtained from classical bases of symmetric
functions. More subtlety, it is also possible to
define numerous multiplicative graphs from the affine Grassmannian associated
to an affine Weyl group. These graphs are
then related to interesting probabilistic models (random walks in alcoves,
TASEP etc.). The talk will consist in an introduction
to these notions and problems.
(Joint work with J. Guilhot and P. Tarrago)
Paolo Papi
Minimal inversion complete sets in finite reflection groups
I will discuss the generalization of a problem in extremal combinatorics of
the symmetric group, arising in Operational Research, to finite reflection
groups and its relationships with the theory of abelian ideals in Borel
subalgebras. The talk is based on a joint work with Malvenuto, Orsina,
Moseneder; Panyushev's deepening of our work will also be discussed.
James Parkinson
Cone types, automata, and regular partitions in Coxeter groups
In 1993 Brink and Howlett showed that finitely generated Coxeter groups are
automatic. One ingredient was the
construction of a finite state automaton recognising the language of reduced
words in the Coxeter group using the remarkable
(finite!) set of roots of "elementary roots" of the associated root
system. Recently Dehornoy, Dyer, and Hohlweg introduced the
notion of a Garside shadow in a Coxeter group. Their work resulting in
further constructions of automata recognising the language of
reduced words, and interesting conjectures arose.
In this talk we outline recent joint work with Y. Yau directed towards these
conjectures. This work centres around the notion of a
"regular partition" of a Coxeter group, and we show that such
partitions are essentially equivalent to the class of automata recognising
the language of reduced words. Moreover, we use this framework to
prove some fundamental facts about the minimal automata recognising the
language of reduced words, and study the associated
"cone types" in Coxeter groups.
(Joint work with Yeeka Yau)
Frédéric Patras
The Magnus formula revisited
The BCH and Magnus formula solve the BCH problem for matrix linear
differential
equations: find a combinatorial expression for the logarithm of a solution.
Whereas the BCH formula is usually expressed using descents in symmetric
groups or a
basis of the free Lie algebra, the Magnus formula has a different structure
and is
best expressed by lifting computations to preLie algebras. In this talk we
present
general combinatorial methods in preLie algebras that allow to provide closed
expression for the Magnus and other similar formulas.
(Joint work with Adrian Celestino)
Claudio Procesi
Swap Polynomials
We discuss several constructions of swap polynomials, that is non-commutative
polynomials in matrix variables $x_i\in M_d(\mathbb Q)$ with values
in $M_d(\mathbb Q)^{\otimes 2}$ which are multiples of the transposition
operator $(1,2)$.
Christophe Reutenauer
On the three-distance theorem
Let a be a real number and k be a natural integer.
Reorder the remainders modulo 1 of the numbers
0, a , 2a, ... , ka; one obtains a sequence
0 = a_0 < a_1 < a_2 < ... < a_k < a_{k+1} = 1.
Then the gaps a_{i+1} - a_i take at most three values, and if there are three, then the largest is the sum of the two smallest
(Theorem of Vera Sos and others, 1958, solving a conjecture of Steinhaus).
In this talk we give a noncommutative interpretation of this result, using the
notion of Christoffel word.
(Work in collaboration with Valérie Berthé.)
Paolo Sentinelli
Special idempotents and projections
We introduce the notions of special idempotent and projection for a finite
graded poset,
as elements of the monoid of regressive endomorphisms (in the category of
posets).
The most important examples of special idempotents are the projections on
the parabolic
quotients (and double quotients) of a finite
Coxeter group. For symmetric groups with the Bruhat order, these are the
only examples
of special idempotents.
The notion of special projection is stronger and implies, for Eulerian posets,
that its image and the complement of its image (as induced subposets)
are graded. To be a projection is the same as the existence of a parabolic
map, in the
meaning of Billey, Fan e Losonczy.
Important examples of special projections are the projections on the
parabolic
quotients of a finite Coxeter group.
Viola Siconolfi
On the Cohomology of the Toric Compact Model of Type A_n
I will start by presenting the wonderful compactification of a toric
arrangement and some results on its cohomology ring, this first part is
based on a paper from De Concini and Gaiffi. I will then focus on the
toric arrangement of type A_n and give a combintorial description of a
monomial basis for the cohomology ring of its wonderful model.
Such a description offers a geometrical point of view on the relation
between some Eulerian statistics on the symmetric group.
(Joint work with G. Gaiffi and O. Papini)
Hugh Thomas
Realizing quotients of weak order as one-skeleta of polytopes via
representation theory
Given a quotient of weak order on a Coxeter group, the problem of producing a polytope whose one-skeleton, suitably ordered, yields the Hasse diagram of the poset, has attracted considerable attention over the past two decades. One notable motivation was the problem of constructing generalized associahedra, which yield realizations of the Cambrian lattices. In recent work, Padrol, Pilaud, and Ritter show that, in types A and B, it is possible to associate a "shard polytope" to each join-irreducible of weak order on the Coxeter group, such that the Hasse diagram of any quotient can be realized as the Minkowski sum of the shard polytopes corresponding to the non-contracted join-irreducibles. In my talk, I will explain how this construction can be understood in terms of the representation theory of preprojective algebras of type A. We conjecture that the same representation-theoretic approach will work in the other simply-laced cases. With Eric Hanson, I am actively working on resolving this conjecture.
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