Proofs of the statments in the minicourse
The general purpose of the minicourse is to show that the framework of
Garside families is natural, efficient, and enables one to recover (and
extend) the results about Garside groups at no extra cost. In practice, we
shall summarize the main known properties of Garside families, with an
emphasis on their various characterizations. Introduced as "what is needed
to guarantee the existence of a greedy normal form", Garside families admit
both extrinsic definitions (recognizing that a subfamily of an given monoid
or category is a Garside family) and intrinsic ones (recognizing that a
family generates a monoid or a category in which it embeds as a Garside
family).
The work of Bessis, Broué, Digne, Dudas, Malle, Michel, Rouquier in the
last 20 years has connected modular character theory of finite reductive
groups (in particular the Broué derived equivalence conjectures) with
questions about conjugacy and centralizers in braid groups and ribbon
categories.
This goes through the cohomology of Deligne-Lusztig varieties, and the
action of some braid monoids on them.
My goal is to explain this picture in the case of GLn(Fq), but in a
way which can be extended to other finite reductive groups.
Abstract: This is joint work with Volker Gebhardt. We will show that every
Garside monoid admits a finite state automaton which accepts the language
of short-lex representatives (with atoms as generators). Moreover this
automaton is the smallest possible one, and in the case of braid monoids
its size is exponential with respect to the braid index. We will also give
new simple formulae for the growth function of Artin-Tits monoids of
spherical type, and we will see how to produce random elements in braid
monoids.
Abstract:
Every element of a Garside group admits a Garside normal form,
and the infimum is a basic invariant arising from the normal form.
In this talk we discuss the behavior of the infimum of an element
under taking powers.
For finite-type Garside groups, there is a well-established theory
which uses an asymptotic analysis on the infimums of powers of elements.
There are applications to discreteness of translation numbers.
Some of the results cannot be generalized to infinite-type Garside groups.
Abstract:
The Garside structures of geometric groups are often derived
from positive presentations of the groups.
In the talk we discuss positive presentations of
the Artin braid groups,
the Artin groups of finite type and
the braid groups of the complex reflection groups,
and the Garside structures derived from those presentations.
Abstract: In this talk I will describe on-going work with Noel
Brady, John Crisp and Robert Sulway that clarifies the relationship
between Garside structures and the dual presentations of euclidean
Artin groups.
Abstract:
Mapping Class Groups are known to enjoy the
Linearly Bounded Conjugator Property
(i.e. given any generating set,
any two conjugated elements of length $l$ admit a conjugator
whose length is linearly bounded by $l$). In the special case of braid groups,
this property together with
Garside theory leads to new
polynomial-time algorithms : to decide the Nielsen-Thurston type of
braids, and to solve the conjugacy problem in the four-strand braid group.
Abstract:
Garside structures are described for the braid groups of the affine type
and those of the infinite family of imprimitive reflection groups G(de,e,r). They
are related to the Garside structures for type (e,e,r) of Mathieu Picantin and
myself. Unlike that (e,e,r) construction, the new ones are not generated group
constructions. However a related Garside structure for what we call a "fake"
braid group of type (de,e,r) is conjectured to be a generated group construction.
Diagrams are proposed for these structures, similar to Artin or Coxeter diagrams,
and natural maps between various braid and reflection groups are described and
interpreted in terms of these diagrams. This also gives rise to a way of
visualizing elements of type (de,e,r) as geometric braids on a cylinder with
winding number 0 mod e.
Résumé:
Parmi les problèmes posés par l'existence de groupes
de tresses associées aux groupes de réflexions complexes, on trouve
une conjecture de Broué Malle et Rouquier, qui prédit que les
algèbres de Hecke associées à ces groupes sont de dimension
finie. On essaiera d'expliquer sur des exemples pourquoi certaines
propriétés des monoides de Garside sont utiles pour
démontrer certains cas
de cette conjecture.
Abstract:
In recent investigations there has been considerable interest in $n$-generated
monoids with a presentation defined by at most $n\\choose 2$
relations that are homogeneous and quadratic, i.e. of the type $xy=zt$ where $x,
y, z, t$ are generators.
Examples of such monoids are divisibility monoids and monoids of $I$-type. The
latter are a monoid interpretation of non-degenerate
involutive set-theoretic solutions of the Yang-Baxter equation.
In this lecture we present some recent results on the so called monoids $S$ of
quadratic type, i.e.
S= <x_1, ..., x_n > where R is a set of $n \choose 2$
relations that are
quadratic and homogeneous. We investigate when they are Noetherian
and discover a strong link with the divisibility
monoids, monoids of $I$-type and Garside monoids. This is joint work with J.
Okni'nski and E. Jespers.
Abstract : (Joint work with Tetsuya Ito) We show that the span of the variable $q$ in the
Lawrence-Krammer-Bigelow representation matrix of a braid is equal to the twice of the dual Garside
length of
the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow\'s geometric
approach. The
key observation is that the dual Garside length of a braid can be read off a certain labeling of its
curve
diagram.
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