Math papers by David Bessis

The main theorem in this paper says that, for any finite complex reflection group W, all complex representations of W may be realized over the field K(W) generated by the character of the natural reflection representation (note that K(W) is always a very small algebraic extension of Q, in most cases a cyclotomic or quadratic extension.) This had almost been obtained before by Mark Benard, who had shown that the Schur indices over K(W) of the characters of W are 1. The theorem then follows from the observation (probably known to Benard, but not made explicit) that the characters indeed have values in K(W). Benard's proof was based on hand-computed character tables, used arguments of many different natures, and had a couple of typos. My argument does not rely on Benard's and is simpler. Still, it is case-by-case, with exceptional groups handled by a general algorithm implemented in GAP.

This paper was written after my first year of doctoral work and, to be honest, it is not very exciting. Although valid, the proof doesn't fulfill my initial goal, which was to produce a conceptual argument. Such an argument might be difficult to find: just consider the case when K(W)=Q (i.e. the Weyl group case), where the only known conceptual proof is Springer's approach (constructing the representations in some cohomology spaces, and even this proof contains some minor case-by-case arguments).

Further developments: Gunter Malle proved a related but deeper theorem about the associated cyclotomic Hecke algebras.

This paper is my first attempt at understanding how Springer's theory of regular elements can be lifted to the level of braid groups. If W is a finite complex reflection group, and if w is a regular element of order d, then the centralizer W' of w in W may be viewed as a complex reflection group, acting on some eigenspace V'. Michel Broué and Jean Michel had observed that w can be naturally lifted to some element b in the braid group B(W), such that b^d is central. They had also remarked that B(W') naturally maps to the centralizer of b in B(W), and asked whether this map was injective. The paper answers this question in almost all cases.

The article is long but has low density and should be fairly easy to read. It might have some educational interest, since it explains a couple of nice tricks for computing in type A braid groups (Artin's lemma for homotoping braids "one strand at a time", Sergiescu presentations, and the many ways of viewing type B braid groups inside type A braid groups) but apart from that, the scientific content is now more or less obsolete.

Further developments: beyond the technical question addressed in this paper, a much wider program was to find braid analogs of Springer's theory of regular elements. Although they lacked confidence and/or evidence to state formal conjecture, this was the underlying fantasy hidden in Broué-Michel's question. The whole program has now been carried out (up to minor details), see items 5, 14, 15 below.

This article explains a general method for constructing nice generating sets for generalised braid groups: if W is a complex reflection group and B(W) is the associated braid group, there exists Artin-like presentations for B(W), that is, positive homogeneous presentations where generators are of a certain type (braid reflections, aka generators-of-the-monodromy, aka meridiens), such that the product of the generators has a central power, and such that, by adding relations expressing that the associated reflections have certain finite orders, one gets a presentation for W. The existence of such presentations was conjectured by Broué-Malle-Rouquier, who had collected case-by-case examples in almost all cases. They used Coxeter-like diagrams to symbolise these presentations (Coxeter and Cohen had previously written down diagrams for W, but not for B(W)).

The idea here is to construct generating sets by taking one-dimensional complex linear slices of the regular orbit space (whose fundamental group is B(W)). For certain natural directions (related to Springer's regular elements), the embedding of a generic slice is π₁-surjective. This is proved using a Zariski theorem of Lefschetz type. Because I couldn't figure out practical ways to work with Hamm-Le's general local analytic version of Zariski's theorem, the paper explains an elementary affine algebraic version, probably known to specialists (but that I couldn't find in modern litterature -- maybe because it is too easy!) and that beginners may like better than the harder versions.

The main weakness of this article is that it doesn't study the monodromy of the Zariski section. In other words, while the main theorem constructs natural generators and proves the existence of nice presentations with those generators, it does not explicitely find the relators. Don't expect too much from this paper: it suggests a potential new way of understanding complex reflection groups, but doesn't really explore it. My recent work on the K(π,1) property (item 14) continues this approach, but is much more substantial.

Known bugs:
a) In Theorem 0.1, the eigenvalues of c and c^-1 have been mistakenly interchanged.

Follow-ups:
a) In a joint project with Jean Michel, we used computers to find explicit presentations in some exceptional cases (see item 8 below).
b) For well-generated groups and when one slices in the Coxeter direction (d=d_n), the monodromy is now completely understood (see item 14 below).

Questions:
a) In the general case, what can be said about the monodromy?
b) (Rephrasing of a) When slicing in another regular direction, is there a generalised Lyashko-Looijenga theory (and a generalised version of all things special to the Coxeter direction that are done in item 14 below)?

One may find three different streams of results in this paper. First, we reinterpret Birman-Ko-Lee's construction of a new presentation for the classical braid group B_n on n strands. We rephrase their results in the language of Garside monoids (introduced by Dehornoy-Paris) and identify the lattice of simples of the Birman-Ko-Lee monoid with the lattice of non-crossing partitions of a regular n-gon. Second, we give an account of Garside monoids from a viewpoint that is slightly different from that of Dehornoy-Paris: instead of starting with a presentation by generators and relations, we axiomatise Garside monoids via intrinsic properties of their lattice of simples; this allows us, for example, to prove that the Birman-Ko-Lee monoid is Garside without having to check some cube condition (this allows us to get rid of the many computations in BKL's paper). Third, using the symmetries of Birman-Ko-Lee's normal form and some lucky diagram chasing, we compute the centralisers of roots of central elements in B_n and prove particular instances of the braid version of Springer's theory (see item 3 above).

About the first stream, it should be noted that the interpretation of the Birman-Ko-Lee in terms of non-crossing partitions was independently obtained by Tom Brady, and also by Daan Krammer.

The second stream (Section 2 in our paper) still has some relevance today, although it is horribly written. Out of laziness, and because we didn't foresee the subsequent developments of Garside theory, we failed a good opportunity to rewrite the theory from scratch and to make it user-friendly. Hopefully, this will be rewritten someday, as Dehornoy, Digne, Krammer and Michel have a joint project to write fundations for Garside theory. Without going into details, the main difference between Dehornoy's syntactic approach and the intrinsic strategy explained in this paper can be summarized as follows. In the syntactic approach, one studies presentations. This is especially useful with freak examples where one lacks geometric insight. Our idea is to directly axiomatize the lattice of simples. Whenever the simples have a geometric interpretation, the intrinsic approach is much simpler to carry out. A real life illustration of these advantages is given by items 10 & 11 below, where the same Garside property is successively proved using both approaches.

Follow-ups:
a) The generalization of BKL's construction to other spherical type Artin groups is done in item 7. This itself was generalized to other situations.
b) Our conjecture 0.1 has now been proved in many settings (see item 14).

Work on this project started in early 1997 and some partial results were included in my PhD. Our motivation was to understand a class of natural morphisms between reflection groups that are visible at the level of Coxeter diagrams (or Coxeter-like diagrams, in the complex case.) For example, "deleting the double-bar" gives rise to a natural diagram epimorphism :
F₄ s —— t ==== u —— v ———> > A₂ x A₂ s —— t u ——v
Morphisms of this type are quite rare between Coxeter groups, but are much more frequent when including the complex cases. In some examples, they relate poorly-understood complex groups to easy-to-work-with real groups, and our hope was that understanding these morphisms might shed some light on the corresponding complex groups.

Our explanation is in terms of invariant theory. The kernel K of a diagram epimorphism W —>> W' (where W and W' are reflection groups) has the property that its invariant ring is a complete intersection (this is the next thing to ask when the invariant ring is not a polynomial ring). In particular, K is generated by double-reflections (elements with two non-identity eigenvalues). These natural generators of K can be associated with rank 2 parabolic subgroups of W and, actually, with relations (symbolised by double-bars or other features of the diagram) in a standard presentation for W.

We study what happens at the level of hyperplane arrangements and braid groups. We also give a classification of triples (K,W,W') involved in such diagram morphisms.

As Ossip Shvartsman points out in his MathSciNet review of our paper, an independent article by George Maxwell studies related (but distinct) questions.

The main result in this article is a generalization of Birman-Ko-Lee's construction (which is the type A case) to the other spherical type Artin groups.

The guiding intuition behind this paper inspired many of my subsequent works (especially items 11, 12 and 14): I suspect that there should exist, besides Coxeter theory, another way to look at reflection groups, and that this dual Coxeter theory should be as precise and rigid as the standard Coxeter theory. In particular, one should expect to find analogs of every key construction and statement in Coxeter theory. Ultimately, one should be able to study algebraic groups and/or generalized structures by replacing key structural theorems (such as, for example, Bruhat decomposition) by some dual analogs.

Let (W,S) be a Coxeter system, with W finite. Let R be the closure of S under conjugacy, i.e., the set of reflections in the natural representation of W. The starting point of the dual theory is to replace (W,S) by (W,R), and to study reduced R-decompositions of elements of W. This may sound naive but, as explained in this article, the pair (W,R) does exhibit remarkable combinorial properties. Geometrically, these properties should be encoded by a dual Tits geometry, based on objects replacing walls, galleries and chambers.

The paper focusses on the combinatorial aspects (some of which were idependently obtained by Brady-Watt), although some preliminary geometric information is also included. The adjective dual has no serious motivation, it was only chosen because of some numerological evidence: for example, c and w₀ plays symmetric roles in both theories, as it appears in the table below (which is a tentative dictionary):

	Classical Coxeter theory	"Dual Coxeter theory"
basic choice	a chamber C (this selects S in R)	a Coxeter element c
generators	S	R
number of generators	n (the rank of W)	N (the number of reflections)
simple elements	W	NCP(W)
number of simples	\|W\| = d₁...d_n	Catalan(W) = (d₁+h)...(d_n+h)/\|W\|
Δ (Garside element)	w₀	c
length of Δ (Garside dimension)	N	n
product of generators	c	w₀
natural basepoint	a point in C	an eigenvector for c
lemma behind Tits-like section	Tits' lemma (Matsumoto property)	transitivity of Hurwitz action
braid relations	sts... = tst...	st = tu
braid monoid	Artin monoid	dual braid monoid
braid monoid (in the type A case)	braids with positive crossings	Birman-Ko-Lee monoid
geometric interpretation of simples	geodesic galleries	tunnels (see item 14)
parabolic theory	intersection lattice of the reflection arrangement	ramification of Lyashko-Looijenga morphism (see item 14)
deformation	Hecke algebra	Temperley-Lieb algebra (?)
the serious stuff	buildings, BN-pairs	??????????

Known bugs.
a) In section 0, the definition of the functor M is incorrect (it brings artificial units). See item 16 below for a patch.

Follow-ups:
a) Tom Brady and Colum Watt now have a mind-blowing direct geometric argument for the lattice property (Vivien Ripoll's master thesis contains a detailed expository account of their proof).
b) Item 14 below provides a rudimentary glimpse of a possible dual version of Tits' language, especially galleries.
c) The theory has been generalised to several infinite Coxeter groups (several cases have been announced by N. Brady, Crisp, Kaul and McCammond; Digne has done the affine type A; the free group case is done in item 12 below)
d) The theory has been generalised to well-generated complex reflection groups (items 11, 14) and, to some extent, to other complex cases (item 14).
e) The combinatorics of generalized non-crossing partitions are now much better understood, thanks to works by Athanasiadis, Armstrong, Chapoton, Fomin, Krattenthaler, Reading, Reiner and many others...
f) Many connections have been established with cluster algebras, free probabilities and other topics. This was at the center of an AIM workshop in Jan. 2005, organized by McCammond, Nica and Reiner. Some material from the workshop (including a problem list) is available here. Much more was obtained since then.
h) The dual braid monoid has been implemented by Jean Michel as part of the development version of the GAP3 package CHEVIE.

Question. Section 4 of my paper contains a geometric interpretation of the dual braid monoid, which I find very natural — although I have failed to make any interesting use of it. What is the meaning of this interpretation? Can it be refined and made fertile? I suspect one might use it to prove the lattice property and/or the K(π,1) property. Note that, in item 14 below, the proof of the K(π,1) property based on the dual braid monoid uses a totally different geometric interpretation (in the quotient space V/W, rather than in V as here). Just clarifying the connection between the two interpretations, and their relation with Brady-Watt's geometric model, is an interesting question.

Work on this project started in the summer of 2001. Previous results by Artin, Brieskorn, Bannai, Nakamura and Broué-Malle-Rouquier provided presentations for braid groups attached to complex reflection groups, except for six exceptional types. A general existence result (see item 4 above) guaranteed the existence of nice presentations for the missing cases. Even more annoying, seventy years old theorems by Zariski and Van Kampen provided a general strategy to compute the missing presentations. I mentioned this to Jean, immediately adding that the algorithm was intractable by hand, and that there was no hope to turn it into a rigorous and efficient piece of software. Jean wasn't convinced at all by my pessimism and we started to work on a possible implementation (see item 8' just below).

The paper explains the different steps needed to reduce the problem to a purely computational one, and sketches the main aspects of our implementation.

$A diagram for G_{34}$ Our presentations are typically obtained by adding one or two additional relations to a standard Artin presentation. Most of the time, the added relations are circular relations between three generators:

For example, this is our diagram for G₃₄ (the largest non-real exceptional group, in dimension 6). Forgetting about the character "6" in the triangle, one sees a Coxeter diagram (of infinite type). The presentation for the braid group B₃₄ is obtained from the associated Artin presentation by adding the relation
tuwtuw = uwtuwt = wtuwtu
(this circular cycling relation of length 6 is symbolized on the diagram by the character "6").
By adding the relations s² = t² = u² = v² = w² = x² = 1, one gets a presentation for the finite group G₃₄.

Some of the presentations in our paper are stated as conjectures rather than theorems. This missing argument didn't lie with the software part of the project, but with the mathematics: we had to use sections by 2-planes for which we weren't sure that they satisfied the genericity conditions. This is now clarified (see item 14), the sections were valid and all our conjectures are indeed theorems.

Follow-ups: item 14 below explains how to find the presentations without having to rely on computers for monodromy computations.

This is the software package that made the above article possible. Although it was specially written on that occasion, it has on its own as much (maybe more) interest than the particular use we made of it.

Basically, it contains an exact implementation of Van Kampen's method for computing presentations of fundamental groups of complements of complex algebraic curves. It takes as input a two-variable polynomial with coefficients in Q[i], and returns a presentation by generators and relations. Moreover, it includes heuristics for simplifying presentations, which make the output much more usable (Van Kampen presentations are very special presentations and, although we don't have any special theorem to support this, one usually obtains very simple results).

To have a guarantee on the result, we rely not on floating point "real numbers" but on adaptative multiprecision arithmetics. VCURVE also includes support for several basic functionalities that were missing in GAP3 (multivariate polynomials and rational fractions, Newton's method for solving polynomial equations,...)

Surprisingly, the software is quite fast. We haven't encountered interesting examples where we couldn't carry out the computations in a reasonable time.

If needed, there is plenty of room for improvement:

The CPU-intensive part of the algorithm is designed to be easily parallelizable.
It should not be too hard to implement the support of coefficients in any cyclotomic extension of Q (at the cost of slowing down the computation).
To compute the π₁ of the complement of an hypersurface, one may use the software after slicing with a generic 2-plane. Explicit genericity criterions exist (see item 14 below for such a criterion), and it is a straightforward exercise to implement them so that the software can accept as input any multivariate polynomial with coefficients in Q[i].
the software computes many intermediate structures (most notably, monodromy braids) that may be more interesting to some users than the final presentation. We haven't spent much time documenting these structures, but they can be made accessible and user-friendly.

Sources and documentation are available here. An interactive online version was available here until the server died.

This note gives a short but self-contained account of Van Kampen's method for finding presentations of fundamental groups of complements of complex algebraic curves. It also explains a variant designed to handle vertical asymptotes (this has some algorithmic relevance).

Several other accounts may be found in the litterature, notably by Cheniot (for the standard method) and by Artal-Carmona-Cogolludo-Luengo-Melle (for a possible way to handle asymptotes). My interest in writing yet another account was to further simplify the arguments (replacing topology by algebra, wherever possible) and to clarify a few points, with VKCURVE's implementation in mind. This note may be thought as part of the documentation for VKCURVE. It is unfortunate that I have been too lazy to include pictures, as the whole idea behind the variation is very visual.

The official story is that this note is the English translation of my original Russian article, that appeared in Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 117, Geometry, 2004. Whether or not you believe this is your own decision.

This article begins with the construction of a lattice of non-crossing partitions associated with the complex reflection group G(e,e,r). We then show that this lattice is the lattice of simples for a certain Garside structure on the braid group B(e,e,r).

Recall that the classification of irreducible finite complex reflection groups contains an infinite family G(de,e,r). The symmetric group S_n corresponds to G(1,1,n). The standard lattice of non-crossing partitions of a regular n-gon is the lattice of simples of the Birman-Ko-Lee monoid for the associated braid group (see item 5). Other real subtypes of the infinite family are G(2,1,n) (type B_n), G(2,2,n) (type D_n) and G(e,e,2) (type I₂(e)), for which analogs of the BKL monoid were constructed in item 7. The Garside structure studied here generalizes the dual braid monoid construction to the non-real case G(e,e,r).

The common property of the real types and of G(e,e,r) is that they are well-generated (see items 4 or 14). As it appeared later, the construction of the dual braid monoid precisely works for well-generated groups (this is done in item 14). Many numerological aspects of the real case continue to hold. For example, the cardinality of the lattice is a generalized Catalan number expressed in terms of the degrees. One small difference however is that, by contrast with the real case, the generators of the monoid are indexed by a strict subset R_c of the set R of all reflections.

Among applications, we obtain solutions to the word and conjugacy problems for B(e,e,r). Because there was no classical Coxeter theory applicable here, no solutions had been known previously.

The case e = 2 of the lattice construction was independently obtained by Athanasiadis-Reiner.

Follow-ups:
a) I've heard several accounts of people having constructed another Garside structure for B(e,e,r), which might be the analog of the classical braid monoid. As far as I know, none of these constructions have appeared as preprints.

This note is my first attempt at extending the dual braid monoid construction to infinite Coxeter groups and associated Artin group. The free group is the Artin group associated with the universal Coxeter group (the free product of n groups of order 2). Following the pattern observed in items 5, 7 and 11, one constructs a (quasi)-Garside structure, whose simple elements can be understood in terms of "non-crossing objects" — here, non-self-intersecting loops drawn in the inside of a punctured disk.

The results are very easy to obtain. Some general features of the dual monoid structure, such as the transitivity of Hurwitz action, actually coincide on that particular example with basic lemmas from Artin's seminal article on braids, published in 1947. The main interest of the note lies in the general pattern it fits in. By contrast with the spherical types, Artin monoids associated with infinite Coxeter groups are not Garside (generators don't have a common multiple). Whenever dual braid monoids can be constructed, they might provide a way to tackle standard open problems about the associated Artin groups.

Related works:
a) Noel Brady, John Crisp, Jon McCammond, Anton Kaul have announced several results about dual monoids for infinite type Artin groups, including the free group case which they obtained independently, and several affine types. Waiting for their preprints to be released, you may want to check Jon McCammond's nice introduction to Garside structures.
b) Independently, François Digne constructed a dual braid monoid the affine type A braid groups.

Some questions and a conjecture:
a) Why is it so easy? What's so special about the punctured plane model for the free group? (Viewing the free group as an Artin group, one would expect to work in the quotient of the complexified Tits cone -- how does one compare the two models, and which other Artin groups admit a model similar to the punctured plane?)
b) How come we're still able to work with non-crossing thingies? Is there a concept behind that, or is it just a superstition?
c) The lattices of generalized non-crossing partitions associated with well-generated complex reflection groups naturally appear as quotients of the lattice of non-crossing loops studied here (to see this, view the punctured disk within a complex vertical line L_y, in the notations of item 14). More generally, to any subgroup H of the braid group on n strings, one can associate a quotient poset (non-crossing loops modulo the monodromy action of H). When is this quotient a lattice? Conjecture: if H is generated by powers of braid reflections, the quotient is a lattice. Item 14 provide examples of this situation.

This is my most substantial work to date. It contains :

The proof of a conjecture from the 1970s, predicting that the complement of a finite complex reflection hyperplane arrangement is a K(π,1) space. The complexified real case had been settled in a classical 1972 paper by Deligne ; though conceptually similar, my approach is different and, for example, the real case is recovered using totally new geometric arguments.
A generalization to braid groups of Springer's theory of regular elements. This includes, for most cases, a classification of periodic elements in B(W) and a computation of their centralizers.
Many new results about B(W), especially for the previously poorly understood high-dimensional exceptional cases: presentations by generators and relations, Garside structure, solution to the word and conjugacy problems, centers, cohomological dimension,...

The paper actually contains the basic steps of a comprehensive geometrico-combinatorial approach to complex reflection groups. Although this approach isn't Coxeter theory (even in the real case, it is different), it is somehow "analog" to it, in the sense that it retains important algebraic and homotopy-theoretic aspects.

Let W be a finite irreducible complex reflection group, acting on a complex vector space V. Denote by V^reg the complement in V of the reflection hyperplanes. Because the quotient map V^reg—> V^reg/W is a covering, V^reg and V^reg/W have the same higher homotopy groups. Deligne's approach uses the real structure on V, and the semi-algebraic objects (chambers, walls, facets,...) whose incidence geometry is controlled by Coxeter theory. By contrast, the approach here is to work in the quotient space V/W and to construct new semi-algebraic objects, whose incidence geometry is encoded by the combinatorics of the dual braid monoid (see item 7). In both approaches however, the semi-algebraic objects are used to construct simplicial models of the universal cover of V^reg, whose contractility is proved by Garside theory.

A huge part of the paper is devoted to the well-generated case, where the geometry of the quotient space V/W exhibits features that are reminiscent of Kyoji Saito's "flat" (Frobenius manifold) structure. In particular, a Lyashko-Looijenga morphism is used to compare V/W with a type A reflection orbifold. This is the key observation that leads to the construction of natural semi-algebraic objects, and to an interpretation of the dual braid monoid in terms of Zariski slices (providing the missing link between items 4 and 7).

The remaining cases are understood as relative versions of the well-generated cases. This is where Springer's theory comes into play: just like the non-simply-laced Weyl groups may be seen as fixed subgroups under diagram automorphisms of simply-laced Weyl groups (e.g., F₄ in E₆), many non-well-generated complex reflection groups may be viewed as fixed subgroups under particular automorphisms of well-generated complex reflection groups (e.g., the exceptional group G₃₁ may be viewed in E₈). Unfortunately, the dual braid monoid is not preserved by these automorphisms. All previous constructions have to be replaced by equivariant versions. The key ingredient here is the notion of divided Garside categories, which is explained in a separate paper (item 15). Basically, by simplicial abstract non-sense, it is possible to replace the dual braid monoid by a certain category whose groupoid of fractions is category-equivalent to the braid group — by contrast with the monoid, the category is stable by the needed automorphisms and can be used to work out the relative version of the theory.

The latest version (v3, April 2007) contains many improvements over previous releases.