Long fully commutative elements in affine Coxeter groups

An element of a Coxeter group $W$ is called fully commutative if any two of its reduced decompositions can be related by a series of transpositions of adjacent commuting generators. In the preprint "Fully commutative elements in finite and affine Coxeter groups" (arXiv:1402.2166), R. Biagioli and the authors proved among other things that, for each irreducible affine Coxeter group, the sequence counting fully commutative elements with respect to length is ultimately periodic. In the present work, we study this sequence in its periodic part for each of these groups, and in particular we determine the minimal period. We also observe that in type $A$ affine we get an instance of the cyclic sieving phenomenon.Comment: 17 pages, 9 figure

Duality relations for hypergeometric series

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for q-hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by N. M. Bailey in the 1930's on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.Comment: 16 page

pℓp^\ell-Torsion Points In Finite Abelian Groups And Combinatorial Identities

The main aim of this article is to compute all the moments of the number of $p^\ell$-torsion elements in some type of nite abelian groups. The averages involved in these moments are those de ned for the Cohen-Lenstra heuristics for class groups and their adaptation for Tate-Shafarevich groups. In particular, we prove that the heuristic model for Tate-Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of $p$-Selmer groups of elliptic curves. For our purpose, we are led to de ne certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context.Comment: 24 page

Combinatorics of fully commutative involutions in classical Coxeter groups

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type $A$, we connect our results to Fan--Green's cell structure of the corresponding Temperley--Lieb algebra.Comment: 25 page

The Cohen-Lenstra heuristics, moments and pjp^j-ranks of some groups

This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to E. Fouvry and J. Kl\"uners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^j$-ranks of Selmer groups of elliptic curves. This is compatible with some theoretical works and other classical conjectures

Bilateral Bailey lattices and Andrews-Gordon type identities

We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs related to $a$ into bilateral Bailey pairs related to $a/q$. Using this lemma and similar ones, we give bilateral versions and simple proofs of other (new and known) Bailey lattices, among which a Bailey lattice of Warnaar and the inverses of Bailey lattices of Lovejoy. As consequences of our bilateral point of view, we derive new $m$-versions of the Andrews-Gordon identities, Bressoud's identities, a new companion to Bressoud's identities, and the Bressoud-G\"ollnitz-Gordon identities. Finally, we give a new elementary proof of another very general identity of Bressoud using one of our Bailey lattices.Comment: 27 pages v2: new identities adde

Fully commutative elements and lattice walks

International audienceAn element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge in the finite case. In this work we deal with any finite or affine Coxeter group $W$, and we enumerate fully commutative elements according to their Coxeter length. Our approach consists in encoding these elements by various classes of lattice walks, and we then use recursive decompositions of these walks in order to obtain the desired generating functions. In type $A$, this reproves a theorem of Barcucci et al.; in type $\tilde{A}$, it simplifies and refines results of Hanusa and Jones. For all other finite and affine groups, our results are new.Un élément d’un groupe de Coxeter $W$ est dit totalement commutatif si deux de ses décompositions réduites peuvent toujours être reliées par une suite de transpositions de générateurs adjacents qui commutent. Ces éléments ont été étudiés en détail par Stembridge dans le cas où $W$ est fini. Dans ce travail, nous considérons $W$ fini ou affine, et énumérons les éléments totalement commutatifs selon leur longueur de Coxeter. Notre approche consiste à encoder ces éléments par diverses classes de chemins du plan que nous décomposons récursivement pour obtenir les fonctions génératrices voulues. Pour le type $A$ cela redonne un théorème de Barcucci et al.; pour $\tilde{A}$, cela simplifie et précise des résultats de Hanusa et Jones. Pour tous les autres groupes finis et affines, nos résultats sont nouveaux

New Finite Rogers-Ramanujan Identities

We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by specialization or through Bailey's method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice.Comment: 19 pages. to appear in Ramanujan

The Cyst-Dividing Bacterium Ramlibacter tataouinensis TTB310 Genome Reveals a Well-Stocked Toolbox for Adaptation to a Desert Environment

Ramlibacter tataouinensis TTB310T (strain TTB310), a betaproteobacterium isolated from a semi-arid region of South Tunisia (Tataouine), is characterized by the presence of both spherical and rod-shaped cells in pure culture. Cell division of strain TTB310 occurs by the binary fission of spherical “cyst-like” cells (“cyst-cyst” division). The rod-shaped cells formed at the periphery of a colony (consisting mainly of cysts) are highly motile and colonize a new environment, where they form a new colony by reversion to cyst-like cells. This unique cell cycle of strain TTB310, with desiccation tolerant cyst-like cells capable of division and desiccation sensitive motile rods capable of dissemination, appears to be a novel adaptation for life in a hot and dry desert environment. In order to gain insights into strain TTB310's underlying genetic repertoire and possible mechanisms responsible for its unusual lifestyle, the genome of strain TTB310 was completely sequenced and subsequently annotated. The complete genome consists of a single circular chromosome of 4,070,194 bp with an average G+C content of 70.0%, the highest among the Betaproteobacteria sequenced to date, with total of 3,899 predicted coding sequences covering 92% of the genome. We found that strain TTB310 has developed a highly complex network of two-component systems, which may utilize responses to light and perhaps a rudimentary circadian hourglass to anticipate water availability at the dew time in the middle/end of the desert winter nights and thus direct the growth window to cyclic water availability times. Other interesting features of the strain TTB310 genome that appear to be important for desiccation tolerance, including intermediary metabolism compounds such as trehalose or polyhydroxyalkanoate, and signal transduction pathways, are presented and discussed