44 research outputs found
-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
-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 -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
Long fully commutative elements in affine Coxeter groups
An element of a Coxeter group 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 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
Combinatorics of fully commutative involutions in classical Coxeter groups
An element of a Coxeter group 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 , 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 -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 -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 into bilateral Bailey pairs
related to . 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 -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 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 , 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 , this reproves a theorem of Barcucci et al.; in type , 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 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 est fini. Dans ce travail, nous considérons 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 cela redonne un théorème de Barcucci et al.; pour , cela simplifie et précise des résultats de Hanusa et Jones. Pour tous les autres groupes finis et affines, nos résultats sont nouveaux
Fully commutative elements and lattice walks
An element of a Coxeter group 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 , 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 , this reproves a theorem of Barcucci et al.; in type , it simplifies and refines results of Hanusa and Jones. For all other finite and affine groups, our results are new
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