461 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
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