27,552 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
Schur Superpolynomials: Combinatorial Definition and Pieri Rule
Schur superpolynomials have been introduced recently as limiting cases of the
Macdonald superpolynomials. It turns out that there are two natural
super-extensions of the Schur polynomials: in the limit and
, corresponding respectively to the Schur
superpolynomials and their dual. However, a direct definition is missing. Here,
we present a conjectural combinatorial definition for both of them, each being
formulated in terms of a distinct extension of semi-standard tableaux. These
two formulations are linked by another conjectural result, the Pieri rule for
the Schur superpolynomials. Indeed, and this is an interesting novelty of the
super case, the successive insertions of rows governed by this Pieri rule do
not generate the tableaux underlying the Schur superpolynomials combinatorial
construction, but rather those pertaining to their dual versions. As an aside,
we present various extensions of the Schur bilinear identity
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