Degenerations and orbits in finite abelian groups

Abstract

A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads to an intuitive self-contained exposition of some of the basic facts concerning these orbits, including their enumeration. Given a partition $\lambda$, the lattice parametrizing orbits in a finite abelian p-group of type $\lambda$ is found to be independent of p. The order of the orbit corresponding to each parameter, which turns out to be a polynomial in p, is calculated. The description of orbits is extended to subquotients by certain characteristic subgroups. Each such characteristic subquotient is shown to have a unique maximal orbit.Comment: 14 pages, 5 figure

Optimal Line Packings From Finite Group actions

“…Our proof of Theorem 5.3 requires a detailed understanding of the orbits of Sp(K) on K, which we now review. The following is due to Dutta and Prasad [16,17]. By the Fundamental Theorem of Finitely Generated Abelian Groups, we can write A = p A p , where the direct product is over all primes p dividing |A|, and each A p has the form A p ∼ = Z p λ p,1 × · · · × Z p λ p,lp for a sequence λ p = (λ p,1 , . .…”

“…[16, Theorem 5.4]. The order ideals of P λp are in one-to-one correspondence with the Aut(A p )-orbits of A p [17,.…”

Combinatorics of finite abelian groups and Weil representations

“…We first recall the theory of orbits (under the full automorphism group) and characteristic subgroups in a finite abelian group from [6]. We then see how it applies to Sp(K)-orbits in K.…”

Permutation representations of the orbits of the automorphism group of a finite module over discrete valuation ring

“…A description of the orbits for this group action has been available for more than a hundred years (see Miller [5], Birkhoff [2], and Dutta-Prasad [3]). …”