The Birkhoff problem: How to classify subgroups of finite abelian groups?


Claus Michael Ringel

Universitšt Bielefeld

Northeastern University

Thursday, October 7, 2004


Talk at 4:30 p.m. in 509 Lake Hall

Tea at 4:00 p.m. in 544 Nightingale Hall


Abstract:   The structure of finite abelian groups is well-known and easy to describe: they are direct sums of indecomposables, and the indecomposables are cyclic groups of order pn, where p is a prime number. It follows that the isomorphism classes of finite abelian p-groups correspond bijectively to the partitions. If we look at subgroups of such groups, they can be described again in this way. But we want to keep track of the embedding, thus we want to find normal forms for the pairs (A0,A1) where A0 is a finite abelian p-group and A1 a subgroup of A0. This problem was posed in 1934 by Birkhoff: he showed that the difficulties increase with the growth of the exponent of A0 and he pointed out that the first real problem arises in case the exponent of A0 is equal to 6. There is a related problem: to classify invariant subspaces of a nilpotent operator on a vector space. The general situation to be discussed is the following: Let L be a commutative uniserial ring of length n (for example Z/pn or k[T]/Tn, where k is a field). Consider the category S(L) of pairs (A0,A1), where A0 is a L-module and A1 a submodule. In case n5, the category S(L) contains only finitely many indecomposable objects, namely 5, 10, 20, or 50, in case n = 2,3,4, or 5, respectively. It is of interest that these cases are related to the Dynkin diagrams A2, D4, E6 and E8, but also that the classification is independent of L. The case n=6 is of special interest: for L = k[T]/T6 a full classication of the indecomposable objects can be given, using two parameters (in Q and in N) which are purely combinatorially, and in addition one which takes values in the projective line over the field k. One may wonder whether there is a similar classification for L = Z/p6. For any n, the categories S(L) are (non-abelian) Frobenius categories and the corresponding stable categories seem to be of special interest. The distinction between the cases n<6, n=6 and n>6 (finite, tame and wild representation type) may be formulated as elliptic, parabolic or hyperbolic behavior. The results presented are joint work with Markus Schmidmeier (Florida Atlantic University).

Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall.

There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from Maxim Braverman.

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Posted: March 17, 2004    URL: