Markus Schmidmeier

Invariant subspaces of nilpotent linear operators, I

Abstract Let k be a field. We consider triples (V, U, T), where V is a finite dimensional k-space, U a subspace of V and T : V → V a linear operator with Tn = 0 for some n, and such that T(U) U. Thus, T is a nilpotent operator on V, and U is an invariant subspace with respect to T. We will discuss the question whether it is possible to classify these triples. These triples (V, U, T) are the objects of a category with the Krull-Remak-Schmidt property, thus it will be suffcient to deal with indecomposable triples. Obviously, the classification problem depends on n, and it will turn out that the decisive case is n = 6. For n < 6, there are only finitely many isomorphism classes of indecomposable triples, whereas for n > 6 we deal with what is called “wild” representation type, so no complete classification can be expected. For n = 6, we will exhibit a complete description of all the indecomposable triples.

The Auslander-Reiten translation in submodule categories

Let A be an artin algebra or, more generally, a locally bounded associative algebra, and S(Λ) the category of all embeddings (A C B) where B is a finitely generated A-module and A is a submodule of B. Then S(Λ) is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in S(Λ) can be computed within mod A by using our construction of minimal monomorphisms. If in addition A is uniserial, then any indecomposable nonprojective object in S(Λ) is invariant under the sixth power of the Auslander-Reiten translation.

Submodule categories of wild representation type

Abstract Let Λ be a commutative local uniserial ring with radical factor field k . We consider the category S ( Λ ) of embeddings of all possible submodules of finitely generated Λ -modules. In case Λ = Z / 〈 p n 〉 , where p is a prime, the problem of classifying the objects in S ( Λ ) , up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that Λ has Loewy length at least seven. We show that S ( Λ ) is controlled k -wild with a single control object I ∈ S ( Λ ) . It follows that each finite dimensional k -algebra can be realized as a quotient End ( X ) / End ( X ) I of the endomorphism ring of some object X ∈ S ( Λ ) modulo the ideal End ( X ) I of all maps which factor through a finite direct sum of copies of I .

Bounded submodules of modules

Abstract Let m , n be positive integers such that m ⩽ n and k be a field. We consider all pairs ( B , A ) where B is a finite dimensional T n -bounded k [ T ] -module and A is a submodule of B which is T m -bounded. They form the objects of the submodule category S m ( k [ T ] / T n ) which is a Krull–Schmidt category with Auslander–Reiten sequences. The case m = n deals with submodules of k [ T ] / T n -modules and has been studied well. In this paper we determine the representation type of the categories S m ( k [ T ] / T n ) also for the cases where m n : It turns out that there are only finitely many indecomposables in S m ( k [ T ] / T n ) if either m 3 , n 6 , or ( m , n ) = ( 3 , 6 ) ; the category is tame if ( m , n ) is one of the pairs ( 3 , 7 ) , ( 4 , 6 ) , ( 5 , 6 ) , or ( 6 , 6 ) ; otherwise, S m ( k [ T ] / T n ) has wild representation type. Moreover, in each of the finite or tame cases we describe the indecomposables and picture the Auslander–Reiten quiver.

Operations on arc diagrams and degenerations for invariant subspaces of linear operators

Abstract: We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described. MSC 2010: Primary: 14L30, 16G20, Secondary: 16G70, 05C85, 47A15

Hall Polynomials via Automorphisms of Short Exact Sequences

We present a sum-product formula for the classical Hall polynomial which is based on tableaux that have been introduced by T. Klein in 1969. In the formula, each summand corresponds to a Klein tableau, while the product is taken over the cardinalities of automorphism groups of short exact sequences which are derived from the tableau. For each such sequence, one can read off from the tableau the summands in an indecomposable decomposition, and the size of their homomorphism and automorphism groups. Klein tableaux are refinements of Littlewood–Richardson tableaux in the sense that each entry ℓ ≥ 2 carries a subscript r. We describe module theoretic and categorical properties shared by short exact sequences which have the same symbol ℓr in a given row in their Klein tableau. Moreover, we determine the interval in the Auslander–Reiten quiver in which the indecomposable sequences of pn-bounded groups which carry such a symbol occur.

THE ENTRIES IN THE LR-TABLEAU

Let Γ be the Littlewood–Richardson tableau corresponding to an embedding M of a subgroup in a finite abelian p-group. Each individual entry in Γ yields information about the homomorphisms from M into a particular subgroup embedding, and hence determines the position of M within the category of subgroup embeddings. Conversely, this category provides a categorification for LR-tableaux in the sense that all subgroup embeddings corresponding to a given LR-tableau share certain homological properties.

Extensions of Simple Modules and the Converse of Schur’s Lemma

The converse of Schur’s lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL condition governs modules of finite length.

A family of noetherian rings with their finite length modules under control

We investigate the category modΛ of finite length modules over the ring Λ=A ⊗k Σ, where Σ is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module Ej gives rise to a quasiprogenerator Pj = A ⊗ Ej. By a result of K. Fuller, Pj induces a category equivalence from which we deduce that modΛ ≃ ∐j mod EndPj. As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring Λ such that modA ≃ modΛ(2) find twisted versions Λ of algebras of wild representation type such that Λ itself is of finite or tame representation type (in mod)(3) describe for certain rings Λ the minimal almost split morphisms in modΛ and observe that almost all of these maps are not almost split in ModΛ.

A Construction of Metabelian Groups

Abstract.In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of finite metabelian groups of order p22 tends to infinity with p. More precisely, for each prime number p there is a family (Mλ)λ=0,...,p−1 of indecomposable and pairwise nonisomorphic metabelian p-groups of the given order. In this manuscript we use recent results on the classification of possible embeddings of a subgroup in a finite abelian p-group to construct families of indecomposable metabelian groups, indexed by several parameters, which have upper bounds on the exponents of the center and the commutator subgroup.