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We consider the category $\mathcal S(n)$ of all pairs $X = (U,V)$, where $V$ is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n = 0$, and $U$ is a subspace of $V$ such that $T(U) \subseteq U$. Our main interest in an object $X=(U,V)$ are the three numbers $uX=\dim U$ (for the subspace), $wX=\dim V/U$ (for the factor) and $bX=\dim {\rm Ker} T$ (for the operator). Actually, instead of looking at the reference space $\Bbb R^3$ with the triples $(uX,wX,bX)$, we will focus the attention to the corresponding projective space $\Bbb T(n)$ which contains for a non-zero object $X$ the level-colevel pair {\bf pr}$X = (uX/bX,wX/bX)$ supporting the object $X$. We use $\Bbb T(n)$ to visualize part of the categorical structure of $\mathcal S(n)$: The action of the duality $D$ and the square $\tau_n^2$ of the Auslander-Reiten translation are represented on $\Bbb T(n)$ by a reflection and a rotation by $120^\circ$ degrees, respectively. Moreover for $n\geq 6$, each component of the Auslander-Reiten quiver of $\mathcal S(n)$ has support either contained in the center of $\Bbb T(n)$ or with the center as its only accumulation point. We show that the only indecomposable objects $X$ in $\mathcal S(n)$ with support having boundary distance smaller than 1 are objects with $bX=1$ which lie on the boundary, whereas any rational vector in $\Bbb T(n)$ with boundary distance at least 2 supports infinitely many indecomposable objects. At present, it is not clear at all what happens for vectors with boundary distance between 1 and 2. The use of $\Bbb T(n)$ provides even in the (quite well-understood) case $n = 6$ some surprises: In particular, we will show that any indecomposable object in $\mathcal S(6)$ lies on one of 12 central lines in $\Bbb T(6)$. The paper is essentially self-contained, all prerequisites which are needed are outlined in detail., Comment: The manuscript has 137 illustrations

In his 1934 paper, G.Birkhoff poses the problem of classifying pairs $(G,U)$, where $G$ is an abelian group and $U\subset G$ a subgroup, up to automorphisms of $G$. In general, Birkhoff's Problem is not considered feasible. In this note, we fix a prime number $p$ and assume that $G$ is a direct sum of cyclic $p$-groups and $U\subset G$ is a subgroup. Under the assumption that the factor group $G/U$ is $p$-bounded, we show that each such pair $(G,U)$ has a unique direct sum decomposition into pairs of type $(\mathbb Z/(p^n),\mathbb Z/(p^n))$ or $(\mathbb Z/(p^n), (p))$. On the other hand, if we assume $U$ is a $p$-bounded subgroup of $G$ and in addition that $G/U$ is a direct sum of cyclic $p$-groups, we show that each such pair $(G,U)$ has a unique direct sum decomposition into pairs of type $(\mathbb Z/(p^n),0)$ or $(\mathbb Z/(p^n), (p^{n-1}))$. We generalize the above results to modules over commutative discrete valuation rings.

The Kernel Complex Lemma states that given commutative diagram with exact rows and exact columns which covers the region under a $\Gamma$-shape, then the kernel sequence on the top and the kernel sequence at the left have in each position isomorphic homology. The dual version, the Cokernel Complex Lemma, has been used to determine the support of a finitely presented functor on the Auslander-Reiten quiver. We note that the Snake Lemma is a consequence of the two results combined., Comment: 7 pages

For $\Lambda$ a selfinjective algebra, and $Q$ a finite quiver without oriented cycles, the algebra $\Lambda Q$ is a Gorenstein algebra and the category ${\rm Gproj}\Lambda Q$ of Gorenstein-projective $\Lambda Q$-modules is a Frobenius category. For a sink $v$ of $Q$, we define a functor $F(v) : \underline{\rm Gproj}\Lambda Q\to \underline{\rm Gproj}\Lambda Q(v)$ between the stable categories modulo projectives, where $Q(v)$ is obtained from $Q$ by changing the direction of each arrow ending in $v$. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that $F(v)$ is an equivalence of categories. In the case where the underlying graph of $Q$ is a tree, we deduce that the stable category $\underline{\rm Gproj}\Lambda Q$ does not depend on the orientation of $Q$. Moreover, if $Q$ is a quiver of type $\mathbb A_3$ and $\Lambda=k[T]/(T^n)$ the bounded polynomial algebra, we use the symmetry of the octahedron in the octahedral axiom to verify that the composition of twelve reflections yields the identity on objects., Comment: 35 pages, 8 figures

Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered a module over a subalgebra. When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor. The key tool is the Cokernel Complex Lemma which links the values of the hammock function to the Auslander-Reiten structure of the category. We are also interested in exact subcategories of module categories which have Auslander-Reiten sequences. Our examples include quiver representations and invariant subspaces of nilpotent linear operators., Comment: 28 pages, 6 figures picturing hammocks on Auslander-Reiten quivers

Like the LR-tableau, a socle tableau is given as a skew diagram with certain entries. Unlike in the LR-tableau, the entries in the socle tableau are weakly increasing in each row, strictly increasing in each column and satisfy a modified lattice permutation property. In the study of embeddings of a subgroup in a finite abelian $p$-group, socle tableaux occur as isomorphism invariants, they are given by the socle series of the subgroup. We show that each socle tableau can be realized by some embedding. Moreover, the socle tableau of an embedding and the LR-tableau of the dual embedding determine each other.

In this paper we generalize Kaplansky's combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group given in his 1951 book ``Infinite Abelian Groups''. For this we introduce partial maps on Littlewood-Richardson tableaux and show that they characterize the isomorphism types of finite direct sums of such cyclic embeddings., Comment: arXiv admin note: substantial text overlap with arXiv:1607.05640

The PLR-moves of neo-Riemannian theory, when considered as reflections on the edges of an equilateral triangle, define the Coxeter group $\widetilde S_3$. The elements are in a natural one-to-one correspondence with the triangles in the infinite Tonnetz. The left action of $\widetilde S_3$ on the Tonnetz gives rise to interesting chord sequences. We compare the system of transformations in $\widetilde S_3$ with the system of Schritte and Wechsel introduced by Hugo Riemann in 1880. Finally, we consider the point reflection group as it captures well the transition from Riemann's infinite Tonnetz to the finite Tonnetz of neo-Riemannian theory., Comment: 14 pages for the Mathematics and Computation in Music Conference in June 2019 in Madrid, the revised version extends the music theoretic discussion

In the Pythagorean tuning system, the fifth is used to generate a scale of 12 notes per octave. In this paper, we use the octave to generate a scale of 19 notes per tritave; one can play this scale on a traditional piano. In this system, the octave becomes a proper interval and the 2:3:4 chord a proper chord. We study harmonic properties obtained from the 2:3:4 chord, in particular composition elements using dominants, subdominants, higher dominants, associated minor chords, inversions, and diminished chords. The Tonnetz (array notation) turns out to be an effective tool to visualize the harmonic development in a composition based on these elements. 2:3:4-harmony may sound pure, yet sparse, as we illustrate in a short piece., Comment: In the new version, some remarks about music perception have been added. 29 pages, 22 figures, 7 tables. To appear in the Journal of Mathematics and Music

For a partition $\beta$, denote by $N_\beta$ the nilpotent linear operator of Jordan type $\beta$. Given partitions $\beta$, $\gamma$, we investigate the representation space ${}_2{\mathbb V}_{\gamma}^\beta$ of all short exact sequences $$ \mathcal E: 0\to N_\alpha \to N_\beta \to N_\gamma \to 0$$ where $\alpha$ is any partition with each part at most 2. Due to the condition on $\alpha$, the isomorphism type of a sequence $\mathcal E$ is given by an arc diagram $\Delta$; denote by ${\mathbb V}_\Delta$ the subset of ${}_2{\mathbb V}_{\gamma}^\beta$ of all sequences isomorphic to $\mathcal E$. Thus, the space ${}_2{\mathbb V}_{\gamma}^\beta$ carries a stratification given by the subsets of type ${\mathbb V}_\Delta$. We compute the dimension of each stratum and show that the boundary of a stratum ${\mathbb V}_\Delta$ consists exactly of those ${\mathbb V}_{\Delta'}$ where $\Delta'$ is obtained from $\Delta$ by a non-empty sequence of arc moves of five possible types {\bf (A) -- (E)}. The case where all three partitions are fixed has been studied in [3] and [4]. There, arc moves of types {\bf (A) -- (D)} suffice to describe the boundary of a ${\mathbb V}_\Delta$ in ${\mathbb V}_{\alpha,\gamma}^\beta$. Our fifth move {\bf (E)}, "explosion", is needed to break up an arc into two poles to allow for changes in the partition $\alpha$.

In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. We then focus on an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.

In this manuscript we show that two partial orders defined on the set of standard Young tableaux of shape $\alpha$ are equivalent. In fact, we give two proofs for the equivalence of the box order and the dominance order for {tableaux}. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of the cover relations. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.

We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\to V$ and two $T$-invariant subspaces $U_1\subset U_2\subset V$. Let $\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $\mathcal S(n)$ for $n\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\leq 4$. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in $(x,y,z)\in (3,1,3)+\mathbb N(2,2,2)$ can be realized, while each neighbor $(x\pm1,y,z), (x,y\pm1,z),(x,y,z\pm1)$ can. Compare this with Bongartz' No-Gap Theorem, which states that for an associative algebra $A$ over an algebraically closed field, there is no gap in the lengths of the indecomposable $A$-modules of finite dimension.

Given partitions $\alpha$, $\beta$, $\gamma$, the short exact sequences $0\to N_\alpha \to N_\beta \to N_\gamma \to 0$ of nilpotent linear operators of Jordan types $\alpha$, $\beta$, $\gamma$, respectively, define a constructible subset $\mathbb V_{\alpha,\gamma}^\beta$ of an affine variety. Geometrically, the varieties $\mathbb V_{\alpha,\gamma}^\beta$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $\Gamma$ of shape $(\alpha,\beta,\gamma)$ contributes one irreducible component $\overline{\mathbb V}_\Gamma$. We consider the partial order $\Gamma\leq_{\sf bound}^*\widetilde{\Gamma}$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetilde{\Gamma}}\cap \overline{\mathbb V}_\Gamma\neq \emptyset$. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of $\alpha$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $\beta\setminus\gamma$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.

Let $k$ be an algebraically closed field and $\alpha$, $\beta$, $\gamma$ be partitions. An algebraic group acts on the constructible set of short exact sequences of nilpotent $k$-linear operators of Jordan types $\alpha$, $\beta$, and $\gamma$, respectively; we are interested in the stratification given by the orbits in the case where all parts of $\alpha$ are at most 2. Geometric properties of the degeneration relation are controlled by the combinatorics of arc diagrams. The extended bubble sort algorithm is used to construct chains of orbits such that subsequent strata have dimension difference equal to one.

For an indecomposable module $M$ over a path algebra of a quiver of type $\widetilde{\mathbb A}_n$, the Gabriel-Roiter measure gives rise to four new numerical invariants; we call them the multiplicity, and the initial, periodic and final parts. We describe how these invariants for $M$ and for its dual specify the position of $M$ in the Auslander-Reiten quiver of the algebra., Comment: 29 pages; 6 figures; references added to Section 1 as per the referee's suggestions; to appear in Communications in Algebra

We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive 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.

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 $\ell \geq 2$ carries a subscript $r$. We describe module theoretic and categorical properties shared by short exact sequences which have the same symbol $\ell_r$ in a given row in their Klein tableau. Moreover, we determine the interval in the Auslander-Reiten quiver in which indecomposable sequences of $p^n$-bounded groups which carry such a symbol occur., Comment: 35 pages

Let $\Gamma$ be the Littlewood-Richardson tableau corresponding to an embedding $M$ of a subgroup in a finite abelian $p$-group. Each individual entry in $\Gamma$ 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., Comment: 14 pages, the proof of Theorem 1 is simplified

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., Comment: 8 pages, no figures, to appear, Proceedings of the International Conference on Algebra and its Applications, Ohio University, Athens, Ohio, June 18-21, 2008 (Birkh\"auser). Expository changes suggested by referee have been incorporated

Let $\alpha$, $\beta$, and $\gamma$ be partitions describing the isomorphism types of the finite abelian $p$-groups $A$, $B$, and $C$. From theorems by Green and Klein it is well-known that there is a short exact sequence $0\to A\to B\to C\to 0$ of abelian groups if and only if there is a Littlewood-Richardson sequence of type $(\alpha,\beta,\gamma)$. Starting from the observation that a sequence of partitions has the LR property if and only if every subsequence of length 2 does, we demonstrate how LR-sequences of length two correspond to embeddings of a $p^2$-bounded subgroup in a finite abelian $p$-group. Using the known classification of all such embeddings we derive short proofs of the theorems by Green and Klein., Comment: 5 pages

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 $T \:V \to V$ a linear operator with $T^n = 0$ for some $n$, and such that $T(U) \subseteq U$. 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 sufficient 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 indecomposables 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., Comment: 55 pages, minor modification in (0.1.3), to appear in: Journal fuer die reine und angewandte Mathematik

Let $R$ be a commutative local uniserial ring of length $n$, $p$ a generator of the maximal ideal, and $k$ the radical factor field. The pairs $(B,A)$ where $B$ is a finitely generated $R$-module and $A\subset B$ a submodule of $B$ such that $p^mA=0$ form the objects in the category $S_m(R)$. We show that in case $m=2$ the categories $S_m(R)$ are in fact quite similar to each other: If also $R'$ is a commutative local uniserial ring of length $n$ and with radical factor field $k$, then the categories $S_2(R)/\mathcal N_R$ and $S_2(R')/\mathcal N_{R'}$ are equivalent for certain nilpotent categorical ideals $N_R$ and $N_{R'}$. As an application, we recover the known classification of all pairs $(B,A)$ where $B$ is a finitely generated abelian group and $A\subset B$ a subgroup of $B$ which is $p^2$-bounded for a given prime number $p$., Comment: 14 pages, to appear in Journal of Algebra and its Applications

Fix a poset $P$ and a natural number $n$. For various commutative local rings $\Lambda$, each of Loewy length $n$, consider the category $\textrm{sub}_\Lambda P$ of $\Lambda$-linear submodule representations of $P$. We give a criterion for when the underlying translation quiver of a connected component of the Auslander-Reiten quiver of $\textrm{sub}_\Lambda P$ is independent of the choice of the base ring $\Lambda$. If $\mathcal P$ is the one-point poset and $\Lambda=\mathbb Z/p^n$ for $p$ a prime number, then $\textrm{sub}_\Lambda P$ consists of all pairs $(B;A)$ where $B$ is a finite abelian $p^n$-bounded group and $A\subset B$ a subgroup. We can respond to a remark by M. C. R. Butler concerning the first occurence of parametrized families of such subgroup embeddings., Comment: 25 pages, nicer diagrams

Let $\Lambda$ be an artin algebra and $S(\Lambda)$ the category of all embeddings $(A\subseteq B)$ where $B$ is a finitely generated $\Lambda$-module and $A$ is a submodule of $B$. Then $S(\Lambda)$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $S(\Lambda)$ can be computed within the category of $\Lambda$-modules by using our construction of minimal monomorphisms. If in addition $\Lambda$ is uniserial then any nonprojective indecomposable object in $\Cal S(\Lambda)$ is invariant under the sixth power of the Auslander-Reiten translation., Comment: Dedicated to Idun Reiten

Let $\Lambda$ be a commutative local uniserial ring of length at least seven with radical factor ring $k$. We consider the category $S(\Lambda)$ of all possible embeddings of submodules of finitely generated $\Lambda$-modules and show that $S(\Lambda)$ is controlled $k$-wild with a single control object $I\in S(\Lambda)$. In particular, 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 $X\in S(\Lambda)$ modulo the ideal $\End(X)_I$ of all maps which factor through a finite direct sum of copies of $I$., Comment: 13 pages

Let $m$, $n$ be positive integers such that $m\leq n$. 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 manuscript we determine the representation type of the categories $S_m(k[T]/T^n)$ also for the cases where $m

In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of finite metabelian groups of order $p^{22}$ tends to infinity with $p$. More precisely, for each prime number $p$ there is a family $(M_\lambda)_{\lambda=0,...,p-1}$ 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., Comment: 5 pages; to appear in Archiv der Mathematik