Searcharxiv⌕ Search

arXiv subjects

Markus Schmidmeier

Publications and source records attributed to Markus Schmidmeier.

At least 19 recordsLinked to original sources
arXiv2025

Gorenstein-Projective Modules over the Ring of Dual Integers

The ring of dual integers is the bounded polynomial ring $\mathbb Z[\eps]=\mathbb Z[T]/(T^2)$ with integer coefficients. We describe the (finitely generated) Gorenstein-projective $\mathbb Z[\eps]$-modules as the torsionless $\mathbb Z[\eps]$-modules, while the stable category of $\Gproj\mathbb Z[\eps]$ modulo projectives is shown to be equivalent to the orbit category $\mathcal D^b(\mathbb Z)/[1]$ of the derived category of the integers. It follows that the latter carries the structure of a triangulated category. \smallskip The category $\Gproj\mathbb Z[\eps]$ is related to the embeddings of a subgroup in a free abelian group and has a quotient which is equivalent to the category of finite abelian groups. In fact, we present a cube which has as vertices eight related categories and as edges in each of the three directions functors which are related to push-down functors modulo the shift; canonical functors to stable categories; and homology functors, respectively. We note that in $\Gproj\mathbb Z[\eps]$ uniqueness of direct sum decomposition fails.

math.RT↗
arXiv2024

Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle $\Bbb T(n)$

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.

math.RT↗
arXiv2023

Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient

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 an elementary abelian $p$-group, we show that the pair $(G,U)$ always has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),\mathbb Z/(p^n))$ or $(\mathbb Z/(p^n), (p))$. Surprisingly, in the dual situation we need an additional condition. If we assume that $U$ itself is an elementary subgroup of $G$, then we show that the pair $(G,U)$ has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),0)$ or $(\mathbb Z/(p^n), (p^{n-1}))$ if and only if $G/U$ is a~direct sum of cyclic $p$-groups. We generalize the above results to modules over commutative discrete valuation rings.

math.GR↗
arXiv2023

Snake lemma variations

The Kernel Complex Lemma states that given commutative diagram with exact rows and exact columns which covers the region under a $Γ$-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.

math.RT↗
arXiv2022

A reflection equivalence for Gorenstein-projective quiver representations

For $Λ$ a selfinjective algebra, and $Q$ a finite quiver without oriented cycles, the algebra $ΛQ$ is a Gorenstein algebra and the category ${\rm Gproj}ΛQ$ of Gorenstein-projective $ΛQ$-modules is a Frobenius category. For a sink $v$ of $Q$, we define a functor $F(v) : \underline{\rm Gproj}ΛQ\to \underline{\rm Gproj}ΛQ(v)$ 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 $\underline{\rm Gproj}ΛQ$ does not depend on the orientation of $Q$. Moreover, if $Q$ is a quiver of type $\mathbb A_3$ and $Λ=k[T]/(T^n)$ 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.

math.RT↗
arXiv2021

Hammocks to visualize the support of finitely presented functors

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.

math.RT↗
arXiv2020

The socle tableau as a dual version of the Littlewood-Richardson tableau

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.

math.RT↗
arXiv2019

Finite direct sums of cyclic embeddings

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.

math.RT↗
arXiv2019

Box moves on Littlewood-Richardson tableaux and an application to invariant subspace varieties

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.

math.RT↗
arXiv2019

2:3:4-Harmony within the Tritave

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.

cs.SD↗
arXiv2019

From Schritte and Wechsel to Coxeter Groups

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.

math.CO↗
arXiv2016

The boundary of the irreducible components for invariant subspace varieties

Given partitions $α$, $β$, $γ$, the short exact sequences $0\to N_α\to N_β\to N_γ\to 0$ of nilpotent linear operators of Jordan types $α$, $β$, $γ$, respectively, define a constructible subset $\mathbb V_{α,γ}^β$ of an affine variety. Geometrically, the varieties $\mathbb V_{α,γ}^β$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $Γ$ of shape $(α,β,γ)$ contributes one irreducible component $\overline{\mathbb V}_Γ$. We consider the partial order $Γ\leq_{\sf bound}^*\widetildeΓ$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetildeΓ}\cap \overline{\mathbb V}_Γ\neq \emptyset$. 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 $α$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $β\setminusγ$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.

math.RT↗
arXiv2016

Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators. Part II

For a partition $β$, denote by $N_β$ the nilpotent linear operator of Jordan type $β$. Given partitions $β$, $γ$, we investigate the representation space ${}_2{\mathbb V}_γ^β$ of all short exact sequences $$ \mathcal E: 0\to N_α\to N_β\to N_γ\to 0$$ where $α$ is any partition with each part at most 2. Due to the condition on $α$, the isomorphism type of a sequence $\mathcal E$ is given by an arc diagram $Δ$; denote by ${\mathbb V}_Δ$ the subset of ${}_2{\mathbb V}_γ^β$ of all sequences isomorphic to $\mathcal E$. Thus, the space ${}_2{\mathbb V}_γ^β$ carries a stratification given by the subsets of type ${\mathbb V}_Δ$. We compute the dimension of each stratum and show that the boundary of a stratum ${\mathbb V}_Δ$ consists exactly of those ${\mathbb V}_{Δ'}$ where $Δ'$ is obtained from $Δ$ 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 ${\mathbb V}_Δ$ in ${\mathbb V}_{α,γ}^β$. Our fifth move {\bf (E)}, "explosion", is needed to break up an arc into two poles to allow for changes in the partition $α$.

math.RT↗
arXiv2015

Two Partial Orders for Standard Young Tableaux

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.

math.RT↗
arXiv2014

The Swiss Cheese Theorem for Linear Operators with Two Invariant Subspaces

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.

math.RT↗
arXiv2013

Arc diagram varieties

Let $k$ be an algebraically closed field and $α$, $β$, $γ$ be partitions. An algebraic group acts on the constructible set of short exact sequences of nilpotent $k$-linear operators of Jordan types $α$, $β$, and $γ$, respectively; we are interested in the stratification given by the orbits in the case where all parts of $α$ 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.

math.RT↗
arXiv2012

The Auslander-Reiten Components in the Rhombic Picture

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.

math.RT↗