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Lukas Katthän

Publications and source records attributed to Lukas Katthän.

At least 37 records · Page 2Linked to original sources
arXiv2016

Algebraic properties of ideals of poset homomorphisms

Given finite posets $P$ and $Q$, we consider a specific ideal $L(P,Q)$, whose minimal monomial generators correspond to order-preserving maps $ϕ:P\rightarrow Q$. We study algebraic invariants of those ideals. In particular, sharp lower and upper bounds for the Castelnuovo-Mumford regularity and the projective dimension are provided. Precise formulas are obtained for a large subclass of these ideals. Moreover, we provide complete characterizations for several algebraic properties of $L(P,Q)$, including being Buchsbaum, Cohen-Macaulay, Gorenstein and having a linear resolution. We also give a partial characterization for Golod property of $L(P,Q)$. Using those results, we derive applications for other important classes of monomial ideals, such as initial ideals of determinantal ideals and multichain ideals.

math.AC↗
arXiv2016

The Symmetric signature of cyclic quotient singularities

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

math.AC↗
arXiv2016

Which series are Hilbert series of graded modules over polynomial rings?

Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules. Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.

math.AC↗
arXiv2016

Betti posets and the Stanley depth

Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that \[ \operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\operatorname{sdepth} S/I = \operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion is best possible.

math.CO↗
arXiv2016

A Canonical module characterization of Serre's $(\mathrm{R}_1)$

In this short note, we give a characterization of domains satisfying Serre's condition $(\mathrm{R}_1)$ in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author (K. Yanagawa, Dualizing complexes of seminormal affine semigroup rings and toric face rings, J. Algebra 425 (2015).) where the normality is described in terms of the "shape" of the canonical module.

math.AC↗
arXiv2016

The Golod property for Stanley-Reisner rings in varying characteristic

We show that the Golod property of a Stanley-Reisner ring can depend on the characteristic of the base field. More precisely, for every finite set $T$ of prime numbers we construct simplicial complexes $Δ$ and $Γ$, such that $\mathbb{K}[Δ]$ is Golod exactly in the characteristics in $T$ and $\mathbb{K}[Γ]$ is Golod exactly in the characteristics not in $T$. Along the way, we show that a one-dimensional simplicial complex is Golod if and only if it is chordal.

math.AC↗
arXiv2015

How to compute the Stanley depth of a module

In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $M$ over the polynomial ring $\mathbb{K}[X_1, \ldots, X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog. Moreover, we show that the question whether $M$ has Stanley depth at least $r$ can be reduced to the question whether a certain combinatorially defined polytope $\mathscr{P}$ contains a $\mathbb{Z}^n$-lattice point.

math.AC↗
arXiv2015

Non-normal affine monoids

We give a geometric description of the set of holes in a non-normal affine monoid $Q$. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of $k[Q]$. From this, we see how various properties of $k[Q]$ like local normality and Serre's conditions $(R_1)$ and $(S_2)$ are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra $k[Q]$ is obtained and some cases where equality holds are identified. We apply this results to seminormal affine monoids.

math.AC↗
arXiv2015

Stanley depth and simplicial spanning trees

We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex. We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators we obtain a partial result.

math.AC↗
arXiv2015

Lcm-lattices and Stanley depth: a first computational approach

Let $\mathbb{K}$ be a field, and let $S=\mathbb{K}[X_1, ..., X_n]$ be the polynomial ring. Let $I$ be a monomial ideal of $S$ with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that $\mathrm{depth}_S S/I = \mathrm{sdepth}_S S/I < \mathrm{sdepth}_S I$. This shows that the Stanley conjecture is true for $S/I$ and $I$, if $I$ can be generated by at most 5 monomials. The result also brings additional computational evidence for a conjecture made by Herzog.

math.AC↗
arXiv2014

The Stanley Depth in the Upper Half of the Koszul Complex

Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this paper, we prove that the $k$-th syzygy module of the residue class field $K$ of $R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than $1$. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.

math.CO↗
arXiv2014

The behavior of Stanley depth under polarization

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\ {I^p/J^p}$, where $\mathrm{sdepth}\ I/J$ denotes the Stanley depth and $I^p$ denotes the polarization. This solves a conjecture by Herzog and reduces the famous Stanley conjecture (for modules of the form $I/J$) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form $R/I$ and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent.

math.AC↗
arXiv2014

Two lower bounds for the Stanley depth of monomial ideals

Let $J\varsubsetneq I$ be two monomial ideals of the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n]$. In this paper, we provide two lower bounds for the Stanley depth of $I/J$. On the one hand, we introduce the notion of lcm number of $I/J$, denoted by $l(I/J)$, and prove that the inequality ${\rm sdepth}(I/J)\geq n-l(I/J)+1$ hold. On the other hand, we show that ${\sdepth}(I/J)\geq n-\dim L_{I/J}$, where $\dim L_{I/J}$ denotes the order dimension of the lcm lattice of $I/J$. We show that $I$ and $S/I$ satisfy Stanley's conjecture, if either the lcm number of $I$ or the order dimension of the lcm lattice of $I$ is small enough. Among other results, we also prove that the Stanley--Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.

math.AC↗
arXiv2013

Normal cyclic polytopes and cyclic polytopes that are not very ample

Let $d$ and $n$ be positive integers with $n \geq d + 1$ and $τ_{1}, ..., τ_{n}$ integers with $τ_{1} < ... < τ_{n}$. Let $C_{d}(τ_{1}, ..., τ_{n}) \subset \RR^{d}$ denote the cyclic polytope of dimension $d$ with $n$ vertices $(τ_{1},τ_{1}^{2},...,τ_{1}^{d}), ..., (τ_{n},τ_{n}^{2},...,τ_{n}^{d})$. We are interested in finding the smallest integer $γ_{d}$ such that if $τ_{i+1} - τ_{i} \geq γ_{d}$ for $1 \leq i < n$, then $C_{d}(τ_{1}, ..., τ_{n})$ is normal. One of the known results is $γ_{d} \leq d (d + 1)$. In the present paper a new inequality $γ_{d} \leq d^{2} - 1$ is proved. Moreover, it is shown that if $d \geq 4$ with $τ_{3} - τ_{2} = 1$, then $C_{d}(τ_{1}, ..., τ_{n})$ is not very ample.

math.CO↗
arXiv2013

Polytopal affine semigroups with holes deep inside

Given a non-negative integer k, we construct a lattice 3-simplex P with the following property: The affine semigroup Q_P associated to P is not normal, and every element $q \in \sat{Q}_P \setminus Q_P$ has lattice distance at least k above every facet of Q_P.

math.CO↗
arXiv2012

Decomposing Sets of Inversions

In this note we consider the question how the set of inversions of a permutation $π\in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a graph theoretical result. For this we establish a correspondence between the substitution decomposition of $π$ and the modular decomposition of its inversion graph. We also consider the special case of multiplicative decompositions.

math.CO↗
arXiv2012

Network representations of non-equilibrium steady states: Cycle decompositions, symmetries and dominant paths

Non-equilibrium steady states (NESS) of Markov processes give rise to non-trivial cyclic probability fluxes. Cycle decompositions of the steady state offer an effective description of such fluxes. Here, we present an iterative cycle decomposition exhibiting a natural dynamics on the space of cycles that satisfies detailed balance. Expectation values of observables can be expressed as cycle "averages", resembling the cycle representation of expectation values in dynamical systems. We illustrate our approach in terms of an analogy to a simple model of mass transit dynamics. Symmetries are reflected in our approach by a reduction of the minimal number of cycles needed in the decomposition. These features are demonstrated by discussing a variant of an asymmetric exclusion process (TASEP). Intriguingly, a continuous change of dominant flow paths in the network results in a change of the structure of cycles as well as in discontinuous jumps in cycle weights.

math-ph↗