# Equivariant Euler characteristics of unitary buildings

Journal of Algebraic Combinatorics

Published On 2021/11

The (p-primary) equivariant Euler characteristics of the buildings for the general unitary groups over finite fields are determined.

Journal

Journal of Algebraic Combinatorics

Published On

2021/11

Volume

54

Issue

3

Page

915-946

## Authors

#### Jesper Møller

##### Aalborg Universitet

Position

Professor in Statistics

H-Index(all)

46

H-Index(since 2020)

23

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Statistics

Probability Theory

University Profile Page

### Other Articles from authors

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.09525

##### Coupling results and Markovian structures for number representations of continuous random variables

A general setting for nested subdivisions of a bounded real set into intervals defining the digits of a random variable with a probability density function is considered. Under the weak condition that is almost everywhere lower semi-continuous, a coupling between and a non-negative integer-valued random variable is established so that have an interpretation as the ``sufficient digits'', since the distribution of conditioned on does not depend on . Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, becomes a Markov chain of a certain order . If then are IID with a known distribution. When and the Markov chain is uniformly geometric ergodic, a coupling is established between and a random time so that the chain after time is stationary and follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base- expansions, generalized L\"uroth series, -expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.

*2024/4/15*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.08387

##### The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let be the base- expansion of a continuous random variable on the unit interval where is the positive solution to for an integer (i.e., is a generalization of the golden mean for which ). We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2024/4/12*

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

##### How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

*2024/3*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2312.09652

##### The asymptotic distribution of the remainder in a certain base- expansion

Let be the base- expansion of a continuous random variable on the unit interval where is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2023/12/15*

Jesper Møller

Aalborg Universitet

Proceedings of the London Mathematical Society

##### Realizability and tameness of fusion systems

A saturated fusion system over a finite p$p$‐group S$S$ is a category whose objects are the subgroups of S$S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over S$S$ is realized by a finite group G$G$ if S$S$ is a Sylow p$p$‐subgroup of G$G$ and morphisms in the category are those induced by conjugation in G$G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the …

*2023/12*

Jesper Møller

Aalborg Universitet

ACM Transactions on Spatial Algorithms and Systems

##### Stochastic Routing with Arrival Windows

Arriving at a destination within a specific time window is important in many transportation settings. For example, trucks may be penalized for early or late arrivals at compact terminals, and early and late arrivals at general practitioners, dentists, and so on, are also discouraged, in part due to COVID. We propose foundations for routing with arrival-window constraints. In a setting where the travel time of a road segment is modeled by a probability distribution, we define two problems where the aim is to find a route from a source to a destination that optimizes or yields a high probability of arriving within a time window while departing as late as possible. In this setting, a core challenge is to enable comparison between paths that may potentially be part of a result path with the goal of determining whether a path is uninteresting and can be disregarded given the existence of another path. We show that existing solutions …

*2023/11/21*

Jesper Møller

Aalborg Universitet

Spatial Statistics

##### Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel …

*2023/6/1*

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

##### Singular distribution functions for random variables with stationary digits

Let F be the cumulative distribution function (CDF) of the base-q expansion , where is an integer and is a stationary stochastic process with state space . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

*2023/3*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2212.08402

##### Cox processes driven by transformed Gaussian processes on linear networks

There is a lack of point process models on linear networks. For an arbitrary linear network, we use isotropic covariance functions with respect to the geodesic metric or the resistance metric to construct new models for isotropic Gaussian processes and hence new models for various Cox processes with isotropic pair correlation functions. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

*2022/12/16*

Jesper Møller

Aalborg Universitet

Stat

##### Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

*2022/12*

Jesper Møller

Aalborg Universitet

International Statistical Review

##### Should we condition on the number of points when modelling spatial point patterns?

We discuss the practice of directly or indirectly assuming a model for the number of points when modelling spatial point patterns even though it is rarely possible to validate such a model in practice because most point pattern data consist of only one pattern. We therefore explore the possibility to condition on the number of points instead when fitting and validating spatial point process models. In a simulation study with different popular spatial point process models, we consider model validation using global envelope tests based on functional summary statistics. We find that conditioning on the number of points will for some functional summary statistics lead to more narrow envelopes and thus stronger tests and that it can also be useful for correcting for some conservativeness in the tests when testing composite hypothesis. However, for other functional summary statistics, it makes little or no difference to condition …

*2022/12*

Jesper Møller

Aalborg Universitet

Journal of Applied Probability

##### Characterization of random variables with stationary digits

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

*2022/12*

Jesper Møller

Aalborg Universitet

Spatial Statistics

##### Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step …

*2022/10/1*

Jesper Møller

Aalborg Universitet

Translational psychiatry

##### Layer III pyramidal cells in the prefrontal cortex reveal morphological changes in subjects with depression, schizophrenia, and suicide

Brodmann Area 46 (BA46) has long been regarded as a hotspot of disease pathology in individuals with schizophrenia (SCH) and major depressive disorder (MDD). Pyramidal neurons in layer III of the Brodmann Area 46 (BA46) project to other cortical regions and play a fundamental role in corticocortical and thalamocortical circuits. The AutoCUTS-LM pipeline was used to study the 3-dimensional structural morphology and spatial organization of pyramidal cells. Using quantitative light microscopy, we used stereology to calculate the entire volume of layer III in BA46 and the total number and density of pyramidal cells. Volume tensors estimated by the planar rotator quantified the volume, shape, and nucleus displacement of pyramidal cells. All of these assessments were carried out in four groups of subjects: controls (C, n = 10), SCH (n = 10), MDD (n = 8), and suicide subjects with a history of depression (SU …

*2022/9/5*

Jesper Møller

Aalborg Universitet

Graphs and Combinatorics

##### Equivariant Euler characteristics of symplectic buildings

We compute the equivariant Euler characteristics of the buildings for the symplectic groups over finite fields.

*2022/6*

Jesper Møller

Aalborg Universitet

Journal of Computational and Graphical Statistics

##### MCMC computations for Bayesian mixture models using repulsive point processes

Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, repulsive mixture models produce a smaller number of well-separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the “cluster centers” is a finite repulsive point process depending on a hyperparameter, specified by a density which may depend on an intractable normalizing constant. By investigating the posterior characterization of this class of mixture models, we derive a MCMC algorithm which avoids the well-known difficulties associated to reversible jump MCMC computation. In particular, we use an ancillary variable method, which eliminates the problem of having intractable …

*2022/4/3*

Jesper Møller

Aalborg Universitet

Scandinavian Journal of Statistics

##### Approximate Bayesian inference for a spatial point process model exhibiting regularity and random aggregation

In this article, we propose a doubly stochastic spatial point process model with both aggregation and repulsion. This model combines the ideas behind Strauss processes and log Gaussian Cox processes. The likelihood for this model is not expressible in closed form but it is easy to simulate realizations under the model. We therefore explain how to use approximate Bayesian computation (ABC) to carry out statistical inference for this model. We suggest a method for model validation based on posterior predictions and global envelopes. We illustrate the ABC procedure and model validation approach using both simulated point patterns and a real data example.

*2022/3*

Jesper Møller

Aalborg Universitet

Communications Biology

##### Cellular 3D-reconstruction and analysis in the human cerebral cortex using automatic serial sections

Techniques involving three-dimensional (3D) tissue structure reconstruction and analysis provide a better understanding of changes in molecules and function. We have developed AutoCUTS-LM, an automated system that allows the latest advances in 3D tissue reconstruction and cellular analysis developments using light microscopy on various tissues, including archived tissue. The workflow in this paper involved advanced tissue sampling methods of the human cerebral cortex, an automated serial section collection system, digital tissue library, cell detection using convolution neural network, 3D cell reconstruction, and advanced analysis. Our results demonstrated the detailed structure of pyramidal cells (number, volume, diameter, sphericity and orientation) and their 3D spatial organization are arranged in a columnar structure. The pipeline of these combined techniques provides a detailed analysis of tissues …

*2021/9/2*

Jesper Møller

Aalborg Universitet

AMERICAN MATHEMATICAL SOCIETY

##### THE NUMBER OF p-ELEMENTS IN FINITE GROUPS OF LIE TYPE OF CHARACTERISTIC p

The combinatorics of the poset of p-radical p-subgroups of a finite group is used to count the number of p-elements.

*2021/7/1*

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Kolja Knauer

Aix-Marseille Université

Journal of Algebraic Combinatorics

##### Beyond symmetry in generalized Petersen graphs

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley …

*2024/1/24*

Samet Sarıoğlan

Hacettepe Üniversitesi

Journal of Algebraic Combinatorics

##### Basis condition for generalized spline modules

A generalized spline on an edge-labeled graph is defined as a vertex labeling, such that the difference of labels on adjacent vertices lies in the ideal generated by the edge label. We study generalized splines over greatest common divisor domains and present a determinantal basis condition for generalized spline modules on arbitrary graphs. The main result of the paper answers a conjecture that appeared in several papers.

*2024/2/15*

Michela Ceria

Università degli Studi di Milano

Journal of Algebraic Combinatorics

##### The direct sum of q-matroids

For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.

*2024/2/17*

Shoujun Xu (Shou-Jun Xu)

Lanzhou University

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##### On regular sets in Cayley graphs

Let be a graph and a, b nonnegative integers. An (a, b)-regular set in is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in has exactly b neighbours in D. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset D of a group G is called an (a, b)-regular set of G if it is an (a, b)-regular set in some Cayley graph of G, and an (a, b)-regular set in a Cayley graph of G is called a subgroup (a, b)-regular set if it is also a subgroup of G. In this paper, we study (a, b)-regular sets in Cayley graphs with a focus on (0, k)-regular sets, where is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, k)-regular set of the group. We also determine all subgroup (0, k)-regular sets of dihedral groups and generalized quaternion groups. We …

*2024/3/2*

Gavril Farkas

Humboldt-Universität zu Berlin

Journal of Algebraic Combinatorics

##### Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra A of finite type over a field\(\Bbbk\) of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When\(A=\Bbbk\langle\Delta\rangle\) is the exterior Stanley–Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

*2024/3/29*

Benjamin Sambale

Leibniz Universität Hannover

Journal of Algebraic Combinatorics

##### Common transversals and complements in abelian groups

Given a finite abelian group G and cyclic subgroups A, B, C of G of the same order, we find necessary and sufficient conditions for A, B, C to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups, we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.

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Université Libre de Bruxelles

Journal of Algebraic Combinatorics

##### A rank augmentation theorem for rank three string C-group representations of the symmetric groups

We give a rank augmentation technique for rank three string C-group representations of the symmetric group and list the hypotheses under which it yields a valid string C-group representation of rank four thereof.

*2024/2/16*

Rijubrata Kundu

Indian Institute of Science Education and Research, Pune

Journal of Algebraic Combinatorics

##### Alternating groups as products of cycle classes-II

Given integers , where either l is odd or k is even, let n(k, l) denote the largest integer n such that each element of is a product of k many l-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that [Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008]. It is known that the conjecture holds when . Moreover, it is also true when . In this article, we determine the exact value of n(k, l) when and . As an immediate consequence, we get that when and , which shows that the above conjecture is not true in general. In fact in this case, the difference between the exact value of n(k, l) and the conjectured value grows linearly in terms of k. Our results complete the determination of n(k, l) for all values of k and l.

*2024/2/26*

Roberto Civino

Università degli Studi dell'Aquila

Journal of Algebraic Combinatorics

##### An ultimately periodic chain in the integral Lie ring of partitions

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

*2024/4/9*

Do Trong Hoang

Hanoi University of Science and Technology

Journal of Algebraic Combinatorics

##### Asymptotic regularity of invariant chains of edge ideals

We study chains of nonzero edge ideals that are invariant under the action of the monoid of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of -invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of -invariant chains of edge ideals.

*2024/1*

Giovanni Longobardi

Università degli Studi di Padova

Journal of Algebraic Combinatorics

##### A standard form for scattered linearized polynomials and properties of the related translation planes

In this paper, we present results concerning the stabilizer in of the subspace, f (x) a scattered linearized polynomial in. Each contains the maps,. By virtue of the results of Beard (Duke Math J, 39: 313–321, 1972) and Willett (Duke Math J 40 (3): 701–704, 1973), the matrices in are simultaneously diagonalizable. This has several consequences:(i) the polynomials such that have a standard form of type for some s and t such that, a divisor of n;(ii) this standard form is essentially unique;(iii) for and, the translation plane associated with f (x) admits nontrivial affine homologies if and only if, and in that case those with axis through the origin form two groups of cardinality that exchange axes and coaxes;(iv) no plane of type, f (x) a scattered polynomial not of pseudoregulus type, is a …

*2024/4/9*

Daniel Hawtin

Sveucilište u Rijeci

Journal of Algebraic Combinatorics

##### Using mixed dihedral groups to construct normal Cayley graphs and a new bipartite 2-arc-transitive graph which is not a Cayley graph

A mixed dihedral group is a group H with two disjoint subgroups X and Y, each elementary abelian of order , such that H is generated by , and . In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph is equal to , where A(H, X, Y) is the setwise stabiliser in of . We use this criterion to resolve a question of Li et al. (J Aust Math Soc 86:111-122, 2009), by constructing a 2-arc-transitive normal cover of order of the complete bipartite graph and prove that it is not a Cayley graph.

*2024/2/23*

Riccardo Aragona

Università degli Studi dell'Aquila

Journal of Algebraic Combinatorics

##### An ultimately periodic chain in the integral Lie ring of partitions

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

*2024/4/9*

Marian Aprodu

Universitatea din Bucuresti

Journal of Algebraic Combinatorics

##### Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra A of finite type over a field\(\Bbbk\) of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When\(A=\Bbbk\langle\Delta\rangle\) is the exterior Stanley–Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

*2024/3/29*

Li Guo

Rutgers, The State University of New Jersey

Journal of Algebraic Combinatorics

##### Quivers and path semigroups characterized by locality conditions

Path algebras from quivers are a fundamental class of algebras with wide applications. Yet it is challenging to describe their universal properties since their underlying path semigroups are only partially defined. A new notion, called locality structures, was recently introduced to deal with partially defined operation, with motivation from locality in convex geometry and quantum field theory. We show that there is a natural correspondence between locality sets and quivers which leads to a concrete class of locality semigroups, called Brandt locality semigroups, which can be obtained by the paths of quivers. Further these path Brandt locality semigroups are precisely the free objects in the category of Brandt locality semigroups with a rigidity condition. This characterization gives a universal property of path algebras and at the same time a combinatorial realization of free rigid Brandt locality semigroups.

*2024/1*

Heiko Dietrich

Monash University

Journal of Algebraic Combinatorics

##### Derangements in wreath products of permutation groups

Given a finite group G acting on a set X let denote the proportion of elements in G that have exactly k fixed points in X. Let denote the symmetric group acting on . For and , the permutational wreath product has two natural actions and we give formulas for both, and . We prove that for the values of these proportions are dense in the intervals and . Among further results, we provide estimates for for subgroups containing .

*2024/2*

Yuval Filmus

Technion - Israel Institute of Technology

Journal of Algebraic Combinatorics

##### A note on “Largest independent sets of certain regular subgraphs of the derangement graph”

Let be the set of all permutations of the symmetric group that have no cycles of length i for all . In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph is equal to the set of all the largest independent sets in the derangement graph , provided n is sufficiently large in terms of k. We give a simpler proof that holds for all n, k and also applies to the alternating group.

*2024/2/23*

Tamizh Chelvam

Manonmaniam Sundaranar University

Journal of Algebraic Combinatorics

##### Generalized non-coprime graphs of groups

Let G be a finite group with identity e and be a subgroup of G. The generalized non-coprime graph of with respect to is the simple undirected graph with as the vertex set and two distinct vertices and are adjacent if and only if and either or , where |x| is the order of . In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, triangle-free, complete bipartite, complete, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime …

*2024/4/1*

Alessio Sammartano

Politecnico di Milano

Journal of Algebraic Combinatorics

##### Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra A of finite type over a field\(\Bbbk\) of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When\(A=\Bbbk\langle\Delta\rangle\) is the exterior Stanley–Reisner algebra associated to a finite simplicial complex, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

*2024/3/29*

Stoyan Dimitrov

Eberhard Karls Universität Tübingen

Journal of Algebraic Combinatorics

##### Chess tableaux, powers of two and affine Lie algebras

Chess tableaux are a special kind of standard Young tableaux where, in the chessboard coloring of the Young diagram, even numbers always appear in white cells and odd numbers in black cells. If, for a partition of n, denotes the number of chess tableaux of shape , then Chow, Eriksson and Fan observed that is divisible by unusually large powers of 2. In this paper, we give an explanation for this phenomenon, proving a lower bound of for the 2-adic valuation of this sum and a generalization of it. We do this by exploiting a connection with a certain representation of the affine Lie algebra on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of with coefficients taken from the ring of rational numbers with odd denominators.

*2023/8*