Around Solomon's descent algebras

We study different problems related to the Solomon's descent algebra $\Sigma(W)$ of a finite Coxeter group $(W,S)$: positive elements, morphisms between descent algebras, Loewy length... One of the main result is that, if $W$ is irreducible and if the longest element is central, then the Loewy length of $\Sigma(W)$ is equal to $\lceil |S| / 2 \rceil$.Comment: 21 page

The Varchenko Determinant of a Coxeter Arrangement

The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.Comment: 12 page

A Note on Element Centralizers in Finite Coxeter Groups

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.Comment: final version, 18 pages, to appear in J. Group Theor

Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in a series of three dealing with groups of rank up to eight (and in particular, all exceptional Coxeter groups), we prove the conjecture for finite Coxeter groups of rank five and six, further developing the algorithmic tools described in the previous article. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik-Solomon characters of the groups considered.Comment: Final Version. 17 page

On reflection subgroups of finite Coxeter groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy

Coxeter arrangements and Solomon's descent algebra

In our recent paper [3], we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for nite Coxeter groups of rank up to 2

On the table of marks of a direct product of finite groups

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S_3

Ich sehe was, was du auch siehst. Über die Möglichkeiten von Augmented und Virtual Reality für die digitale Beteiligung von Bürger:innen in der Bau- und Stadtplanung

Digital Government eröffnet Möglichkeiten, Verwaltungs- und Regierungsprozesse kritisch zu reflektieren und sie entsprechend neu zu denken. Oblagen Bürgerbeteiligungsprozesse in der Vergangenheit zahlreichen Hürden, bietet die e‑Partizipation Möglichkeiten, sie mit modernen Technologien zu verbinden, die eine niedrigschwellige Teilhabe ermöglichen. In dem Forschungsprojekt Take Part, gefördert durch das Bundesministerium für Bildung und Forschung, werden innovative Formen der Beteiligung von Bürger:innen in der Stadt- und Bauplanung mithilfe von Augmented und Virtual Reality (AR und VR) erforscht. Dabei geht es vor allem darum, neue Anreize zu schaffen, Bürger:innen zur Beteiligung zu motivieren und durch diese das Konfliktpotential um Bauprojekte zu reduzieren. Mithilfe der innerhalb von Take Part entwickelten App können Bürger:innen Bauvorhaben diskutieren, Feedback geben oder über sie abstimmen, während sie dabei den Beteiligungsgegenstand anschaulich in AR und VR präsentiert bekommen. Zugleich können auch Initiator:innen mithilfe eines Partizipationsökosystems die Beteiligung im jeweiligen Bauvorhaben konfigurieren, indem sie sowohl vorhandene Module kombinieren und konfigurieren, als auch passende Dienstleistungen, wie beispielsweise 3D-Modellierungen, einkaufen. In diesem Beitrag sollen die konkreten technologischen Entwicklungen (u. a. Outdoor-AR-Tracking und räumlich verankerte Diskussionen), sowie das Partizipationsökosystem (Dienstentwicklungs- und Ausführungsplattform) vorgestellt werden. Erstmalig soll so der entwickelte Prototyp umfassend dargestellt werden. Auf die Herausforderung, eine e‑Partizipations-App zu entwickeln, die die Möglichkeit bietet, verschiedene Interaktionskonzepte ineinander zu integrieren und gleichzeitig eine überzeugende User-Experience bietet, soll ebenfalls eingegangen werden. Anschließend wird das Potenzial einer solchen Lösung für die digitale Mitbestimmung in lokaler Verwaltung vor allem in Bezug auf gesteigerte Vorstellungskraft und Motivation zur Teilhabe für Nutzer:innen diskutiert und in den Kontext der Covid-19 Pandemie gesetzt

The role of carcinoembryonic antigen-related cell adhesion molecule 1 in cancer

The Carcinoembryonic antigen-related cell adhesion molecule 1 (CEACAM1), also known as CD66a, is a member of the immunoglobulin superfamily. CEACAM1 was shown to be a prognostic marker in patients suffering from cancer. In this review, we summarize pre-clinical and clinical evidence linking CEACAM1 to tumorigenicity and cancer progression. Furthermore, we discuss potential CEACAM1-based mechanisms that may affect cancer biology