138 research outputs found
Around Solomon's descent algebras
We study different problems related to the Solomon's descent algebra
of a finite Coxeter group : positive elements, morphisms
between descent algebras, Loewy length... One of the main result is that, if
is irreducible and if the longest element is central, then the Loewy length
of is equal to .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
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 acting on the th
graded component of its Orlik-Solomon algebra as a sum of characters induced
from linear characters of centralizers of elements of . Our refined
conjecture relates the character above to a component of a decomposition of the
regular character of related to Solomon's descent algebra of . 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
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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
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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
On Reflection Subgroups of Finite Coxeter Groups
Let be a finite Coxeter group. We classify the reflection subgroups of
up to conjugacy and give necessary and sufficient conditions for the map
that assigns to a reflection subgroup of the conjugacy class of its
Coxeter elements to be injective, up to conjugacy.Comment: 17 pages; final version, to appear in Comm. Algebr
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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
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