scholarly journals Generalized Non-Crossing Partitions and Buildings

10.37236/7200 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Julia Heller ◽  
Petra Schwer
Keyword(s):  
Lower Bound ◽  
Coxeter Group ◽  
Order Complex ◽  
Spherical Building ◽  
Partition Lattice ◽  
Finite Coxeter Group

For any finite Coxeter group $W$ of rank $n$ we show that the order complex of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give a new proof of the fact that the non-crossing partition lattice in type $A_n$ is supersolvable for all $n$. Moreover, we show that in case $B_n$, this is only the case if $n<4$. We also obtain a lower bound on the radius of the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the radius is $\binom{n}{2}$. A Corrigendum for this paper was added on May 17, 2018.

scholarly journals Some remarks on the algebraic structure of the finite Coxeter groupF4

1999 ◽  
Vol 22 (1) ◽  
pp. 81-84
Author(s):  
Muhammad A. Albar ◽  
Norah Al-Saleh
Keyword(s):  
Coxeter Group ◽  
Algebraic Structure ◽  
Finite Coxeter Group

We consider in this paper the algebraic structure and some properties of the finite Coxeter groupF4.

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

2019 ◽  
Vol 75 (3) ◽  
pp. 541-550
Author(s):  
Emmanuel Bourret ◽  
Zofia Grabowiecka
Keyword(s):  
Coxeter Group ◽  
Dynkin Diagram ◽  
Geometrical Structure ◽  
Three Dimensions ◽  
Icosahedral Group ◽  
Finite Coxeter Group

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H 3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H 3 is given and presented as a `pancake structure'.

scholarly journals Generalized Descent Algebras

2007 ◽  
Vol 50 (4) ◽  
pp. 535-546
Author(s):  
Christophe Hohlweg
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Descent Algebra ◽  
Finite Coxeter Group ◽  
Solomon Descent Algebra ◽  
Descent Algebras

AbstractIf A is a subset of the set of reflections of a finite Coxeter group W, we define a sub-ℤ-module of the group algebra ℤW. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if W is of type B, the Mantaci–Reutenauer algebra.

scholarly journals Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

2020 ◽  
Author(s):  
Fabrizio Caselli ◽  
Michele D’Adderio ◽  
Mario Marietti
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Combinatorial Properties ◽  
Coxeter Matroid ◽  
Finite Coxeter Group ◽  
Bruhat Intervals

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

scholarly journals Special involutions and bulky parabolic subgroups in finite Coxeter groups

2005 ◽  
Vol 79 (1) ◽  
pp. 141-147 ◽  
Author(s):  
Götz Pfeiffer ◽  
Gerhard Röhrle
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Hyperplane Arrangement ◽  
Conjugacy Classes ◽  
Parabolic Subgroups ◽  
Finite Coxeter Group

AbstractThe conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

scholarly journals Leading coefficients and cellular bases of Hecke algebras

2009 ◽  
Vol 52 (3) ◽  
pp. 653-677 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Weyl Group ◽  
Coxeter Group ◽  
Cellular Structure ◽  
Hecke Algebras ◽  
Point Of View ◽  
Reflection Groups ◽  
Cellular Basis ◽  
Complex Reflection Groups ◽  
Finite Coxeter Group ◽  
New Construction

AbstractLet H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.

scholarly journals On the Hurwitz action in finite Coxeter groups

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Barbara Baumeister ◽  
Thomas Gobet ◽  
Kieran Roberts ◽  
Patrick Wegener
Keyword(s):  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Coxeter Element ◽  
Necessary And Sufficient Condition ◽  
Reduced Decompositions ◽  
Finite Coxeter Group ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Hurwitz Action

AbstractWe provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

scholarly journals Geometry of certain finite Coxeter group actions

2018 ◽  
Vol 118 (2) ◽  
pp. 351-378
Author(s):  
M. J. Dyer ◽  
G. I. Lehrer
Keyword(s):  
Coxeter Group ◽  
Group Actions ◽  
Finite Coxeter Group
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