THE BRAID ROOK MONOID

2008 ◽  
Vol 18 (04) ◽  
pp. 779-802 ◽  
Author(s):  
EDDY GODELLE
Keyword(s):  
Group Theory ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Hecke Algebra ◽  
Linear Algebraic Group ◽  
Weyl Groups ◽  
Rook Monoid ◽  
Algebraic Monoid ◽  
Tits Group

In linear algebraic monoid theory, the Renner monoids play the role of the Weyl groups in linear algebraic group theory. It is well known that Weyl groups are Coxeter groups, and that we can associate a Hecke algebra and an Artin–Tits group to each Coxeter group. The question of the existence of a Hecke algebra associated with each Renner monoid has been positively answered. In this paper we discuss the question of the existence of an equivalent of the Artin–Tits groups in the framework of Renner monoids. We consider the seminal case of the rook monoid and introduce a new length function.

scholarly journals Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

2019 ◽  
Vol 72 (1) ◽  
pp. 1-55
Author(s):  
Pramod N. Achar ◽  
Simon Riche ◽  
Cristian Vay
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Grothendieck Group ◽  
General Setting ◽  
Hecke Algebra ◽  
Abelian Category ◽  
Flag Varieties ◽  
Perverse Sheaves

AbstractIn this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

scholarly journals -GRAPHS AND GYOJA’S -GRAPH ALGEBRA

2017 ◽  
Vol 226 ◽  
pp. 1-43 ◽  
Author(s):  
JOHANNES HAHN
Keyword(s):  
Irreducible Representation ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Hecke Algebra ◽  
Path Algebra ◽  
Explicit Description ◽  
Graph Algebra ◽  
Finite Coxeter Group ◽  
General Conjecture

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

scholarly journals Linear syzygies, hyperbolic Coxeter groups and regularity

2019 ◽  
Vol 155 (6) ◽  
pp. 1076-1097 ◽  
Author(s):  
Alexandru Constantinescu ◽  
Thomas Kahle ◽  
Matteo Varbaro
Keyword(s):  
Group Theory ◽  
Coxeter Group ◽  
Commutative Algebra ◽  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Geometric Group Theory ◽  
Monomial Ideals ◽  
Linear Syzygies ◽  
High Regularity ◽  
Virtual Cohomological Dimension

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.

scholarly journals The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

2014 ◽  
Vol 66 (2) ◽  
pp. 354-372 ◽  
Author(s):  
Ruth Kellerhals ◽  
Alexander Kolpakov
Keyword(s):  
Growth Rate ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Triangle Group ◽  
Minimal Growth ◽  
Salem Number

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

scholarly journals Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

2011 ◽  
Vol 16 ◽  
pp. 82 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Muna Al-Shueili
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Linear Combinations ◽  
Octahedral Group ◽  
Snub Cube ◽  
Proper Rotation

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand  respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.  

scholarly journals Gamma Positivity of the Descent Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

10.37236/9037 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Hiranya Kishore Dey ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Coxeter Groups ◽  
Weyl Groups ◽  
Alternating Group ◽  
Type B ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Type D ◽  
Positive Elements

The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ is known to be gamma positive for all $n$. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all $n$. We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively.  We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$.  We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.

On the generation of Coxeter groups and their alternating subgroups by involutions

2019 ◽  
Vol 22 (6) ◽  
pp. 1001-1013
Author(s):  
Rupert W. T. Yu
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Minimum Number

Abstract In this note, we determine the minimum number of involutions required to generate a finite irreducible Coxeter group, and also whenever such generation is possible, its alternating subgroup. Explicit generators are given.

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