scholarly journals The Hecke group algebra of a Coxeter group and its representation theory

2009 ◽  
Vol 321 (8) ◽  
pp. 2230-2258 ◽  
Author(s):  
Florent Hivert ◽  
Nicolas M. Thiéry
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Coxeter Group ◽  
Hecke Group

scholarly journals The biHecke monoid of a finite Coxeter group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Florent Hivert ◽  
Anne Schilling ◽  
Nicolas M. Thiéry
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Coxeter Group ◽  
Projective Modules ◽  
Permutation Matrices ◽  
Hecke Group ◽  
Finite Coxeter Group ◽  
Nous Montrons ◽  
Combinatorial Model ◽  
International Audience

arXiv : http://arxiv.org/abs/0912.2212 International audience For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0-Hecke$ algebra $H_0(W)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each $w∈W$ a combinatorial module $T_w$ whose support is the interval $[1,w]_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset. Pour tout groupe de Coxeter fini $W$, nous définissons deux nouveaux objets : son ordre de coupures et son monoïde de Hecke double. L'ordre de coupures, construit au moyen d'une généralisation de la notion de bloc dans les matrices de permutations, est presque un treillis sur $W$. La construction du monoïde de Hecke double s'appuie sur le modèle combinatoire usuel de la $0-algèbre$ de Hecke $H_0(W)$, pour le groupe symétrique, l'algèbre (ou le monoïde) engendré par les opérateurs de tri par bulles élémentaires. Les auteurs ont introduit précédemment l'algèbre de Hecke-groupe, construite comme l'algèbre engendrée conjointement par les opérateurs de tri et d'anti-tri, et décrit sa théorie des représentations. Dans cet article, nous considérons le monoïde engendré par ces opérateurs. Nous montrons qu'il admet $|W|$ modules simples et projectifs. Afin de construire ses modules simples, nous introduisons pour tout $w∈W$ un module combinatoire $T_w$ dont le support est l'intervalle [$1,w]_R$ pour l'ordre faible droit. Ce module détermine une algèbre dont la théorie des représentations généralise celle de l'algèbre de Hecke groupe, en remplaçant la combinatoire des descentes par celle des blocs et de l'ordre de coupures.

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 Gröbner–Shirshov bases for non-crystallographic Coxeter groups

2019 ◽  
Vol 75 (3) ◽  
pp. 584-592 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Dong-il Lee
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Standard Monomials ◽  
Natural Operation

For the group algebra of the finite non-crystallographic Coxeter group of type H 4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.

scholarly journals On the radical of the group algebra of a p-nilpotent group

1972 ◽  
Vol 13 (1) ◽  
pp. 119-123 ◽  
Author(s):  
R. J. Clarke
Keyword(s):  
Representation Theory ◽  
Nilpotent Group ◽  
Group Algebra ◽  
Ordinary Representation

In this note we give a basis for the radical of the group algebra of a p-nilpotent group over a field of characteristic p in terms of the ordinary representation theory of the group. We use our result to calculate the exponent of the radical for such a group.

scholarly journals LOCALIZED ENDOMORPHISMS IN KITAEV'S TORIC CODE ON THE PLANE

2011 ◽  
Vol 23 (04) ◽  
pp. 347-373 ◽  
Author(s):  
PIETER NAAIJKENS
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Ground State ◽  
Algebraic Approach ◽  
Quantum Spin ◽  
Spin Systems ◽  
Quantum Spin Systems ◽  
Quantum Double ◽  
Code Model ◽  
Toric Code

We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of [Formula: see text], i.e. Drinfel'd's quantum double of the group algebra of ℤ2.

scholarly journals The role of minimal idempotents in the representation theory of locally compact groups

1980 ◽  
Vol 23 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Bruce A. Barnes
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Finite Groups ◽  
Compact Groups ◽  
Irreducible Matrix ◽  
Matrix Representations ◽  
Locally Compact ◽  
Irreducible Modules ◽  
Direct Correspondence

In the representation theory of finite groups, the minimal idempotents of the group algebra play a central role. In this case the minimal idempotents determine irreducible modules over the group algebra, which in turn are in direct correspondence with the irreducible matrix representations of the group; see Chapter IV of the book of C. Curtis and I. Reiner (2). Many of the same ideas generalise to the situation where the group is compact. In addition, minimal idempotents are involved in some important parts of the theory of Hubert algebras; see M. Rieffel's paper (20).

The structure of the full icosahedral group

2014 ◽  
Vol 70 (a1) ◽  
pp. C527-C527
Author(s):  
Dong-il Lee ◽  
Jeong-Yup Lee
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Algebra Structure ◽  
Groebner Basis ◽  
Icosahedral Group ◽  
Standard Monomials ◽  
Dynkin Diagrams ◽  
And Algebra

We consider the full icosahedral group, which is the Coxeter group of type H_3. The Coxeter groups appear naturally in geometry and algebra. In 1935, the finite Coxeter groups were classified by Coxeter in terms of Coxeter-Dynkin diagrams. We remark that the affine extensions of the Coxeter groups of types H are related to quasicrystals with tenfold symmetry. Our approach to understanding the structure of Coxeter groups is the noncommutative Groebner basis theory, which is called the Groebner-Shirshov basis theory. By completing the relations coming from a presentation of the Coxeter group, we find a Groebner-Shirshov basis to obtain a set of standard monomials. Especially, for the Coxeter group of type H_3, its Groebner-Shirshov basis and the corresponding standard monomials are constructed. Thus, we understand the algebra structure of the group algebra C[H_3], which is not commutative.

scholarly journals Hook representations of the symmetric groups

1971 ◽  
Vol 12 (2) ◽  
pp. 136-149 ◽  
Author(s):  
M. H. Peel
Keyword(s):  
Representation Theory ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Modular Representation ◽  
Splitting Field ◽  
Arbitrary Field ◽  
Modular Representation Theory ◽  
Prime Fields

In this paper we are concerned with the representation theory of the symmetric groupsover a field K of characteristic p. Every field is a splitting field for the symmetric groups. Consequently, in order to study the modular representation theory of these groups, it is sufficient to work over the prime fields. However, we take K to be an arbitrary field of characteristic p, since the presentation of the results is not affected by this choice. Sn denotes the group of permutations of {x1, …, xn], where x1,…,xn are independent indeterminates over K. The group algebra of Sn with coefficients in K is denoted by Фn.

scholarly journals On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Representation Theory ◽  
Generating Function ◽  
Coxeter Group ◽  
Upper Bound ◽  
Open Problem ◽  
Special Class ◽  
Exact Enumeration ◽  
Restricted Permutations ◽  
International Audience ◽  
Freely Braided

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

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