scholarly journals The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

10.37236/8874 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Lawrence H. Harper ◽  
Gene B. Kim ◽  
Neal Livesay
Keyword(s):  
Coxeter Groups ◽  
Symmetric Groups ◽  
Product Theorem ◽  
Reflection Groups ◽  
Dihedral Groups ◽  
Alternate Proof ◽  
Sperner Property ◽  
Noncrossing Partition ◽  
The Absolute ◽  
Partition Lattices

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders).  In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner.  In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups.  Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

Automorphism groups of simply reducible groups

2020 ◽  
pp. 2150088
Author(s):  
Yongzhi Luan
Keyword(s):  
Computer Algebra System ◽  
Coxeter Groups ◽  
Eigenvalue Problems ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Molecular Symmetry ◽  
Dihedral Groups ◽  
Inner Automorphism ◽  
The Moment

Simply reducible groups are closely related to the eigenvalue problems in quantum theory and molecular symmetry in chemistry. Classification of simply reducible groups is still an open problem which is interesting to physicists. Since there are not many examples of simply reducible groups in literature at the moment, we try to find some examples of simply reducible groups as candidates for the classification. By studying the automorphism and inner automorphism groups of symmetric groups, dihedral groups, Clifford groups and Coxeter groups, we find some new examples of candidates. We use the computer algebra system GAP to get most of these automorphism and inner automorphism groups.

scholarly journals Coxeter groups as automorphism groups of solid transitive 3-simplex tilings

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 557-577 ◽  
Author(s):  
Milica Stojanovic
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Space ◽  
Coxeter Groups ◽  
Automorphism Groups ◽  
Order Parameters ◽  
Reflection Groups ◽  
Isometry Groups ◽  
Polar Plane ◽  
Side Face ◽  
The Absolute

In the papers of I.K. Zhuk, then more completely of E. Moln?r, I. Prok, J. Szirmai all simplicial 3-tilings have been classified, where a symmetry group acts transitively on the simplex tiles. The involved spaces depends on some rotational order parameters. When a vertex of a such simplex lies out of the absolute, e.g. in hyperbolic space H3, then truncation with its polar plane gives a truncated simplex or simply, trunc-simplex. Looking for symmetries of these tilings by simplex or trunc-simplex domains, with their side face pairings, it is possible to find all their group extensions, especially Coxeter?s reflection groups, if they exist. So here, connections between isometry groups and their supergroups is given by expressing the generators and the corresponding parameters. There are investigated simplices in families F3, F4, F6 and appropriate series of trunc-simplices. In all cases the Coxeter groups are the maximal ones.

Coxeter Groups

2017 ◽  
Author(s):  
Adam Piggott
Keyword(s):  
Finite Groups ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Research Projects ◽  
Reflection Groups ◽  
Euclidean Spaces ◽  
Geometric Space

This chapter considers Coxeter groups and how to find a space on which a group acts by building a space using combinatorics from the group. It first describes groups generated by reflections, focusing on Euclidean spaces and showing that some natural, beautiful, and important subsets of Euclidean spaces have symmetric groups that are discrete and are generated by reflections. It then explores discrete groups generated by reflections, beginning with irreducible finite groups generated by reflections followed by infinite reflection groups. It also looks at relations in finite groups generated by reflections before concluding with an analysis of special subgroups of the Coxeter group and how to construct a geometric space for a Coxeter group. The discussion includes exercises and research projects.

scholarly journals DETOUR ENERGY OF COMPLEMENT OF SUBGROUP GRAPH OF DIHEDRAL GROUP

2017 ◽  
Vol 1 (2) ◽  
Author(s):  
Abdussakir Abdussakir
Keyword(s):  
Dihedral Group ◽  
Normal Subgroups ◽  
Dihedral Groups ◽  
The Absolute

Study on the energy of a graph becomes a topic of great interest. One is the detour energy which is the sum of the absolute values of all eigenvalue of the detour matrix of a graph. Graphs obtained from a group also became a study that attracted the attention of many researchers. This article discusses the subgroup graph for several normal subgroups of dihedral groups. The discussion focused on the detour energy of complement of subgroup graph of dihedral group

scholarly journals Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

scholarly journals Specht modules for finite reflection groups

1995 ◽  
Vol 37 (3) ◽  
pp. 279-287 ◽  
Author(s):  
S. HalicioǦlu
Keyword(s):  
Present Author ◽  
Symmetric Groups ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Young Tableaux ◽  
Reflection Groups ◽  
Specht Modules ◽  
Irreducible Modules ◽  
Original Approach

Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A. Young, and of irreducible modules, called Specht modules, due to W. Specht, for the symmetric groups Sn which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [13]). James [12] extended these ideas to construct irreducible representations and modules over an arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to cover the Weyl groups of type Bn. In [14] Morris described a possible extension of James' work for Weyl groups in general. Later, the present author and Morris [8] gave an alternative generalisation of James' work which is an extended improvement and extension of the original approach suggested by Morris. We now give a possible extension of James' work for finite reflection groups in general.

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