scholarly journals The conjugacy problem for Coxeter groups

2009 ◽  
pp. 71-171 ◽  
Author(s):  
Daan Krammer
Keyword(s):  
Coxeter Groups ◽  
Conjugacy Problem

scholarly journals Posets from Admissible Coxeter Sequences

10.37236/684 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Matthew Macauley ◽  
Henning S. Mortveit
Keyword(s):  
Coxeter Groups ◽  
Conjugacy Problem ◽  
Equivalence Classes ◽  
Quiver Representations ◽  
Edge Deletion ◽  
Acyclic Orientations ◽  
Edge Contraction ◽  
Combinatorial Invariant ◽  
Source To Sink ◽  
Admissible Sequences

We study the equivalence relation on the set of acyclic orientations of an undirected graph $\Gamma$ generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver representations, and asynchronous graph dynamical systems. To each equivalence class we associate a poset, characterize combinatorial properties of these posets, and in turn, the admissible sequences. This allows us to construct an explicit bijection from the equivalence classes over $\Gamma$ to those over $\Gamma'$ and $\Gamma"$, the graphs obtained from $\Gamma$ by edge deletion and edge contraction of a fixed cycle-edge, respectively. This bijection yields quick and elegant proofs of two non-trivial results: $(i)$ A complete combinatorial invariant of the equivalence classes, and $(ii)$ a solution to the conjugacy problem of Coxeter elements for simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and K. Eriksson using a much different approach.

ON ALGORITHMIC SOLVABILITY OF THE PROBLEM OF GENERALIZED CONJUGACY OF SUBGROUPS IN COXETER GROUPS WITH A TREE STRUCTURE

2021 ◽  
Vol 1 (2) ◽  
pp. 7-14
Author(s):  
I. V. Dobrynina ◽  
◽  
E. L. Turenova ◽  
Keyword(s):  
Coxeter Groups ◽  
Conjugacy Problem ◽  
Tree Structure ◽  
The Other ◽  
Combinatorial Group Theory ◽  
Finitely Generated ◽  
Algorithmic Problems ◽  
Algorithmic Solvability ◽  
The One ◽  
Combinatorial Group

The main algorithmic problems of combinatorial group theory posed by M. Den and G. Titze at the beginning of the twentieth century are the problems of word, word conjugacy and of group isomorphism. However, these problems, as follows from the results of P.S. Novikov and S.I. Adyan, turned out to be unsolvable in the class of finitely defined groups. Therefore, algorithmic problems began to be considered in specific classes of groups. The word conjugacy problem allows for two generalizations. On the one hand, we consider the problem of conjugacy of subgroups, that is, the problem of constructing an algorithm that allows for any two finitely generated subgroups to determine whether they are conjugate or not. On the other hand, the problem of generalized conjugacy of words is posed, that is, the problem of constructing an algorithm that allows for any two finite sets of words to determine whether they are conjugated or not. Combining both of these generalizations into one, we obtain the problem of generalized conjugacy of subgroups. Coxeter groups were introduced in the 30s of the last century, and the problems of equality and conjugacy of words are algorithmically solvable in them. To solve other algorithmic problems, various subclasses are distinguished. This is partly due to the unsolvability in Coxeter groups of another important problem – the problem of occurrence, that is, the problem of the existence of an algorithm that allows for any word and any finitely generated subgroup of a certain group to determine whether this word belongs to this subgroup or not. The paper proves the algorithmic solvability of the problem of generalized conjugacy of subgroups in Coxeter groups with a tree structure.

scholarly journals Automorphisms of odd Coxeter groups

2021 ◽  
Author(s):  
Tushar Kanta Naik ◽  
Mahender Singh
Keyword(s):  
Coxeter Groups

scholarly journals The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Aleksander J. Cianciara ◽  
S. James Gates ◽  
Yangrui Hu ◽  
Renée Kirk
Keyword(s):  
Coxeter Groups ◽  
Auxiliary Field ◽  
Permutation Groups ◽  
Weight Space ◽  
Field Problem ◽  
Truncated Octahedron ◽  
Color Problem ◽  
Minimal Representations

Abstract A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

Kazhdan-Lusztig cells in some weighted Coxeter groups

2017 ◽  
Vol 61 (2) ◽  
pp. 325-352 ◽  
Author(s):  
Jianyi Shi ◽  
Gao Yang
Keyword(s):  
Coxeter Groups

scholarly journals Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1330
Author(s):  
Raeyong Kim
Keyword(s):  
Group Structure ◽  
Conjugacy Problem ◽  
High Dimensional ◽  
Fundamental Groups ◽  
Graph Manifold ◽  
Solvable Word Problem ◽  
Algorithmic Problems ◽  
Graph Manifolds ◽  
Dimensional Graph ◽  
Solvable Word

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

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