scholarly journals Study of conditions of existence of irreducible representations of finite groups by means of computer algebra system GAP

2021 ◽  
Vol 15 ◽  
pp. 105
Author(s):  
K.S. Mordak
Keyword(s):  
Computer Algebra ◽  
Finite Groups ◽  
Computer Algebra System ◽  
Irreducible Representations ◽  
Representations Of Finite Groups

We consider the criterion of existence of exact irreducible representations of finite groups and construct the algorithm that allows to verify it by means of computer algebra system GAP.

scholarly journals Characterizing the universal rigidity of generic tensegrities

2021 ◽  
Author(s):  
Ryoshun Oba ◽  
Shin-ichi Tanigawa
Keyword(s):  
Finite Groups ◽  
Stability Condition ◽  
Point Group ◽  
Group Symmetry ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Point Group Symmetry ◽  
Block Diagonalization ◽  
Representations Of Finite Groups ◽  
Universal Rigidity

AbstractA tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

On the integral and globally irreducible representations of finite groups

2018 ◽  
Vol 17 (05) ◽  
pp. 1850087
Author(s):  
Dmitry Malinin
Keyword(s):  
Number Field ◽  
Finite Groups ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Integral Representations ◽  
Irreducible Representations ◽  
Minimal Realization ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Representations Of Finite Groups

We consider the arithmetic of integral representations of finite groups over algebraic integers and the generalization of globally irreducible representations introduced by Van Oystaeyen and Zalesskii. For the ring of integers [Formula: see text] of an algebraic number field [Formula: see text] we are interested in the question: what are the conditions for subgroups [Formula: see text] such that [Formula: see text], the [Formula: see text]-span of [Formula: see text], coincides with [Formula: see text], the ring of [Formula: see text]-matrices over [Formula: see text], and what are the minimal realization fields.

scholarly journals Algorithms for permutability in finite groups

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Adolfo Ballester-Bolinches ◽  
Enric Cosme-Llópez ◽  
Ramón Esteban-Romero
Keyword(s):  
Computer Algebra ◽  
Finite Groups ◽  
Computer Algebra System ◽  
Permutable Subgroups ◽  
Sylow Permutable

AbstractIn this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.

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